Mathematics for Computer Science

Authors Albert R Meyer Eric Lehman F Thomson Leighton

Plaintext

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Mathematics for Computer Science
revised Friday 10th March, 2017, 22:22

Eric Lehman
Google Inc.

F Thomson Leighton
Department of Mathematics
and the Computer Science and AI Laboratory,
Massachussetts Institute of Technology;
Akamai Technologies

Albert R Meyer
Department of Electrical Engineering and Computer Science
and the Computer Science and AI Laboratory,
Massachussetts Institute of Technology

2016, Eric Lehman, F Tom Leighton, Albert R Meyer. This work is available under the
terms of the Creative Commons Attribution-ShareAlike 3.0 license.
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Contents

I Proofs
Introduction 3
0.1 References 4
1 What is a Proof? 5
1.1 Propositions 5
1.2 Predicates 8
1.3 The Axiomatic Method 8
1.4 Our Axioms 9
1.5 Proving an Implication 11
1.6 Proving an “If and Only If” 13
1.7 Proof by Cases 15
1.8 Proof by Contradiction 16
1.9 Good Proofs in Practice 17
1.10 References 19
2 The Well Ordering Principle 29
2.1 Well Ordering Proofs 29
2.2 Template for Well Ordering Proofs 30
2.3 Factoring into Primes 32
2.4 Well Ordered Sets 33
3 Logical Formulas 47
3.1 Propositions from Propositions 48
3.2 Propositional Logic in Computer Programs 52
3.3 Equivalence and Validity 54
3.4 The Algebra of Propositions 57
3.5 The SAT Problem 62
3.6 Predicate Formulas 63
3.7 References 68
4 Mathematical Data Types 95
4.1 Sets 95
4.2 Sequences 100
4.3 Functions 101
4.4 Binary Relations 103
4.5 Finite Cardinality 107
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iv Contents

5 Induction 127
5.1 Ordinary Induction 127
5.2 Strong Induction 136
5.3 Strong Induction vs. Induction vs. Well Ordering 143
6 State Machines 163
6.1 States and Transitions 163
6.2 The Invariant Principle 164
6.3 Partial Correctness & Termination 172
6.4 The Stable Marriage Problem 177
7 Recursive Data Types 207
7.1 Recursive Definitions and Structural Induction 207
7.2 Strings of Matched Brackets 211
7.3 Recursive Functions on Nonnegative Integers 215
7.4 Arithmetic Expressions 217
7.5 Games as a Recursive Data Type 222
7.6 Induction in Computer Science 226
8 Infinite Sets 253
8.1 Infinite Cardinality 254
8.2 The Halting Problem 263
8.3 The Logic of Sets 267
8.4 Does All This Really Work? 270

II Structures
Introduction 295
9 Number Theory 297
9.1 Divisibility 297
9.2 The Greatest Common Divisor 302
9.3 Prime Mysteries 309
9.4 The Fundamental Theorem of Arithmetic 311
9.5 Alan Turing 314
9.6 Modular Arithmetic 318
9.7 Remainder Arithmetic 320
9.8 Turing’s Code (Version 2.0) 323
9.9 Multiplicative Inverses and Cancelling 325
9.10 Euler’s Theorem 329
9.11 RSA Public Key Encryption 334
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v Contents

9.12 What has SAT got to do with it? 336
9.13 References 337
10 Directed graphs & Partial Orders 375
10.1 Vertex Degrees 377
10.2 Walks and Paths 378
10.3 Adjacency Matrices 381
10.4 Walk Relations 384
10.5 Directed Acyclic Graphs & Scheduling 385
10.6 Partial Orders 393
10.7 Representing Partial Orders by Set Containment 397
10.8 Linear Orders 398
10.9 Product Orders 398
10.10 Equivalence Relations 399
10.11 Summary of Relational Properties 401
11 Communication Networks 433
11.1 Routing 433
11.2 Routing Measures 434
11.3 Network Designs 437
12 Simple Graphs 453
12.1 Vertex Adjacency and Degrees 453
12.2 Sexual Demographics in America 455
12.3 Some Common Graphs 457
12.4 Isomorphism 459
12.5 Bipartite Graphs & Matchings 461
12.6 Coloring 466
12.7 Simple Walks 471
12.8 Connectivity 473
12.9 Forests & Trees 478
12.10 References 486
13 Planar Graphs 525
13.1 Drawing Graphs in the Plane 525
13.2 Definitions of Planar Graphs 525
13.3 Euler’s Formula 536
13.4 Bounding the Number of Edges in a Planar Graph 537
13.5 Returning to K5 and K3;3 538
13.6 Coloring Planar Graphs 539
13.7 Classifying Polyhedra 541
13.8 Another Characterization for Planar Graphs 544
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vi Contents

III Counting
Introduction 553
14 Sums and Asymptotics 555
14.1 The Value of an Annuity 556
14.2 Sums of Powers 562
14.3 Approximating Sums 564
14.4 Hanging Out Over the Edge 568
14.5 Products 574
14.6 Double Trouble 577
14.7 Asymptotic Notation 580
15 Cardinality Rules 605
15.1 Counting One Thing by Counting Another 605
15.2 Counting Sequences 606
15.3 The Generalized Product Rule 609
15.4 The Division Rule 613
15.5 Counting Subsets 616
15.6 Sequences with Repetitions 618
15.7 Counting Practice: Poker Hands 621
15.8 The Pigeonhole Principle 626
15.9 Inclusion-Exclusion 635
15.10 Combinatorial Proofs 641
15.11 References 645
16 Generating Functions 683
16.1 Infinite Series 683
16.2 Counting with Generating Functions 685
16.3 Partial Fractions 691
16.4 Solving Linear Recurrences 694
16.5 Formal Power Series 699
16.6 References 702

IV Probability
Introduction 721
17 Events and Probability Spaces 723
17.1 Let’s Make a Deal 723
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vii Contents

17.2 The Four Step Method 724
17.3 Strange Dice 733
17.4 The Birthday Principle 740
17.5 Set Theory and Probability 742
17.6 References 746
18 Conditional Probability 755
18.1 Monty Hall Confusion 755
18.2 Definition and Notation 756
18.3 The Four-Step Method for Conditional Probability 758
18.4 Why Tree Diagrams Work 760
18.5 The Law of Total Probability 768
18.6 Simpson’s Paradox 770
18.7 Independence 772
18.8 Mutual Independence 774
18.9 Probability versus Confidence 778
19 Random Variables 807
19.1 Random Variable Examples 807
19.2 Independence 809
19.3 Distribution Functions 810
19.4 Great Expectations 819
19.5 Linearity of Expectation 830
20 Deviation from the Mean 861
20.1 Markov’s Theorem 861
20.2 Chebyshev’s Theorem 864
20.3 Properties of Variance 868
20.4 Estimation by Random Sampling 874
20.5 Confidence in an Estimation 877
20.6 Sums of Random Variables 879
20.7 Really Great Expectations 888
21 Random Walks 913
21.1 Gambler’s Ruin 913
21.2 Random Walks on Graphs 923

V Recurrences
Introduction 941
22 Recurrences 943
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viii Contents

22.1 The Towers of Hanoi 943
22.2 Merge Sort 946
22.3 Linear Recurrences 950
22.4 Divide-and-Conquer Recurrences 957
22.5 A Feel for Recurrences 964

Bibliography 971
Glossary of Symbols 975
Index 979
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I Proofs
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Introduction
This text explains how to use mathematical models and methods to analyze prob-
lems that arise in computer science. Proofs play a central role in this work because
the authors share a belief with most mathematicians that proofs are essential for
genuine understanding. Proofs also play a growing role in computer science; they
are used to certify that software and hardware will always behave correctly, some-
thing that no amount of testing can do.
Simply put, a proof is a method of establishing truth. Like beauty, “truth” some-
times depends on the eye of the beholder, and it should not be surprising that what
constitutes a proof differs among fields. For example, in the judicial system, legal
truth is decided by a jury based on the allowable evidence presented at trial. In the
business world, authoritative truth is specified by a trusted person or organization,
or maybe just your boss. In fields such as physics or biology, scientific truth is
confirmed by experiment.1 In statistics, probable truth is established by statistical
analysis of sample data.
Philosophical proof involves careful exposition and persuasion typically based
on a series of small, plausible arguments. The best example begins with “Cogito
ergo sum,” a Latin sentence that translates as “I think, therefore I am.” This phrase
comes from the beginning of a 17th century essay by the mathematician/philosopher,
René Descartes, and it is one of the most famous quotes in the world: do a web
search for it, and you will be flooded with hits.
Deducing your existence from the fact that you’re thinking about your existence
is a pretty cool and persuasive-sounding idea. However, with just a few more lines
1 Actually, only scientific falsehood can be demonstrated by an experiment—when the experiment
fails to behave as predicted. But no amount of experiment can confirm that the next experiment won’t
fail. For this reason, scientists rarely speak of truth, but rather of theories that accurately predict past,
and anticipated future, experiments.
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4 0.1. References

of argument in this vein, Descartes goes on to conclude that there is an infinitely
beneficent God. Whether or not you believe in an infinitely beneficent God, you’ll
probably agree that any very short “proof” of God’s infinite beneficence is bound
to be far-fetched. So even in masterful hands, this approach is not reliable.
Mathematics has its own specific notion of “proof.”

Definition. A mathematical proof of a proposition is a chain of logical deductions
leading to the proposition from a base set of axioms.

The three key ideas in this definition are highlighted: proposition, logical deduc-
tion, and axiom. Chapter 1 examines these three ideas along with some basic ways
of organizing proofs. Chapter 2 introduces the Well Ordering Principle, a basic
method of proof; later, Chapter 5 introduces the closely related proof method of
induction.
If you’re going to prove a proposition, you’d better have a precise understand-
ing of what the proposition means. To avoid ambiguity and uncertain definitions
in ordinary language, mathematicians use language very precisely, and they often
express propositions using logical formulas; these are the subject of Chapter 3.
The first three Chapters assume the reader is familiar with a few mathematical
concepts like sets and functions. Chapters 4 and 8 offer a more careful look at
such mathematical data types, examining in particular properties and methods for
proving things about infinite sets. Chapter 7 goes on to examine recursively defined
data types.

0.1 References
[12], [46], [1]
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1 What is a Proof?

1.1 Propositions
Definition. A proposition is a statement (communication) that is either true or
false.

For example, both of the following statements are propositions. The first is true,
and the second is false.

Proposition 1.1.1. 2 + 3 = 5.

Proposition 1.1.2. 1 + 1 = 3.

Being true or false doesn’t sound like much of a limitation, but it does exclude
statements such as “Wherefore art thou Romeo?” and “Give me an A!” It also ex-
cludes statements whose truth varies with circumstance such as, “It’s five o’clock,”
or “the stock market will rise tomorrow.”
Unfortunately it is not always easy to decide if a claimed proposition is true or
false:

Claim 1.1.3. For every nonnegative integer n the value of n2 C n C 41 is prime.

(A prime is an integer greater than 1 that is not divisible by any other integer
greater than 1. For example, 2, 3, 5, 7, 11, are the first five primes.) Let’s try some
numerical experimentation to check this proposition. Let

p.n/ WWD n2 C n C 41:1 (1.1)
We begin with p.0/ D 41, which is prime; then

p.1/ D 43; p.2/ D 47; p.3/ D 53; : : : ; p.20/ D 461

are each prime. Hmmm, starts to look like a plausible claim. In fact we can keep
checking through n D 39 and confirm that p.39/ D 1601 is prime.
But p.40/ D 402 C 40 C 41 D 41 41, which is not prime. So Claim 1.1.3
is false since it’s not true that p.n/ is prime for all nonnegative integers n. In
fact, it’s not hard to show that no polynomial with integer coefficients can map all
1 The symbol WWD means “equal by definition.” It’s always ok simply to write “=” instead of WWD,
but reminding the reader that an equality holds by definition can be helpful.
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6 Chapter 1 What is a Proof?

nonnegative numbers into prime numbers, unless it’s a constant (see Problem 1.26).
But this example highlights the point that, in general, you can’t check a claim about
an infinite set by checking a finite sample of its elements, no matter how large the
sample.
By the way, propositions like this about all numbers or all items of some kind are
so common that there is a special notation for them. With this notation, Claim 1.1.3
would be
8n 2 N: p.n/ is prime: (1.2)
Here the symbol 8 is read “for all.” The symbol N stands for the set of nonnegative
integers: 0, 1, 2, 3, . . . (ask your instructor for the complete list). The symbol “2”
is read as “is a member of,” or “belongs to,” or simply as “is in.” The period after
the N is just a separator between phrases.
Here are two even more extreme examples:
Conjecture. [Euler] The equation

a4 C b 4 C c 4 D d 4

has no solution when a; b; c; d are positive integers.
Euler (pronounced “oiler”) conjectured this in 1769. But the conjecture was
proved false 218 years later by Noam Elkies at a liberal arts school up Mass Ave.
The solution he found was a D 95800; b D 217519; c D 414560; d D 422481.
In logical notation, Euler’s Conjecture could be written,

8a 2 ZC 8b 2 ZC 8c 2 ZC 8d 2 ZC : a4 C b 4 C c 4 ¤ d 4 :

Here, ZC is a symbol for the positive integers. Strings of 8’s like this are usually
abbreviated for easier reading:

8a; b; c; d 2 ZC : a4 C b 4 C c 4 ¤ d 4 :

Here’s another claim which would be hard to falsify by sampling: the smallest
possible x; y; z that satisfy the equality each have more than 1000 digits!
False Claim. 313.x 3 C y 3 / D z 3 has no solution when x; y; z 2 ZC .
It’s worth mentioning a couple of further famous propositions whose proofs were
sought for centuries before finally being discovered:
Proposition 1.1.4 (Four Color Theorem). Every map can be colored with 4 colors
so that adjacent2 regions have different colors.
2 Two regions are adjacent only when they share a boundary segment of positive length. They are
not considered to be adjacent if their boundaries meet only at a few points.
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1.1. Propositions 7

Several incorrect proofs of this theorem have been published, including one that
stood for 10 years in the late 19th century before its mistake was found. A laborious
proof was finally found in 1976 by mathematicians Appel and Haken, who used a
complex computer program to categorize the four-colorable maps. The program
left a few thousand maps uncategorized, which were checked by hand by Haken
and his assistants—among them his 15-year-old daughter.
There was reason to doubt whether this was a legitimate proof—the proof was
too big to be checked without a computer. No one could guarantee that the com-
puter calculated correctly, nor was anyone enthusiastic about exerting the effort
to recheck the four-colorings of thousands of maps that were done by hand. Two
decades later a mostly intelligible proof of the Four Color Theorem was found,
though a computer is still needed to check four-colorability of several hundred spe-
cial maps.3

Proposition 1.1.5 (Fermat’s Last Theorem). There are no positive integers x, y
and z such that
xn C yn D zn
for some integer n > 2.

In a book he was reading around 1630, Fermat claimed to have a proof for this
proposition, but not enough space in the margin to write it down. Over the years,
the Theorem was proved to hold for all n up to 4,000,000, but we’ve seen that this
shouldn’t necessarily inspire confidence that it holds for all n. There is, after all,
a clear resemblance between Fermat’s Last Theorem and Euler’s false Conjecture.
Finally, in 1994, British mathematician Andrew Wiles gave a proof, after seven
years of working in secrecy and isolation in his attic. His proof did not fit in any
margin.4
Finally, let’s mention another simply stated proposition whose truth remains un-
known.

Conjecture 1.1.6 (Goldbach). Every even integer greater than 2 is the sum of two
primes.

Goldbach’s Conjecture dates back to 1742. It is known to hold for all numbers
up to 1018 , but to this day, no one knows whether it’s true or false.
3 The story of the proof of the Four Color Theorem is told in a well-reviewed popular (non-

technical) book: “Four Colors Suffice. How the Map Problem was Solved.” Robin Wilson. Princeton
Univ. Press, 2003, 276pp. ISBN 0-691-11533-8.
4 In fact, Wiles’ original proof was wrong, but he and several collaborators used his ideas to arrive

at a correct proof a year later. This story is the subject of the popular book, Fermat’s Enigma by
Simon Singh, Walker & Company, November, 1997.
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8 Chapter 1 What is a Proof?

For a computer scientist, some of the most important things to prove are the
correctness of programs and systems—whether a program or system does what it’s
supposed to. Programs are notoriously buggy, and there’s a growing community
of researchers and practitioners trying to find ways to prove program correctness.
These efforts have been successful enough in the case of CPU chips that they are
now routinely used by leading chip manufacturers to prove chip correctness and
avoid some notorious past mistakes.
Developing mathematical methods to verify programs and systems remains an
active research area. We’ll illustrate some of these methods in Chapter 5.

1.2 Predicates
A predicate can be understood as a proposition whose truth depends on the value
of one or more variables. So “n is a perfect square” describes a predicate, since you
can’t say if it’s true or false until you know what the value of the variable n happens
to be. Once you know, for example, that n equals 4, the predicate becomes the true
proposition “4 is a perfect square”. Remember, nothing says that the proposition
has to be true: if the value of n were 5, you would get the false proposition “5 is a
perfect square.”
Like other propositions, predicates are often named with a letter. Furthermore, a
function-like notation is used to denote a predicate supplied with specific variable
values. For example, we might use the name “P ” for predicate above:

P .n/ WWD “n is a perfect square”;

and repeat the remarks above by asserting that P .4/ is true, and P .5/ is false.
This notation for predicates is confusingly similar to ordinary function notation.
If P is a predicate, then P .n/ is either true or false, depending on the value of n.
On the other hand, if p is an ordinary function, like n2 C1, then p.n/ is a numerical
quantity. Don’t confuse these two!

1.3 The Axiomatic Method
The standard procedure for establishing truth in mathematics was invented by Eu-
clid, a mathematician working in Alexandria, Egypt around 300 BC. His idea was
to begin with five assumptions about geometry, which seemed undeniable based on
direct experience. (For example, “There is a straight line segment between every
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1.4. Our Axioms 9

pair of points”.) Propositions like these that are simply accepted as true are called
axioms.
Starting from these axioms, Euclid established the truth of many additional propo-
sitions by providing “proofs.” A proof is a sequence of logical deductions from
axioms and previously proved statements that concludes with the proposition in
question. You probably wrote many proofs in high school geometry class, and
you’ll see a lot more in this text.
There are several common terms for a proposition that has been proved. The
different terms hint at the role of the proposition within a larger body of work.
Important true propositions are called theorems.
A lemma is a preliminary proposition useful for proving later propositions.
A corollary is a proposition that follows in just a few logical steps from a
theorem.
These definitions are not precise. In fact, sometimes a good lemma turns out to be
far more important than the theorem it was originally used to prove.
Euclid’s axiom-and-proof approach, now called the axiomatic method, remains
the foundation for mathematics today. In fact, just a handful of axioms, called the
Zermelo-Fraenkel with Choice axioms (ZFC), together with a few logical deduction
rules, appear to be sufficient to derive essentially all of mathematics. We’ll examine
these in Chapter 8.

1.4 Our Axioms
The ZFC axioms are important in studying and justifying the foundations of math-
ematics, but for practical purposes, they are much too primitive. Proving theorems
in ZFC is a little like writing programs in byte code instead of a full-fledged pro-
gramming language—by one reckoning, a formal proof in ZFC that 2 C 2 D 4
requires more than 20,000 steps! So instead of starting with ZFC, we’re going to
take a huge set of axioms as our foundation: we’ll accept all familiar facts from
high school math.
This will give us a quick launch, but you may find this imprecise specification
of the axioms troubling at times. For example, in the midst of a proof, you may
start to wonder, “Must I prove this little fact or can I take it as an axiom?” There
really is no absolute answer, since what’s reasonable to assume and what requires
proof depends on the circumstances and the audience. A good general guideline is
simply to be up front about what you’re assuming.
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10 Chapter 1 What is a Proof?

1.4.1 Logical Deductions
Logical deductions, or inference rules, are used to prove new propositions using
previously proved ones.
A fundamental inference rule is modus ponens. This rule says that a proof of P
together with a proof that P IMPLIES Q is a proof of Q.
Inference rules are sometimes written in a funny notation. For example, modus
ponens is written:

Rule.
P; P IMPLIES Q
Q
When the statements above the line, called the antecedents, are proved, then we
can consider the statement below the line, called the conclusion or consequent, to
also be proved.
A key requirement of an inference rule is that it must be sound: an assignment
of truth values to the letters P , Q, . . . , that makes all the antecedents true must
also make the consequent true. So if we start off with true axioms and apply sound
inference rules, everything we prove will also be true.
There are many other natural, sound inference rules, for example:

Rule.
P IMPLIES Q; Q IMPLIES R
P IMPLIES R
Rule.
NOT .P / IMPLIES NOT .Q/
Q IMPLIES P
On the other hand,

Non-Rule.
NOT .P / IMPLIES NOT .Q/
P IMPLIES Q
is not sound: if P is assigned T and Q is assigned F, then the antecedent is true
and the consequent is not.
As with axioms, we will not be too formal about the set of legal inference rules.
Each step in a proof should be clear and “logical”; in particular, you should state
what previously proved facts are used to derive each new conclusion.
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1.5. Proving an Implication 11

1.4.2 Patterns of Proof
In principle, a proof can be any sequence of logical deductions from axioms and
previously proved statements that concludes with the proposition in question. This
freedom in constructing a proof can seem overwhelming at first. How do you even
start a proof?
Here’s the good news: many proofs follow one of a handful of standard tem-
plates. Each proof has it own details, of course, but these templates at least provide
you with an outline to fill in. We’ll go through several of these standard patterns,
pointing out the basic idea and common pitfalls and giving some examples. Many
of these templates fit together; one may give you a top-level outline while others
help you at the next level of detail. And we’ll show you other, more sophisticated
proof techniques later on.
The recipes below are very specific at times, telling you exactly which words to
write down on your piece of paper. You’re certainly free to say things your own
way instead; we’re just giving you something you could say so that you’re never at
a complete loss.

1.5 Proving an Implication
Propositions of the form “If P , then Q” are called implications. This implication
is often rephrased as “P IMPLIES Q.”
Here are some examples:
(Quadratic Formula) If ax 2 C bx C c D 0 and a ¤ 0, then
p
xD b ˙ b 2 4ac =2a:

(Goldbach’s Conjecture 1.1.6 rephrased) If n is an even integer greater than
2, then n is a sum of two primes.
If 0 x 2, then x 3 C 4x C 1 > 0.
There are a couple of standard methods for proving an implication.

1.5.1 Method #1
In order to prove that P IMPLIES Q:
1. Write, “Assume P .”
2. Show that Q logically follows.
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12 Chapter 1 What is a Proof?

Example
Theorem 1.5.1. If 0 x 2, then x 3 C 4x C 1 > 0.
Before we write a proof of this theorem, we have to do some scratchwork to
figure out why it is true.
The inequality certainly holds for x D 0; then the left side is equal to 1 and
1 > 0. As x grows, the 4x term (which is positive) initially seems to have greater
magnitude than x 3 (which is negative). For example, when x D 1, we have
4x D 4, but x 3 D 1 only. In fact, it looks like x 3 doesn’t begin to dominate
until x > 2. So it seems the x 3 C 4x part should be nonnegative for all x between
0 and 2, which would imply that x 3 C 4x C 1 is positive.
So far, so good. But we still have to replace all those “seems like” phrases with
solid, logical arguments. We can get a better handle on the critical x 3 C 4x part
by factoring it, which is not too hard:

x 3 C 4x D x.2 x/.2 C x/

Aha! For x between 0 and 2, all of the terms on the right side are nonnegative. And
a product of nonnegative terms is also nonnegative. Let’s organize this blizzard of
observations into a clean proof.

Proof. Assume 0 x 2. Then x, 2 x and 2Cx are all nonnegative. Therefore,
the product of these terms is also nonnegative. Adding 1 to this product gives a
positive number, so:
x.2 x/.2 C x/ C 1 > 0
Multiplying out on the left side proves that

x 3 C 4x C 1 > 0

as claimed.

There are a couple points here that apply to all proofs:
You’ll often need to do some scratchwork while you’re trying to figure out
the logical steps of a proof. Your scratchwork can be as disorganized as you
like—full of dead-ends, strange diagrams, obscene words, whatever. But
keep your scratchwork separate from your final proof, which should be clear
and concise.

Proofs typically begin with the word “Proof” and end with some sort of de-
limiter like or “QED.” The only purpose for these conventions is to clarify
where proofs begin and end.
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1.6. Proving an “If and Only If” 13

1.5.2 Method #2 - Prove the Contrapositive
An implication (“P IMPLIES Q”) is logically equivalent to its contrapositive
NOT .Q/ IMPLIES NOT .P / :

Proving one is as good as proving the other, and proving the contrapositive is some-
times easier than proving the original statement. If so, then you can proceed as
follows:
1. Write, “We prove the contrapositive:” and then state the contrapositive.
2. Proceed as in Method #1.

Example
p
Theorem 1.5.2. If r is irrational, then r is also irrational.
A number is rational when it equals a quotient of integers —that is, if it equals
m=n for some integers m and n. If it’s not rational, then it’s called irrational. So
p
we must show that if r is not a ratio of integers, then r is also not a ratio of
integers. That’s pretty convoluted! We can eliminate both not’s and simplify the
proof by using the contrapositive instead.
p
Proof. We prove the contrapositive: if r is rational, then r is rational.
p
Assume that r is rational. Then there exist integers m and n such that:
p m
rD
n
Squaring both sides gives:
m2
rD 2
n
2 2
Since m and n are integers, r is also rational.

1.6 Proving an “If and Only If”
Many mathematical theorems assert that two statements are logically equivalent;
that is, one holds if and only if the other does. Here is an example that has been
known for several thousand years:
Two triangles have the same side lengths if and only if two side lengths
and the angle between those sides are the same.
The phrase “if and only if” comes up so often that it is often abbreviated “iff.”
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14 Chapter 1 What is a Proof?

1.6.1 Method #1: Prove Each Statement Implies the Other
The statement “P IFF Q” is equivalent to the two statements “P IMPLIES Q” and
“Q IMPLIES P .” So you can prove an “iff” by proving two implications:
1. Write, “We prove P implies Q and vice-versa.”
2. Write, “First, we show P implies Q.” Do this by one of the methods in
Section 1.5.
3. Write, “Now, we show Q implies P .” Again, do this by one of the methods
in Section 1.5.

1.6.2 Method #2: Construct a Chain of Iffs
In order to prove that P is true iff Q is true:
1. Write, “We construct a chain of if-and-only-if implications.”
2. Prove P is equivalent to a second statement which is equivalent to a third
statement and so forth until you reach Q.
This method sometimes requires more ingenuity than the first, but the result can be
a short, elegant proof.

Example
The standard deviation of a sequence of values x1 ; x2 ; : : : ; xn is defined to be:
s
.x1 /2 C .x2 /2 C C .xn /2
(1.3)
n
where is the average or mean of the values:
x1 C x2 C C xn
WWD
n
Theorem 1.6.1. The standard deviation of a sequence of values x1 ; : : : ; xn is zero
iff all the values are equal to the mean.
For example, the standard deviation of test scores is zero if and only if everyone
scored exactly the class average.

Proof. We construct a chain of “iff” implications, starting with the statement that
the standard deviation (1.3) is zero:
s
.x1 /2 C .x2 /2 C C .xn /2
D 0: (1.4)
n
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1.7. Proof by Cases 15

Now since zero is the only number whose square root is zero, equation (1.4) holds
iff
.x1 /2 C .x2 /2 C C .xn /2 D 0: (1.5)
Squares of real numbers are always nonnegative, so every term on the left-hand
side of equation (1.5) is nonnegative. This means that (1.5) holds iff

Every term on the left-hand side of (1.5) is zero. (1.6)

But a term .xi /2 is zero iff xi D , so (1.6) is true iff

Every xi equals the mean.

1.7 Proof by Cases
Breaking a complicated proof into cases and proving each case separately is a com-
mon, useful proof strategy. Here’s an amusing example.
Let’s agree that given any two people, either they have met or not. If every pair
of people in a group has met, we’ll call the group a club. If every pair of people in
a group has not met, we’ll call it a group of strangers.
Theorem. Every collection of 6 people includes a club of 3 people or a group of 3
strangers.
Proof. The proof is by case analysis5 . Let x denote one of the six people. There
are two cases:

1. Among 5 other people besides x, at least 3 have met x.

2. Among the 5 other people, at least 3 have not met x.

Now, we have to be sure that at least one of these two cases must hold,6 but that’s
easy: we’ve split the 5 people into two groups, those who have shaken hands with
x and those who have not, so one of the groups must have at least half the people.
Case 1: Suppose that at least 3 people did meet x.
This case splits into two subcases:
5 Describing your approach at the outset helps orient the reader.
6 Part of a case analysis argument is showing that you’ve covered all the cases. This is often
obvious, because the two cases are of the form “P ” and “not P .” However, the situation above is not
stated quite so simply.
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16 Chapter 1 What is a Proof?

Case 1.1: No pair among those people met each other. Then these
people are a group of at least 3 strangers. The theorem holds in this
subcase.
Case 1.2: Some pair among those people have met each other. Then
that pair, together with x, form a club of 3 people. So the theorem
holds in this subcase.

This implies that the theorem holds in Case 1.
Case 2: Suppose that at least 3 people did not meet x.
This case also splits into two subcases:

Case 2.1: Every pair among those people met each other. Then these
people are a club of at least 3 people. So the theorem holds in this
subcase.
Case 2.2: Some pair among those people have not met each other.
Then that pair, together with x, form a group of at least 3 strangers. So
the theorem holds in this subcase.

This implies that the theorem also holds in Case 2, and therefore holds in all cases.

1.8 Proof by Contradiction
In a proof by contradiction, or indirect proof, you show that if a proposition were
false, then some false fact would be true. Since a false fact by definition can’t be
true, the proposition must be true.
Proof by contradiction is always a viable approach. However, as the name sug-
gests, indirect proofs can be a little convoluted, so direct proofs are generally prefer-
able when they are available.
Method: In order to prove a proposition P by contradiction:

1. Write, “We use proof by contradiction.”

2. Write, “Suppose P is false.”

3. Deduce something known to be false (a logical contradiction).

4. Write, “This is a contradiction. Therefore, P must be true.”
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1.9. Good Proofs in Practice 17

Example
p
We’ll prove by contradiction that 2 is irrational. Remember that a number is ra-
tional if it is equal to a ratio of integers—for example, 3:5 D 7=2 and 0:1111 D
1=9 are rational numbers.
p
Theorem 1.8.1. 2 is irrational.
p
Proof. We use proof byp contradiction. Suppose the claim is false, and 2 is ratio-
nal. Then we can write 2 as a fraction n=d in lowest terms.
Squaring both sides gives 2 D n2 =d 2 and so 2d 2 D n2 . This implies that n is a
multiple of 2 (see Problems 1.15 and 1.16). Therefore n2 must be a multiple of 4.
But since 2d 2 D n2 , we know 2d 2 is a multiple of 4 and so d 2 is a multiple of 2.
This implies that d is a multiple of 2.
So, the numerator and denominator havep 2 as a common factor, which contradicts
the fact that n=d is in lowest terms. Thus, 2 must be irrational.

1.9 Good Proofs in Practice
One purpose of a proof is to establish the truth of an assertion with absolute cer-
tainty, and mechanically checkable proofs of enormous length or complexity can
accomplish this. But humanly intelligible proofs are the only ones that help some-
one understand the subject. Mathematicians generally agree that important mathe-
matical results can’t be fully understood until their proofs are understood. That is
why proofs are an important part of the curriculum.
To be understandable and helpful, more is required of a proof than just logical
correctness: a good proof must also be clear. Correctness and clarity usually go
together; a well-written proof is more likely to be a correct proof, since mistakes
are harder to hide.
In practice, the notion of proof is a moving target. Proofs in a professional
research journal are generally unintelligible to all but a few experts who know all
the terminology and prior results used in the proof. Conversely, proofs in the first
weeks of a beginning course like 6.042 would be regarded as tediously long-winded
by a professional mathematician. In fact, what we accept as a good proof later in
the term will be different from what we consider good proofs in the first couple
of weeks of 6.042. But even so, we can offer some general tips on writing good
proofs:

State your game plan. A good proof begins by explaining the general line of rea-
soning, for example, “We use case analysis” or “We argue by contradiction.”
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18 Chapter 1 What is a Proof?

Keep a linear flow. Sometimes proofs are written like mathematical mosaics, with
juicy tidbits of independent reasoning sprinkled throughout. This is not good.
The steps of an argument should follow one another in an intelligible order.

A proof is an essay, not a calculation. Many students initially write proofs the way
they compute integrals. The result is a long sequence of expressions without
explanation, making it very hard to follow. This is bad. A good proof usually
looks like an essay with some equations thrown in. Use complete sentences.

Avoid excessive symbolism. Your reader is probably good at understanding words,
but much less skilled at reading arcane mathematical symbols. Use words
where you reasonably can.

Revise and simplify. Your readers will be grateful.

Introduce notation thoughtfully. Sometimes an argument can be greatly simpli-
fied by introducing a variable, devising a special notation, or defining a new
term. But do this sparingly, since you’re requiring the reader to remember
all that new stuff. And remember to actually define the meanings of new
variables, terms, or notations; don’t just start using them!

Structure long proofs. Long programs are usually broken into a hierarchy of smaller
procedures. Long proofs are much the same. When your proof needed facts
that are easily stated, but not readily proved, those fact are best pulled out
as preliminary lemmas. Also, if you are repeating essentially the same argu-
ment over and over, try to capture that argument in a general lemma, which
you can cite repeatedly instead.

Be wary of the “obvious.” When familiar or truly obvious facts are needed in a
proof, it’s OK to label them as such and to not prove them. But remember
that what’s obvious to you may not be—and typically is not—obvious to
your reader.
Most especially, don’t use phrases like “clearly” or “obviously” in an attempt
to bully the reader into accepting something you’re having trouble proving.
Also, go on the alert whenever you see one of these phrases in someone else’s
proof.

Finish. At some point in a proof, you’ll have established all the essential facts
you need. Resist the temptation to quit and leave the reader to draw the
“obvious” conclusion. Instead, tie everything together yourself and explain
why the original claim follows.
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1.10. References 19

Creating a good proof is a lot like creating a beautiful work of art. In fact,
mathematicians often refer to really good proofs as being “elegant” or “beautiful.”
It takes a practice and experience to write proofs that merit such praises, but to
get you started in the right direction, we will provide templates for the most useful
proof techniques.
Throughout the text there are also examples of bogus proofs—arguments that
look like proofs but aren’t. Sometimes a bogus proof can reach false conclusions
because of missteps or mistaken assumptions. More subtle bogus proofs reach
correct conclusions, but do so in improper ways such as circular reasoning, leaping
to unjustified conclusions, or saying that the hard part of the proof is “left to the
reader.” Learning to spot the flaws in improper proofs will hone your skills at seeing
how each proof step follows logically from prior steps. It will also enable you to
spot flaws in your own proofs.
The analogy between good proofs and good programs extends beyond structure.
The same rigorous thinking needed for proofs is essential in the design of criti-
cal computer systems. When algorithms and protocols only “mostly work” due
to reliance on hand-waving arguments, the results can range from problematic to
catastrophic. An early example was the Therac 25, a machine that provided radia-
tion therapy to cancer victims, but occasionally killed them with massive overdoses
due to a software race condition. A further example of a dozen years ago (August
2004) involved a single faulty command to a computer system used by United and
American Airlines that grounded the entire fleet of both companies—and all their
passengers!
It is a certainty that we’ll all one day be at the mercy of critical computer systems
designed by you and your classmates. So we really hope that you’ll develop the
ability to formulate rock-solid logical arguments that a system actually does what
you think it should do!

1.10 References
[12], [1], [46], [16], [20]
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20 Chapter 1 What is a Proof?

Problems for Section 1.1
Class Problems
Problem 1.1.
Albert announces to his class that he plans to surprise them with a quiz sometime
next week.
His students first wonder if the quiz could be on Friday of next week. They
reason that it can’t: if Albert didn’t give the quiz before Friday, then by midnight
Thursday, they would know the quiz had to be on Friday, and so the quiz wouldn’t
be a surprise any more.
Next the students wonder whether Albert could give the surprise quiz Thursday.
They observe that if the quiz wasn’t given before Thursday, it would have to be
given on the Thursday, since they already know it can’t be given on Friday. But
having figured that out, it wouldn’t be a surprise if the quiz was on Thursday either.
Similarly, the students reason that the quiz can’t be on Wednesday, Tuesday, or
Monday. Namely, it’s impossible for Albert to give a surprise quiz next week. All
the students now relax, having concluded that Albert must have been bluffing. And
since no one expects the quiz, that’s why, when Albert gives it on Tuesday next
week, it really is a surprise!
What, if anything, do you think is wrong with the students’ reasoning?

Problem 1.2.
The Pythagorean Theorem says that if a and b are the lengths of the sides of a right
triangle, and c is the length of its hypotenuse, then
a2 C b 2 D c 2 :
This theorem is so fundamental and familiar that we generally take it for granted.
But just being familiar doesn’t justify calling it “obvious”—witness the fact that
people have felt the need to devise different proofs of it for milllenia.7 In this
problem we’ll examine a particularly simple “proof without words” of the theorem.
Here’s the strategy. Suppose you are given four different colored copies of a
right triangle with sides of lengths a, b and c, along with a suitably sized square,
as shown in Figure 1.1.
(a) You will first arrange the square and four triangles so they form a c c square.
From this arrangement you will see that the square is .b a/ .b a/.
7 Over a hundred different proofs are listed on the mathematics website http://www.cut-the-
knot.org/pythagoras/.
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1.10. References 21

b c

Figure 1.1 Right triangles and square.

(b) You will then arrange the same shapes so they form two squares, one a a
and the other b b.
You know that the area of an s s square is s 2 . So appealing to the principle that
Area is Preserved by Rearranging,
you can now conclude that a2 C b 2 D c 2 , as claimed.
This really is an elegant and convincing proof of the Pythagorean Theorem, but it
has some worrisome features. One concern is that there might be something special
about the shape of these particular triangles and square that makes the rearranging
possible—for example, suppose a D b?
(c) How would you respond to this concern?

(d) Another concern is that a number of facts about right triangles, squares and
lines are being implicitly assumed in justifying the rearrangements into squares.
Enumerate some of these assumed facts.

Problem 1.3.
What’s going on here?!
p p p p p 2
1D 1 D . 1/. 1/ D 1 1D 1 D 1:
(a) Precisely identify and explain the mistake(s) in this bogus proof.

(b) Prove (correctly) that if 1 D 1, then 2 D 1.
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22 Chapter 1 What is a Proof?

(c) Every positive real number r has two square roots, one positive and the other
p
negative. The standard convention is that the expression r refers to the positive
square root of r. Assuming familiar properties of multiplication of real numbers,
prove that for positive real numbers r and s,
p p p
rs D r s:

Problem 1.4.
Identify exactly where the bugs are in each of the following bogus proofs.8
(a) Bogus Claim: 1=8 > 1=4:

Bogus proof.

3>2
3 log10 .1=2/ > 2 log10 .1=2/
log10 .1=2/3 > log10 .1=2/2
.1=2/3 > .1=2/2 ;

and the claim now follows by the rules for multiplying fractions.

(b) Bogus proof : 1¢ D $0:01 D .$0:1/2 D .10¢/2 D 100¢ D $1:

Bogus proof.

aDb
a2 D ab
a2 b 2 D ab b2
.a b/.a C b/ D .a b/b
aCb Db
a D 0:

8 From [45], Twenty Years Before the Blackboard by Michael Stueben and Diane Sandford
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1.10. References 23

Problem 1.5.
It’s a fact that the Arithmetic Mean is at least as large as the Geometric Mean,
namely,
aCb p
ab
2
for all nonnegative real numbers a and b. But there’s something objectionable
about the following proof of this fact. What’s the objection, and how would you fix
it?

Bogus proof.

aCb ‹ p
ab; so
2
‹ p
a C b 2 ab; so
‹
a2 C 2ab C b 2 4ab; so
‹
a2 2ab C b 2 0; so
2
.a b/ 0 which we know is true.

The last statement is true because a b is a real number, and the square of a real
number is never negative. This proves the claim.

Practice Problems
Problem 1.6.
Why does the “surprise” paradox of Problem 1.1 present a philosophical problem
but not a mathematical one?

Problems for Section 1.5
Homework Problems
Problem 1.7.
Show that log7 n is either an integer or irrational, where n is a positive integer. Use
whatever familiar facts about integers and primes you need, but explicitly state such
facts.
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24 Chapter 1 What is a Proof?

Problems for Section 1.7
Practice Problems
Problem 1.8.
Prove by cases that
max.r; s/ C min.r; s/ D r C s (*)
for all real numbers r; s.

Class Problems
Problem 1.9.
If we raise an irrational number to p
an irrational power, can the result be rational?
p 2
Show that it can by considering 2 and arguing by cases.

Problem 1.10.
Prove by cases that
jr C sj jrj C jsj (1)
for all real numbers r; s.9

Homework Problems
Problem 1.11. (a) Suppose that

a C b C c D d;

where a; b; c; d are nonnegative integers.
Let P be the assertion that d is even. Let W be the assertion that exactly one among
a; b; c are even, and let T be the assertion that all three are even.
Prove by cases that
P IFF ŒW OR T :

(b) Now suppose that
w2 C x2 C y 2 D z2;
9 The absolute value jrj of r equals whichever of r or r is not negative.
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1.10. References 25

where w; x; y; z are nonnegative integers. Let P be the assertion that z is even, and
let R be the assertion that all three of w; x; y are even. Prove by cases that

P IFF R:

Hint: An odd number equals 2m C 1 for some integer m, so its square equals
4.m2 C m/ C 1.

Exam Problems
Problem 1.12. p
Prove that there is an irrational
p number a such that a 3 is rational.
p 3
Hint: Consider 3 2 and argue by cases.

Problems for Section 1.8
Practice Problems
Problem 1.13.
Prove that for any n > 0, if an is even, then a is even.
Hint: Contradiction.

Problem 1.14.
p
Prove that if a b D n, then either a or b must be n, where a; b, and n are
nonnegative real numbers. Hint: by contradiction, Section 1.8.

Problem 1.15.
Let n be a nonnegative integer.
(a) Explain why if n2 is even—that is, a multiple of 2—then n is even.

(b) Explain why if n2 is a multiple of 3, then n must be a multiple of 3.

Problem 1.16.
Give an example of two distinct positive integers m; n such that n2 is a multiple of
m, but n is not a multiple of m. How about having m be less than n?
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26 Chapter 1 What is a Proof?

Class Problems
Problem 1.17. p
How far can you generalize
p the proof of Theorem 1.8.1 that 2 is irrational? For
example, how about 3?

Problem 1.18.
Prove that log4 6 is irrational.

Problem 1.19. p p
Prove by p
contradiction
p p that p3 C 2 is irrational.
Hint: . 3 C 2/. 3 2/

Problem 1.20.
Here is a generalization of Problem 1.17 that you may not have thought of:

Lemma. Let the coefficients of the polynomial

a0 C a1 x C a2 x 2 C C am 1x
m 1
C xm

be integers. Then any real root of the polynomial is either integral or irrational.
p
m
(a) Explain why the Lemma immediately implies that k is irrational whenever
k is not an mth power of some integer.

(b) Carefully prove the Lemma.
You may find it helpful to appeal to:
Fact. If a prime p is a factor of some power of an integer, then it is a factor of that
integer.
You may assume this Fact without writing down its proof, but see if you can explain
why it is true.

Exam Problems
Problem 1.21.
Prove that log9 12 is irrational.
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1.10. References 27

Problem 1.22.
Prove that log12 18 is irrational.

Problem 1.23. p3
A familiar proof that 72 is irrational depends on the fact that a certain equation
among those below is unsatisfiable by integers a; b > 0. Note that more than one
is unsatisfiable. Indicate the equation that would
p appear in the proof, and explain
3 2
why it is unsatisfiable. (Do not assume that 7 is irrational.)

i. a2 D 72 C b 2

ii. a3 D 72 C b 3

iii. a2 D 72 b 2

iv. a3 D 72 b 3

v. a3 D 73 b 3

vi. .ab/3 D 72

Homework Problems
Problem 1.24.
The fact that that there are irrational numbers a; b such that ab is rational was
proved in Problem 1.9 by cases. Unfortunately, that proof was nonconstructive: it
p a specific pair a; b with this property. But in fact, it’s easy to do this:
didn’t reveal
let a WWD 2 and bpWWD 2 log2 3.
We know a D 2 is irrational, and ab D 3 by definition. Finish the proof that
these values for a; b work, by showing that 2 log2 3 is irrational.

Problem 1.25. p
Here is a different proof that 2 is irrational, taken from the American Mathemat-
ical Monthly, v.116, #1, Jan. 2009, p.69:
p
Proof. Suppose for the sake ofcontradiction that 2 is rational, and choose the
p
least integer q > 0 such that 2 1 q is a nonnegative integer. Let q 0 WWD
p p
2 1 q. Clearly 0 < q 0 < q. But an easy computation shows that 2 1 q0
is a nonnegative integer, contradicting the minimality of q.
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28 Chapter 1 What is a Proof?

(a) This proof was written for an audience of college teachers, and at this point it
is a little more concise than desirable. Write out a more complete version which
includes an explanation of each step.

(b) Now that you have justified the steps in this proof, do you have a preference
for one of these proofs over the other? Why? Discuss these questions with your
teammates for a few minutes and summarize your team’s answers on your white-
board.

Problem 1.26.
For n D 40, the value of polynomial p.n/ WWD n2 C n C 41 is not prime, as noted
in Section 1.1. But we could have predicted based on general principles that no
nonconstant polynomial can generate only prime numbers.
In particular, let q.n/ be a polynomial with integer coefficients, and let c WWD q.0/
be the constant term of q.
(a) Verify that q.cm/ is a multiple of c for all m 2 Z.

(b) Show that if q is nonconstant and c > 1, then as n ranges over the nonnegative
integers N there are infinitely many q.n/ 2 Z that are not primes.
Hint: You may assume the familiar fact that the magnitude of any nonconstant
polynomial q.n/ grows unboundedly as n grows.

(c) Conclude that for every nonconstant polynomial q there must be an n 2 N
such that q.n/ is not prime. Hint: Only one easy case remains.
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2 The Well Ordering Principle
Every nonempty set of nonnegative integers has a smallest element.

This statement is known as The Well Ordering Principle. Do you believe it?
Seems sort of obvious, right? But notice how tight it is: it requires a nonempty
set—it’s false for the empty set which has no smallest element because it has no
elements at all. And it requires a set of nonnegative integers—it’s false for the
set of negative integers and also false for some sets of nonnegative rationals—for
example, the set of positive rationals. So, the Well Ordering Principle captures
something special about the nonnegative integers.
While the Well Ordering Principle may seem obvious, it’s hard to see offhand
why it is useful. But in fact, it provides one of the most important proof rules in
discrete mathematics. In this chapter, we’ll illustrate the power of this proof method
with a few simple examples.

2.1 Well Ordering Proofs
We actually
p have already taken the Well Ordering Principle for granted in proving
that 2 is irrational. That proof assumed that for any positive integers m and n,
the fraction m=n can be written in lowest terms, that is, in the form m0 =n0 where
m0 and n0 are positive integers with no common prime factors. How do we know
this is always possible?
Suppose to the contrary that there are positive integers m and n such that the
fraction m=n cannot be written in lowest terms. Now let C be the set of positive
integers that are numerators of such fractions. Then m 2 C , so C is nonempty.
Therefore, by Well Ordering, there must be a smallest integer m0 2 C . So by
definition of C , there is an integer n0 > 0 such that
m0
the fraction cannot be written in lowest terms.
n0
This means that m0 and n0 must have a common prime factor, p > 1. But

m0 =p m0
D ;
n0 =p n0
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30 Chapter 2 The Well Ordering Principle

so any way of expressing the left-hand fraction in lowest terms would also work for
m0 =n0 , which implies
m0 =p
the fraction cannot be in written in lowest terms either.
n0 =p
So by definition of C , the numerator m0 =p is in C . But m0 =p < m0 , which
contradicts the fact that m0 is the smallest element of C .
Since the assumption that C is nonempty leads to a contradiction, it follows that
C must be empty. That is, that there are no numerators of fractions that can’t be
written in lowest terms, and hence there are no such fractions at all.
We’ve been using the Well Ordering Principle on the sly from early on!

2.2 Template for Well Ordering Proofs
More generally, there is a standard way to use Well Ordering to prove that some
property, P .n/ holds for every nonnegative integer n. Here is a standard way to
organize such a well ordering proof:

To prove that “P .n/ is true for all n 2 N” using the Well Ordering Principle:

Define the set C of counterexamples to P being true. Specifically, define

C WWD fn 2 N j NOT.P .n// is trueg:

(The notation fn j Q.n/g means “the set of all elements n for which Q.n/
is true.” See Section 4.1.4.)

Assume for proof by contradiction that C is nonempty.

By the Well Ordering Principle, there will be a smallest element n in C .

Reach a contradiction somehow—often by showing that P .n/ is actually
true or by showing that there is another member of C that is smaller than
n. This is the open-ended part of the proof task.

Conclude that C must be empty, that is, no counterexamples exist.

2.2.1 Summing the Integers
Let’s use this template to prove
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2.2. Template for Well Ordering Proofs 31

Theorem 2.2.1.
1 C 2 C 3 C C n D n.n C 1/=2 (2.1)
for all nonnegative integers n.

First, we’d better address a couple of ambiguous special cases before they trip us
up:

If n D 1, then there is only one term in the summation, and so 1 C 2 C 3 C
C n is just the term 1. Don’t be misled by the appearance of 2 and 3 or by
the suggestion that 1 and n are distinct terms!

If n D 0, then there are no terms at all in the summation. By convention, the
sum in this case is 0.

So, while the three dots notation, which is called an ellipsis, is convenient, you
have to watch out for these special cases where the notation is misleading. In
fact, whenever you see an ellipsis, you should be on the lookout to be sure you
understand the pattern, watching out for the beginning and the end.
We could have eliminated the need for guessing by rewriting the left side of (2.1)
with summation notation:
n
X X
i or i:
i D1 1i n

Both of these expressions denote the sum of all values taken by the expression to
the right of the sigma as the variable i ranges from 1 to n. Both expressions make
it clear what (2.1) means when n D 1. The second expression makes it clear that
when n D 0, there are no terms in the sum, though you still have to know the
convention that a sum of no numbers equals 0 (the product of no numbers is 1, by
the way).
OK, back to the proof:

Proof. By contradiction. Assume that Theorem 2.2.1 is false. Then, some nonneg-
ative integers serve as counterexamples to it. Let’s collect them in a set:

n.n C 1/
C WWD fn 2 N j 1 C 2 C 3 C C n ¤ g:
2
Assuming there are counterexamples, C is a nonempty set of nonnegative integers.
So, by the Well Ordering Principle, C has a minimum element, which we’ll call
c. That is, among the nonnegative integers, c is the smallest counterexample to
equation (2.1).
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32 Chapter 2 The Well Ordering Principle

Since c is the smallest counterexample, we know that (2.1) is false for n D c but
true for all nonnegative integers n < c. But (2.1) is true for n D 0, so c > 0. This
means c 1 is a nonnegative integer, and since it is less than c, equation (2.1) is
true for c 1. That is,
.c 1/c
1 C 2 C 3 C C .c 1/ D :
2
But then, adding c to both sides, we get

.c 1/c c2 c C 2c c.c C 1/
1 C 2 C 3 C C .c 1/ C c D Cc D D ;
2 2 2
which means that (2.1) does hold for c, after all! This is a contradiction, and we
are done.

2.3 Factoring into Primes
We’ve previously taken for granted the Prime Factorization Theorem, also known
as the Unique Factorization Theorem and the Fundamental Theorem of Arithmetic,
which states that every integer greater than one has a unique1 expression as a prod-
uct of prime numbers. This is another of those familiar mathematical facts which
are taken for granted but are not really obvious on closer inspection. We’ll prove
the uniqueness of prime factorization in a later chapter, but well ordering gives an
easy proof that every integer greater than one can be expressed as some product of
primes.

Theorem 2.3.1. Every positive integer greater than one can be factored as a prod-
uct of primes.

Proof. The proof is by well ordering.
Let C be the set of all integers greater than one that cannot be factored as a
product of primes. We assume C is not empty and derive a contradiction.
If C is not empty, there is a least element n 2 C by well ordering. The n can’t
be prime, because a prime by itself is considered a (length one) product of primes
and no such products are in C .
So n must be a product of two integers a and b where 1 < a; b < n. Since a
and b are smaller than the smallest element in C , we know that a; b … C . In other
words, a can be written as a product of primes p1 p2 pk and b as a product of
1 . . . unique up to the order in which the prime factors appear
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2.4. Well Ordered Sets 33

primes q1 ql . Therefore, n D p1 pk q1 ql can be written as a product of
primes, contradicting the claim that n 2 C . Our assumption that C is not empty
must therefore be false.

2.4 Well Ordered Sets
A set of numbers is well ordered when each of its nonempty subsets has a minimum
element. The Well Ordering Principle says, of course, that the set of nonnegative
integers is well ordered, but so are lots of other sets, such as every finite set, or the
sets rN of numbers of the form rn, where r is a positive real number and n 2 N.
Well ordering commonly comes up in computer science as a method for proving
that computations won’t run forever. The idea is to assign a value to the successive
steps of a computation so that the values get smaller at every step. If the values are
all from a well ordered set, then the computation can’t run forever, because if it did,
the values assigned to its successive steps would define a subset with no minimum
element. You’ll see several examples of this technique applied in Chapter 6 to prove
that various state machines will eventually terminate.
Notice that a set may have a minimum element but not be well ordered. The set
of nonnegative rational numbers is an example: it has a minimum element zero,
but it also has nonempty subsets that don’t have minimum elements—the positive
rationals, for example.
The following theorem is a tiny generalization of the Well Ordering Principle.

Theorem 2.4.1. For any nonnegative integer n the set of integers greater than or
equal to n is well ordered.

This theorem is just as obvious as the Well Ordering Principle, and it would
be harmless to accept it as another axiom. But repeatedly introducing axioms gets
worrisome after a while, and it’s worth noticing when a potential axiom can actually
be proved. We can easily prove Theorem 2.4.1 using the Well Ordering Principle:

Proof. Let S be any nonempty set of integers n. Now add n to each of the
elements in S ; let’s call this new set S C n. Now S C n is a nonempty set of
nonnegative integers, and so by the Well Ordering Principle, it has a minimum
element m. But then it’s easy to see that m n is the minimum element of S .

The definition of well ordering states that every subset of a well ordered set
is well ordered, and this yields two convenient, immediate corollaries of Theo-
rem 2.4.1:
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34 Chapter 2 The Well Ordering Principle

Definition 2.4.2. A lower bound (respectively, upper bound) for a set S of real
numbers is a number b such that b s (respectively, b s) for every s 2 S .

Note that a lower or upper bound of set S is not required to be in the set.

Corollary 2.4.3. Any set of integers with a lower bound is well ordered.

Proof. A set of integers with a lower bound b 2 R will also have the integer n D
bbc as a lower bound, where bbc, called the floor of b, is gotten by rounding down
b to the nearest integer. So Theorem 2.4.1 implies the set is well ordered.

Corollary 2.4.4. Any nonempty set of integers with an upper bound has a maximum
element.

Proof. Suppose a set S of integers has an upper bound b 2 R. Now multiply each
element of S by -1; let’s call this new set of elements S . Now, of course, b is a
lower bound of S . So S has a minimum element m by Corollary 2.4.3. But
then it’s easy to see that m is the maximum element of S.

2.4.1 A Different Well Ordered Set (Optional)
Another example of a well ordered set of numbers is the set F of fractions that can
be expressed in the form n=.n C 1/:
0 1 2 3 n
; ; ; ;:::; ;::::
1 2 3 4 nC1
The minimum element of any nonempty subset of F is simply the one with the
minimum numerator when expressed in the form n=.n C 1/.
Now we can define a very different well ordered set by adding nonnegative inte-
gers to numbers in F. That is, we take all the numbers of the form n C f where n is
a nonnegative integer and f is a number in F. Let’s call this set of numbers—you
guessed it—N C F. There is a simple recipe for finding the minimum number in
any nonempty subset of N C F, which explains why this set is well ordered:

Lemma 2.4.5. N C F is well ordered.

Proof. Given any nonempty subset S of N C F, look at all the nonnegative integers
n such that n C f is in S for some f 2 F. This is a nonempty set nonnegative
integers, so by the WOP, there is a minimum one; call it ns .
By definition of ns , there is some f 2 F such that nS C f is in the set S. So
the set all fractions f such that nS C f 2 S is a nonempty subset of F, and since
F is well ordered, this nonempty set contains a minimum element; call it fS . Now
it easy to verify that nS C fS is the minimum element of S (Problem 2.20).
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2.4. Well Ordered Sets 35

The set N C F is different from the earlier examples. In all the earlier examples,
each element was greater than only a finite number of other elements. In N C F,
every element greater than or equal to 1 can be the first element in strictly decreas-
ing sequences of elements of arbitrary finite length. For example, the following
decreasing sequences of elements in N C F all start with 1:

1; 0:
1; 12 ; 0:
1; 23 ; 12 ; 0:
1; 34 ; 23 ; 12 ; 0:
::
:

Nevertheless, since N C F is well ordered, it is impossible to find an infinite de-
creasing sequence of elements in N C F, because the set of elements in such a
sequence would have no minimum.

Problems for Section 2.2
Practice Problems
Problem 2.1.
For practice using the Well Ordering Principle, fill in the template of an easy to
prove fact: every amount of postage that can be assembled using only 10 cent and
15 cent stamps is divisible by 5.
In particular, let the notation “j j k” indicate that integer j is a divisor of integer
k, and let S.n/ mean that exactly n cents postage can be assembled using only 10
and 15 cent stamps. Then the proof shows that

S.n/ IMPLIES 5 j n; for all nonnegative integers n: (2.2)

Fill in the missing portions (indicated by “. . . ”) of the following proof of (2.2).

Let C be the set of counterexamples to (2.2), namely

C WWD fn j : : :g

Assume for the purpose of obtaining a contradiction that C is nonempty.
Then by the WOP, there is a smallest number m 2 C . This m must be
positive because . . . .
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36 Chapter 2 The Well Ordering Principle

But if S.m/ holds and m is positive, then S.m 10/ or S.m 15/
must hold, because . . . .
So suppose S.m 10/ holds. Then 5 j .m 10/, because. . .
But if 5 j .m 10/, then obviously 5 j m, contradicting the fact that m
is a counterexample.
Next, if S.m 15/ holds, we arrive at a contradiction in the same way.
Since we get a contradiction in both cases, we conclude that. . .
which proves that (2.2) holds.

Problem 2.2.
The Fibonacci numbers F .0/; F .1/; F .2/; : : : are defined as follows:
8
<0
ˆ if n D 0;
F .n/ WWD 1 if n D 1;
ˆ
F .n 1/ C F .n 2/ if n > 1:
:

Exactly which sentence(s) in the following bogus proof contain logical errors?
Explain.
False Claim. Every Fibonacci number is even.
Bogus proof. Let all the variables n; m; k mentioned below be nonnegative integer
valued.
1. The proof is by the WOP.

2. Let EF.n/ mean that F .n/ is even.

3. Let C be the set of counterexamples to the assertion that EF.n/ holds for all
n 2 N, namely,
C WWD fn 2 N j NOT.EF.n//g:

4. We prove by contradiction that C is empty. So assume that C is not empty.

5. By WOP, there is a least nonnegative integer m 2 C .

6. Then m > 0, since F .0/ D 0 is an even number.

7. Since m is the minimum counterexample, F .k/ is even for all k < m.

8. In particular, F .m 1/ and F .m 2/ are both even.
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2.4. Well Ordered Sets 37

9. But by the definition, F .m/ equals the sum F .m 1/ C F .m 2/ of two
even numbers, and so it is also even.
10. That is, EF.m/ is true.
11. This contradicts the condition in the definition of m that NOT.EF.m// holds.
12. This contradition implies that C must be empty. Hence, F .n/ is even for all
n 2 N.

Problem 2.3.
In Chapter 2, the Well Ordering Principle was used to show that all positive rational
numbers can be written in “lowest terms,” that is, as a ratio of positive integers with
no common factor prime factor. Below is a different proof which also arrives at this
correct conclusion, but this proof is bogus. Identify every step at which the proof
makes an unjustified inference.

Bogus proof. Suppose to the contrary that there was positive rational q such that q
cannot be written in lowest terms. Now let C be the set of such rational numbers
that cannot be written in lowest terms. Then q 2 C , so C is nonempty. So there
must be a smallest rational q0 2 C . So since q0 =2 < q0 , it must be possible to
express q0 =2 in lowest terms, namely,
q0 m
D (2.3)
2 n
for positive integers m; n with no common prime factor. Now we consider two
cases:
Case 1: [n is odd]. Then 2m and n also have no common prime factor, and
therefore m 2m
q0 D 2 D
n n
expresses q0 in lowest terms, a contradiction.
Case 2: [n is even]. Any common prime factor of m and n=2 would also be a
common prime factor of m and n. Therefore m and n=2 have no common prime
factor, and so
m
q0 D
n=2
expresses q0 in lowest terms, a contradiction.
Since the assumption that C is nonempty leads to a contradiction, it follows that
C is empty—that is, there are no counterexamples.
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38 Chapter 2 The Well Ordering Principle

Class Problems
Problem 2.4.
Use the Well Ordering Principle 2 to prove that
n
X n.n C 1/.2n C 1/
k2 D : (2.4)
6
kD0

for all nonnegative integers n.

Problem 2.5.
Use the Well Ordering Principle to prove that there is no solution over the positive
integers to the equation:
4a3 C 2b 3 D c 3 :

Problem 2.6.
You are given a series of envelopes, respectively containing 1; 2; 4; : : : ; 2m dollars.
Define
Property m: For any nonnegative integer less than 2mC1 , there is a
selection of envelopes whose contents add up to exactly that number
of dollars.
Use the Well Ordering Principle (WOP) to prove that Property m holds for all
nonnegative integers m.
Hint: Consider two cases: first, when the target number of dollars is less than
2 and second, when the target is at least 2m .
m

Homework Problems
Problem 2.7.
Use the Well Ordering Principle to prove that any integer greater than or equal to 8
can be represented as the sum of nonnegative integer multiples of 3 and 5.

Problem 2.8.
Use the Well Ordering Principle to prove that any integer greater than or equal to
50 can be represented as the sum of nonnegative integer multiples of 7, 11, and 13.
2 Proofs by other methods such as induction or by appeal to known formulas for similar sums will
not receive full credit.
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2.4. Well Ordered Sets 39

Problem 2.9.
Euler’s Conjecture in 1769 was that there are no positive integer solutions to the
equation
a4 C b 4 C c 4 D d 4 :
Integer values for a; b; c; d that do satisfy this equation were first discovered in
1986. So Euler guessed wrong, but it took more than two centuries to demonstrate
his mistake.
Now let’s consider Lehman’s equation, similar to Euler’s but with some coeffi-
cients:
8a4 C 4b 4 C 2c 4 D d 4 (2.5)
Prove that Lehman’s equation (2.5) really does not have any positive integer
solutions.
Hint: Consider the minimum value of a among all possible solutions to (2.5).

Problem 2.10.
Use the Well Ordering Principle to prove that

n 3n=3 (2.6)

for every nonnegative integer n.
Hint: Verify (2.6) for n 4 by explicit calculation.

Problem 2.11.
A winning configuration in the game of Mini-Tetris is a complete tiling of a 2 n
board using only the three shapes shown below:

For example, here are several possible winning configurations on a 2 5 board:
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40 Chapter 2 The Well Ordering Principle

(a) Let Tn denote the number of different winning configurations on a 2n board.
Determine the values of T1 , T2 and T3 .

(b) Express Tn in terms of Tn 1 and Tn 2 for n > 2.

(c) Use the Well Ordering Principle to prove that the number of winning configu-
rations on a 2 n Mini-Tetris board is:3
2nC1 C . 1/n
Tn D (*)
3

Problem 2.12.
Mini-Tetris is a game whose objective is to provide a complete “tiling” of a 2 n
board using tiles of specified shapes. In this problem we consider the following set
of five tiles:

For example, there are two possible tilings of a 2 1 board:

Also, here are three tilings for a 2 2 board:
3A good question is how someone came up with equation (*) in the first place. A simple Well
Ordering proof gives no hint about this, but it should be absolutely convincing anyway.
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2.4. Well Ordered Sets 41

Note that tiles may not be rotated, which is why the second and third of the above
tilings count as different, even though one is a 180o rotation of the other. (A 90o
degree rotation of these shapes would not count as a tiling at all.)
(a) There are four more 2 2 tilings in addition to the three above. What are they?
Let Tn denote the number of different tilings of a 2 n board. We know that
T1 D 2 and T2 D 7. Also, T0 D 1 because there is exactly one way to tile a 2 0
board—don’t use any tiles.
(b) Tn can be specified in terms of Tn 1 and Tn 2 as follows:

Tn D 2Tn 1 C 3Tn 2 (2.7)

for n 2.
Briefly explain how to justify this equation.

3nC1 C . 1/n
:
4

Exam Problems
Problem 2.13.
Except for an easily repaired omission, the following proof using the Well Ordering
Principle shows that every amount of postage that can be paid exactly using only
10 cent and 15 cent stamps, is divisible by 5.
Namely, let the notation “j j k” indicate that integer j is a divisor of integer k,
and let S.n/ mean that exactly n cents postage can be assembled using only 10 and
15 cent stamps. Then the proof shows that

S.n/ IMPLIES 5 j n; for all nonnegative integers n: (2.8)

Fill in the missing portions (indicated by “. . . ”) of the following proof of (2.8), and
at the end, identify the minor mistake in the proof and how to fix it.
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42 Chapter 2 The Well Ordering Principle

Let C be the set of counterexamples to (2.8), namely

C WWD fn j S.n/ and NOT.5 j n/g

Assume for the purpose of obtaining a contradiction that C is nonempty.
Then by the WOP, there is a smallest number m 2 C . Then S.m 10/
or S.m 15/ must hold, because the m cents postage is made from 10
and 15 cent stamps, so we remove one.
So suppose S.m 10/ holds. Then 5 j .m 10/, because. . .
But if 5 j .m 10/, then 5 j m, because. . .
contradicting the fact that m is a counterexample.
Next suppose S.m 15/ holds. Then the proof for m 10 carries
over directly for m 15 to yield a contradiction in this case as well.
Since we get a contradiction in both cases, we conclude that C must
be empty. That is, there are no counterexamples to (2.8), which proves
that (2.8) holds.

The proof makes an implicit assumption about the value of m. State the assump-
tion and justify it in one sentence.

Problem 2.14. (a) Prove using the Well Ordering Principle that, using 6¢, 14¢, and
21¢ stamps, it is possible to make any amount of postage over 50¢. To save time,
you may specify assume without proof that 50¢, 51¢, . . . 100¢ are all makeable, but
you should clearly indicate which of these assumptions your proof depends on.

(b) Show that 49¢ is not makeable.

Problem 2.15.
We’ll use the Well Ordering Principle to prove that for every positive integer n, the
sum of the first n odd numbers is n2 , that is,
n
X1
.2i C 1/ D n2 ; (2.9)
i D0

for all n > 0.
Assume to the contrary that equation (2.9) failed for some positive integer n. Let
m be the least such number.
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2.4. Well Ordered Sets 43

(a) Why must there be such an m?

(b) Explain why m 2.

(d) What term should be added to the left-hand side of (2.10) so the result equals
m
X
.2.i 1/ C 1/‹
i D1

(e) Conclude that equation (2.9) holds for all positive integers n.

Problem 2.16.
Use the Well Ordering Principle (WOP) to prove that

2 C 4 C C 2n D n.n C 1/ (2.11)

for all n > 0.

Problem 2.17.
Prove by the Well Ordering Principle that for all nonnegative integers, n:
2
3 3 3 3 n.n C 1/
0 C 1 C 2 C C n D : (2.12)
2

Problem 2.18.
Use the Well Ordering Principle to prove that

n.n C 1/.n C 2/
1 2 C 2 3 C 3 4 C C n.n C 1/ D (*)
3
for all integers n 1.
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44 Chapter 2 The Well Ordering Principle

Problem 2.19.
Say a number of cents is makeable if it is the value of some set of 6 cent and 15
cent stamps. Use the Well Ordering Principle to show that every integer that is a
multiple of 3 and greater than or equal to twelve is makeable.

Problems for Section 2.4
Homework Problems
Problem 2.20.
Complete the proof of Lemma 2.4.5 by showing that the number nS C fS is the
minimum element in S .

Practice Problems
Problem 2.21.
Indicate which of the following sets of numbers have a minimum element and
which are well ordered. For those that are not well ordered, give an example of
a subset with no minimum element.
p
(a) The integers 2.
p
(b) The rational numbers 2.

(d) The set G of rationals of the form m=n where m; n > 0 and n g, where g
is a googol 10100 .

(e) The set F of fractions of the form n=.n C 1/:
0 1 2 3
; ; ; ;::::
1 2 3 4
(f) Let W WWD N [ F be the set consisting of the nonnegative integers along with
all the fractions of the form n=.n C 1/. Describe a length 5 decreasing sequence of
elements of W starting with 1,. . . length 50 decreasing sequence,. . . length 500.

Problem 2.22.
Use the Well Ordering Principle to prove that every finite, nonempty set of real
numbers has a minimum element.
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2.4. Well Ordered Sets 45

Class Problems
Problem 2.23.
Prove that a set R of real numbers is well ordered iff there is no infinite decreasing
sequence of numbers R. In other words, there is no set of numbers ri 2 R such
that
r0 > r1 > r2 > : : : : (2.13)
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3 Logical Formulas
It is amazing that people manage to cope with all the ambiguities in the English
language. Here are some sentences that illustrate the issue:

“You may have cake, or you may have ice cream.”

“If pigs can fly, then your account won’t get hacked.”

“If you can solve any problem we come up with, then you get an A for the
course.”

“Every American has a dream.”

What precisely do these sentences mean? Can you have both cake and ice cream or
must you choose just one dessert? Pigs can’t fly, so does the second sentence say
anything about the security of your account? If you can solve some problems we
come up with, can you get an A for the course? And if you can’t solve a single one
of the problems, does it mean you can’t get an A? Finally, does the last sentence
imply that all Americans have the same dream—say of owning a house—or might
different Americans have different dreams—say, Eric dreams of designing a killer
software application, Tom of being a tennis champion, Albert of being able to sing?
Some uncertainty is tolerable in normal conversation. But when we need to
formulate ideas precisely—as in mathematics and programming—the ambiguities
inherent in everyday language can be a real problem. We can’t hope to make an
exact argument if we’re not sure exactly what the statements mean. So before we
start into mathematics, we need to investigate the problem of how to talk about
mathematics.
To get around the ambiguity of English, mathematicians have devised a spe-
cial language for talking about logical relationships. This language mostly uses
ordinary English words and phrases such as “or,” “implies,” and “for all.” But
mathematicians give these words precise and unambiguous definitions which don’t
always match common usage.
Surprisingly, in the midst of learning the language of logic, we’ll come across
the most important open problem in computer science—a problem whose solution
could change the world.
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48 Chapter 3 Logical Formulas

3.1 Propositions from Propositions
In English, we can modify, combine, and relate propositions with words such as
“not,” “and,” “or,” “implies,” and “if-then.” For example, we can combine three
propositions into one like this:
If all humans are mortal and all Greeks are human, then all Greeks are mortal.
For the next while, we won’t be much concerned with the internals of propositions—
whether they involve mathematics or Greek mortality—but rather with how propo-
sitions are combined and related. So, we’ll frequently use variables such as P and
Q in place of specific propositions such as “All humans are mortal” and “2 C 3 D
5.” The understanding is that these propositional variables, like propositions, can
take on only the values T (true) and F (false). Propositional variables are also
called Boolean variables after their inventor, the nineteenth century mathematician
George—you guessed it—Boole.

3.1.1 NOT , AND , and OR
Mathematicians use the words NOT, AND and OR for operations that change or
combine propositions. The precise mathematical meaning of these special words
can be specified by truth tables. For example, if P is a proposition, then so is
“NOT.P /,” and the truth value of the proposition “NOT.P /” is determined by the
truth value of P according to the following truth table:

P NOT.P /
T F
F T
The first row of the table indicates that when proposition P is true, the proposi-
tion “NOT.P /” is false. The second line indicates that when P is false, “NOT.P /”
is true. This is probably what you would expect.
In general, a truth table indicates the true/false value of a proposition for each
possible set of truth values for the variables. For example, the truth table for the
proposition “P AND Q” has four lines, since there are four settings of truth values
for the two variables:

P Q P AND Q
T T T
T F F
F T F
F F F
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3.1. Propositions from Propositions 49

According to this table, the proposition “P AND Q” is true only when P and Q
are both true. This is probably the way you ordinarily think about the word “and.”
There is a subtlety in the truth table for “P OR Q”:

P Q P OR Q
T T T
T F T
F T T
F F F
The first row of this table says that “P OR Q” is true even if both P and Q are
true. This isn’t always the intended meaning of “or” in everyday speech, but this is
the standard definition in mathematical writing. So if a mathematician says, “You
may have cake, or you may have ice cream,” he means that you could have both.
If you want to exclude the possibility of having both cake and ice cream, you
should combine them with the exclusive-or operation, XOR:

P Q P XOR Q
T T F
T F T
F T T
F F F

3.1.2 If and Only If
Mathematicians commonly join propositions in an additional way that doesn’t arise
in ordinary speech. The proposition “P if and only if Q” asserts that P and Q have
the same truth value. Either both are true or both are false.
P Q P IFF Q
T T T
T F F
F T F
F F T

For example, the following if-and-only-if statement is true for every real number
x:
x 2 4 0 IFF jxj 2:
For some values of x, both inequalities are true. For other values of x, neither
inequality is true. In every case, however, the IFF proposition as a whole is true.
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50 Chapter 3 Logical Formulas

3.1.3 IMPLIES

The combining operation whose technical meaning is least intuitive is “implies.”
Here is its truth table, with the lines labeled so we can refer to them later.

P Q P IMPLIES Q
T T T (tt)
T F F (tf)
F T T (ft)
F F T (ff)
The truth table for implications can be summarized in words as follows:

An implication is true exactly when the if-part is false or the then-part is true.

This sentence is worth remembering; a large fraction of all mathematical statements
are of the if-then form!
Let’s experiment with this definition. For example, is the following proposition
true or false?
If Goldbach’s Conjecture is true, then x 2 0 for every real number x.
We already mentioned that no one knows whether Goldbach’s Conjecture, Proposi-
tion 1.1.6, is true or false. But that doesn’t prevent us from answering the question!
This proposition has the form P IMPLIES Q where the hypothesis P is “Gold-
bach’s Conjecture is true” and the conclusion Q is “x 2 0 for every real number
x.” Since the conclusion is definitely true, we’re on either line (tt) or line (ft) of the
truth table. Either way, the proposition as a whole is true!
Now let’s figure out the truth of one of our original examples:
If pigs fly, then your account won’t get hacked.
Forget about pigs, we just need to figure out whether this proposition is true or
false. Pigs do not fly, so we’re on either line (ft) or line (ff) of the truth table. In
both cases, the proposition is true!

False Hypotheses
This mathematical convention—that an implication as a whole is considered true
when its hypothesis is false—contrasts with common cases where implications are
supposed to have some causal connection between their hypotheses and conclu-
sions.
For example, we could agree—or at least hope—that the following statement is
true:
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3.1. Propositions from Propositions 51

If you followed the security protocal, then your account won’t get
hacked.
We regard this implication as unproblematical because of the clear causal connec-
tion between security protocols and account hackability.
On the other hand, the statement:
If pigs could fly, then your account won’t get hacked,
would commonly be rejected as false—or at least silly—because porcine aeronau-
tics have nothing to do with your account security. But mathematically, this impli-
cation counts as true.
It’s important to accept the fact that mathematical implications ignore causal
connections. This makes them a lot simpler than causal implications, but useful
nevertheless. To illustrate this, suppose we have a system specification which con-
sists of a series of, say, a dozen rules,1
If the system sensors are in condition 1,
then the system takes action 1.
If the system sensors are in condition 2,
then the system takes action 2.
::
:
If the system sensors are in condition 12,
then the system takes action 12.
Letting Ci be the proposition that the system sensors are in condition i , and Ai
be the proposition that system takes action i , the specification can be restated more
concisely by the logical formulas

C1 IMPLIES A1 ;
C2 IMPLIES A2 ;
::
:
C12 IMPLIES A12 :

Now the proposition that the system obeys the specification can be nicely expressed
as a single logical formula by combining the formulas together with ANDs::

ŒC1 IMPLIES A1 AND ŒC2 IMPLIES A2 AND AND ŒC12 IMPLIES A12 : (3.1)

For example, suppose only conditions C2 and C5 are true, and the system indeed
takes the specified actions A2 and A5 . So in this case, the system is behaving
1 Problem 3.16 concerns just such a system.
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52 Chapter 3 Logical Formulas

according to specification, and we accordingly want formula (3.1) to come out true.
The implications C2 IMPLIES A2 and C5 IMPLIES A5 are both true because both
their hypotheses and their conclusions are true. But in order for (3.1) to be true, we
need all the other implications, all of whose hypotheses are false, to be true. This
is exactly what the rule for mathematical implications accomplishes.

3.2 Propositional Logic in Computer Programs
Propositions and logical connectives arise all the time in computer programs. For
example, consider the following snippet, which could be either C, C++, or Java:

if ( x > 0 || (x <= 0 && y > 100) )
::
:
(further instructions)

Java uses the symbol || for “OR,” and the symbol && for “AND.” The further
instructions are carried out only if the proposition following the word if is true.
On closer inspection, this big expression is built from two simpler propositions.
Let A be the proposition that x > 0, and let B be the proposition that y > 100.
Then we can rewrite the condition as

A OR .NOT.A/ AND B/: (3.2)

3.2.1 Truth Table Calculation
A truth table calculation reveals that the more complicated expression 3.2 always
has the same truth value as
A OR B: (3.3)
We begin with a table with just the truth values of A and B:

A B A OR .NOT.A/ AND B/ A OR B
T T
T F
F T
F F
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3.2. Propositional Logic in Computer Programs 53

These values are enough to fill in two more columns:
A B A OR .NOT.A/ AND B/ A OR B
T T F T
T F F T
F T T T
F F T F
Now we have the values needed to fill in the AND column:
A B A OR .NOT.A/ AND B/ A OR B
T T F F T
T F F F T
F T T T T
F F T F F
and this provides the values needed to fill in the remaining column for the first OR:
A B A OR .NOT.A/ AND B/ A OR B
T T T F F T
T F T F F T
F T T T T T
F F F T F F
Expressions whose truth values always match are called equivalent. Since the two
emphasized columns of truth values of the two expressions are the same, they are
equivalent. So we can simplify the code snippet without changing the program’s
behavior by replacing the complicated expression with an equivalent simpler one:
if ( x > 0 || y > 100 )
::
:
(further instructions)
The equivalence of (3.2) and (3.3) can also be confirmed reasoning by cases:
A is T. An expression of the form .T OR anything/ is equivalent to T. Since A is T
both (3.2) and (3.3) in this case are of this form, so they have the same truth
value, namely, T.
A is F. An expression of the form .F OR anything/ will have same truth value as
anything. Since A is F, (3.3) has the same truth value as B.
An expression of the form .T AND anything/ is equivalent to anything, as is
any expression of the form F OR anything. So in this case A OR .NOT.A/ AND
B/ is equivalent to .NOT.A/ AND B/, which in turn is equivalent to B.
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54 Chapter 3 Logical Formulas

Therefore both (3.2) and (3.3) will have the same truth value in this case,
namely, the value of B.
Simplifying logical expressions has real practical importance in computer sci-
ence. Expression simplification in programs like the one above can make a program
easier to read and understand. Simplified programs may also run faster, since they
require fewer operations. In hardware, simplifying expressions can decrease the
number of logic gates on a chip because digital circuits can be described by logical
formulas (see Problems 3.6 and 3.7). Minimizing the logical formulas corresponds
to reducing the number of gates in the circuit. The payoff of gate minimization is
potentially enormous: a chip with fewer gates is smaller, consumes less power, has
a lower defect rate, and is cheaper to manufacture.

3.2.2 Cryptic Notation
Java uses symbols like “&&” and “jj” in place of AND and OR. Circuit designers
use “” and “C,” and actually refer to AND as a product and OR as a sum. Mathe-
maticians use still other symbols, given in the table below.
English Symbolic Notation
NOT .P / :P (alternatively, P )
P AND Q P ^Q
P OR Q P _Q
P IMPLIES Q P !Q
if P then Q P !Q
P IFF Q P !Q
P XOR Q P ˚Q
For example, using this notation, “If P AND NOT.Q/, then R” would be written:
.P ^ Q/ ! R:
The mathematical notation is concise but cryptic. Words such as “AND” and
“OR” are easier to remember and won’t get confused with operations on numbers.
We will often use P as an abbreviation for NOT.P /, but aside from that, we mostly
stick to the words—except when formulas would otherwise run off the page.

3.3 Equivalence and Validity
3.3.1 Implications and Contrapositives
Do these two sentences say the same thing?
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3.3. Equivalence and Validity 55

If I am hungry, then I am grumpy.
If I am not grumpy, then I am not hungry.

We can settle the issue by recasting both sentences in terms of propositional logic.
Let P be the proposition “I am hungry” and Q be “I am grumpy.” The first sentence
says “P IMPLIES Q” and the second says “NOT.Q/ IMPLIES NOT.P /.” Once
more, we can compare these two statements in a truth table:

P Q .P IMPLIES Q/ .NOT.Q/ IMPLIES NOT.P //
T T T F T F
T F F T F F
F T T F T T
F F T T T T

Sure enough, the highlighted columns showing the truth values of these two state-
ments are the same. A statement of the form “NOT.Q/ IMPLIES NOT.P /” is called
the contrapositive of the implication “P IMPLIES Q.” The truth table shows that
an implication and its contrapositive are equivalent—they are just different ways of
saying the same thing.
In contrast, the converse of “P IMPLIES Q” is the statement “Q IMPLIES P .”
The converse to our example is:

If I am grumpy, then I am hungry.

This sounds like a rather different contention, and a truth table confirms this suspi-
cion:
P Q P IMPLIES Q Q IMPLIES P
T T T T
T F F T
F T T F
F F T T
Now the highlighted columns differ in the second and third row, confirming that an
implication is generally not equivalent to its converse.
One final relationship: an implication and its converse together are equivalent to
an iff statement, specifically, to these two statements together. For example,

If I am grumpy then I am hungry, and if I am hungry then I am grumpy.

are equivalent to the single statement:

I am grumpy iff I am hungry.
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56 Chapter 3 Logical Formulas

Once again, we can verify this with a truth table.

P Q .P IMPLIES Q/ AND .Q IMPLIES P / P IFF Q
T T T T T T
T F F F T F
F T T F F F
F F T T T T
The fourth column giving the truth values of
.P IMPLIES Q/ AND .Q IMPLIES P /
is the same as the sixth column giving the truth values of P IFF Q, which confirms
that the AND of the implications is equivalent to the IFF statement.

3.3.2 Validity and Satisfiability
A valid formula is one which is always true, no matter what truth values its vari-
ables may have. The simplest example is
P OR NOT.P /:
You can think about valid formulas as capturing fundamental logical truths. For
example, a property of implication that we take for granted is that if one statement
implies a second one, and the second one implies a third, then the first implies the
third. The following valid formula confirms the truth of this property of implication.
Œ.P IMPLIES Q/ AND .Q IMPLIES R/ IMPLIES .P IMPLIES R/:
Equivalence of formulas is really a special case of validity. Namely, statements
F and G are equivalent precisely when the statement .F IFF G/ is valid. For
example, the equivalence of the expressions (3.3) and (3.2) means that
.A OR B/ IFF .A OR .NOT.A/ AND B//
is valid. Of course, validity can also be viewed as an aspect of equivalence. Namely,
a formula is valid iff it is equivalent to T.
A satisfiable formula is one which can sometimes be true—that is, there is some
assignment of truth values to its variables that makes it true. One way satisfiabil-
ity comes up is when there are a collection of system specifications. The job of
the system designer is to come up with a system that follows all the specs. This
means that the AND of all the specs must be satisfiable or the designer’s job will be
impossible (see Problem 3.16).
There is also a close relationship between validity and satisfiability: a statement
P is satisfiable iff its negation NOT.P / is not valid.
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3.4. The Algebra of Propositions 57

3.4 The Algebra of Propositions
3.4.1 Propositions in Normal Form
Every propositional formula is equivalent to a “sum-of-products” or disjunctive
form. More precisely, a disjunctive form is simply an OR of AND-terms, where
each AND-term is an AND of variables or negations of variables, for example,

.A AND B/ OR .A AND C /: (3.4)

You can read a disjunctive form for any propositional formula directly from its
truth table. For example, the formula

A AND .B OR C / (3.5)

has truth table:
A B C A AND .B OR C /
T T T T
T T F T
T F T T
T F F F
F T T F
F T F F
F F T F
F F F F
The formula (3.5) is true in the first row when A, B and C are all true, that is, where
A AND B AND C is true. It is also true in the second row where A AND B AND C
is true, and in the third row when A AND B AND C is true, and that’s all. So (3.5)
is true exactly when

.A AND B AND C / OR .A AND B AND C / OR .A AND B AND C / (3.6)

is true.
The expression (3.6) is a disjunctive form where each AND-term is an AND of
every one of the variables or their complements in turn. An expression of this form
is called a disjunctive normal form (DNF). A DNF formula can often be simplified
into a smaller disjunctive form. For example, the DNF (3.6) further simplifies to
the equivalent disjunctive form (3.4) above.
Applying the same reasoning to the F entries of a truth table yields a conjunctive
form for any formula—an AND of OR-terms in which the OR-terms are OR’s only
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58 Chapter 3 Logical Formulas

of variables or their negations. For example, formula (3.5) is false in the fourth
row of its truth table (3.4.1) where A is T, B is F and C is F. But this is exactly
the one row where .A OR B OR C / is F! Likewise, the (3.5) is false in the fifth
row which is exactly where .A OR B OR C / is F. This means that (3.5) will be F
whenever the AND of these two OR-terms is false. Continuing in this way with the
OR -terms corresponding to the remaining three rows where (3.5) is false, we get a
conjunctive normal form (CNF) that is equivalent to (3.5), namely,
.A OR B OR C / AND .A OR B OR C / AND .A OR B OR C /AND
.A OR B OR C / AND .A OR B OR C /
The methods above can be applied to any truth table, which implies
Theorem 3.4.1. Every propositional formula is equivalent to both a disjunctive
normal form and a conjunctive normal form.

3.4.2 Proving Equivalences
A check of equivalence or validity by truth table runs out of steam pretty quickly:
a proposition with n variables has a truth table with 2n lines, so the effort required
to check a proposition grows exponentially with the number of variables. For a
proposition with just 30 variables, that’s already over a billion lines to check!
An alternative approach that sometimes helps is to use algebra to prove equiv-
alence. A lot of different operators may appear in a propositional formula, so a
useful first step is to get rid of all but three: AND, OR and NOT. This is easy
because each of the operators is equivalent to a simple formula using only these
three. For example, A IMPLIES B is equivalent to NOT.A/ OR B. Formulas using
onlyAND, OR and NOT for the remaining operators are left to Problem 3.17.
We list below a bunch of equivalence axioms with the symbol “ ! ” between
equivalent formulas. These axioms are important because they are all that’s needed
to prove every possible equivalence. We’ll start with some equivalences for AND’s
that look like the familiar ones for multiplication of numbers:
A AND B ! B AND A (commutativity of AND)
(3.7)
.A AND B/ AND C ! A AND .B AND C / (associativity of AND)
(3.8)
T AND A ! A (identity for AND)
F AND A ! F (zero for AND)
A AND .B OR C / ! .A AND B/ OR .A AND C / (distributivity of AND over OR)
(3.9)
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3.4. The Algebra of Propositions 59

Associativity (3.8) justifies writing A AND B AND C without specifying whether
it is parenthesized as A AND .B AND C / or .A AND B/ AND C . Both ways of
inserting parentheses yield equivalent formulas.
Unlike arithmetic rules for numbers, there is also a distributivity law for “sums”
over “products:”

A OR .B AND C / ! .A OR B/ AND .A OR C / (distributivity of OR over AND)
(3.10)

Three more axioms that don’t directly correspond to number properties are

A AND A ! A (idempotence for AND)
A AND A ! F (contradiction for AND) (3.11)
NOT .A/ ! A (double negation) (3.12)

There are a corresponding set of equivalences for OR which we won’t bother to
list, except for the OR rule corresponding to contradiction for AND (3.11):

A OR A ! T (validity for OR) (3.13)

Finally, there are De Morgan’s Laws which explain how to distribute NOT’s over
AND ’sand OR’s:

NOT.A AND B/ ! A OR B (De Morgan for AND) (3.14)
NOT .A OR B/ ! A AND B (De Morgan for OR) (3.15)

All of these axioms can be verified easily with truth tables.
These axioms are all that’s needed to convert any formula to a disjunctive normal
form. We can illustrate how they work by applying them to turn the negation of
formula (3.5),
NOT..A AND B/ OR .A AND C //: (3.16)
into disjunctive normal form.
We start by applying De Morgan’s Law for OR (3.15) to (3.16) in order to move
the NOT deeper into the formula. This gives

NOT .A AND B/ AND NOT.A AND C /:

Now applying De Morgan’s Law for AND (3.14) to the two innermost AND-terms,
gives
.A OR B/ AND .A OR C /: (3.17)
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60 Chapter 3 Logical Formulas

At this point NOT only applies to variables, and we won’t need De Morgan’s Laws
any further.
Now we will repeatedly apply (3.9), distributivity of AND over OR, to turn (3.17)
into a disjunctive form. To start, we’ll distribute .A OR B/ over AND to get

..A OR B/ AND A/ OR ..A OR B/ AND C /:

Using distributivity over both AND’s we get

..A AND A/ OR .B AND A// OR ..A AND C / OR .B AND C //:

By the way, we’ve implicitly used commutativity (3.7) here to justify distributing
over an AND from the right. Now applying idempotence to remove the duplicate
occurrence of A we get

.A OR .B AND A// OR ..A AND C / OR .B AND C //:

Associativity now allows dropping the parentheses around the terms being OR’d to
yield the following disjunctive form for (3.16):

A OR .B AND A/ OR .A AND C / OR .B AND C /: (3.18)

The last step is to turn each of these AND-terms into a disjunctive normal form
with all three variables A, B and C . We’ll illustrate how to do this for the second
AND -term .B AND A/. This term needs to mention C to be in normal form. To
introduce C , we use validity for OR and identity for AND to conclude that

.B AND A/ ! .B AND A/ AND .C OR C /:

Now distributing .B AND A/ over the OR yields the disjunctive normal form

.B AND A AND C / OR .B AND A AND C /:

Doing the same thing to the other AND-terms in (3.18) finally gives a disjunctive
normal form for (3.5):

.A AND B AND C / OR .A AND B AND C / OR
.A AND B AND C / OR .A AND B AND C / OR
.B AND A AND C / OR .B AND A AND C / OR
.A AND C AND B/ OR .A AND C AND B/ OR
.B AND C AND A/ OR .B AND C AND A/:
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3.4. The Algebra of Propositions 61

Using commutativity to sort the term and OR-idempotence to remove duplicates,
finally yields a unique sorted DNF:

.A AND B AND C / OR
.A AND B AND C / OR
.A AND B AND C / OR
.A AND B AND C / OR
.A AND B AND C /:

This example illustrates a strategy for applying these equivalences to convert any
formula into disjunctive normal form, and conversion to conjunctive normal form
works similarly, which explains:

Theorem 3.4.2. Any propositional formula can be transformed into disjunctive
normal form or a conjunctive normal form using the equivalences listed above.

What has this got to do with equivalence? That’s easy: to prove that two for-
mulas are equivalent, convert them both to disjunctive normal form over the set of
variables that appear in the terms. Then use commutativity to sort the variables and
AND -terms so they all appear in some standard order. We claim the formulas are
equivalent iff they have the same sorted disjunctive normal form. This is obvious
if they do have the same disjunctive normal form. But conversely, the way we read
off a disjunctive normal form from a truth table shows that two different sorted
DNF’s over the same set of variables correspond to different truth tables and hence
to inequivalent formulas. This proves

Theorem 3.4.3 (Completeness of the propositional equivalence axioms). Two propo-
sitional formula are equivalent iff they can be proved equivalent using the equiva-
lence axioms listed above.

The benefit of the axioms is that they leave room for ingeniously applying them
to prove equivalences with less effort than the truth table method. Theorem 3.4.3
then adds the reassurance that the axioms are guaranteed to prove every equiva-
lence, which is a great punchline for this section. But we don’t want to mislead
you: it’s important to realize that using the strategy we gave for applying the ax-
ioms involves essentially the same effort it would take to construct truth tables, and
there is no guarantee that applying the axioms will generally be any easier than
using truth tables.
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62 Chapter 3 Logical Formulas

3.5 The SAT Problem
Determining whether or not a more complicated proposition is satisfiable is not so
easy. How about this one?

.P OR Q OR R/ AND .P OR Q/ AND .P OR R/ AND .R OR Q/

The general problem of deciding whether a proposition is satisfiable is called
SAT. One approach to SAT is to construct a truth table and check whether or not
a T ever appears, but as with testing validity, this approach quickly bogs down
for formulas with many variables because truth tables grow exponentially with the
number of variables.
Is there a more efficient solution to SAT? In particular, is there some brilliant
procedure that determines SAT in a number of steps that grows polynomially—like
n2 or n14 —instead of exponentially—2n —whether any given proposition of size n
is satisfiable or not? No one knows. And an awful lot hangs on the answer.
The general definition of an “efficient” procedure is one that runs in polynomial
time, that is, that runs in a number of basic steps bounded by a polynomial in s,
where s is the size of an input. It turns out that an efficient solution to SAT would
immediately imply efficient solutions to many other important problems involving
scheduling, routing, resource allocation, and circuit verification across multiple dis-
ciplines including programming, algebra, finance, and political theory. This would
be wonderful, but there would also be worldwide chaos. Decrypting coded mes-
sages would also become an easy task, so online financial transactions would be
insecure and secret communications could be read by everyone. Why this would
happen is explained in Section 9.12.
Of course, the situation is the same for validity checking, since you can check for
validity by checking for satisfiability of a negated formula. This also explains why
the simplification of formulas mentioned in Section 3.2 would be hard—validity
testing is a special case of determining if a formula simplifies to T.
Recently there has been exciting progress on SAT-solvers for practical applica-
tions like digital circuit verification. These programs find satisfying assignments
with amazing efficiency even for formulas with millions of variables. Unfortu-
nately, it’s hard to predict which kind of formulas are amenable to SAT-solver
methods, and for formulas that are unsatisfiable, SAT-solvers are generally much
less effective.
So no one has a good idea how to solve SAT in polynomial time, or how to
prove that it can’t be done—researchers are completely stuck. The problem of
determining whether or not SAT has a polynomial time solution is known as the
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3.6. Predicate Formulas 63

“P vs. NP” problem.2 It is the outstanding unanswered question in theoretical
computer science. It is also one of the seven Millenium Problems: the Clay Institute
will award you $1,000,000 if you solve the P vs. NP problem.

3.6 Predicate Formulas
3.6.1 Quantifiers
The “for all” notation 8 has already made an early appearance in Section 1.1. For
example, the predicate
“x 2 0”
is always true when x is a real number. That is,

8x 2 R: x 2 0

is a true statement. On the other hand, the predicate

“5x 2 7 D 0”
p
is only sometimes true; specifically, when x D ˙ 7=5. There is a “there exists”
notation 9 to indicate that a predicate is true for at least one, but not necessarily all
objects. So
9x 2 R: 5x 2 7 D 0
is true, while
8x 2 R: 5x 2 7D0
is not true.
There are several ways to express the notions of “always true” and “sometimes
true” in English. The table below gives some general formats on the left and specific
examples using those formats on the right. You can expect to see such phrases
hundreds of times in mathematical writing!

2P stands for problems whose instances can be solved in time that grows polynomially with the
size of the instance. NP stands for nondeterministtic polynomial time, but we’ll leave an explanation
of what that is to texts on the theory of computational complexity.
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64 Chapter 3 Logical Formulas

Always True
For all x 2 D, P .x/ is true. For all x 2 R, x 2 0.
P .x/ is true for every x in the set D. x 2 0 for every x 2 R.

Sometimes True
There is an x 2 D such that P .x/ is true. There is an x 2 R such that 5x 2 7 D 0.
P .x/ is true for some x in the set D. 5x 2 7 D 0 for some x 2 R.
P .x/ is true for at least one x 2 D. 5x 2 7 D 0 for at least one x 2 R.
All these sentences “quantify” how often the predicate is true. Specifically, an
assertion that a predicate is always true is called a universal quantification, and an
assertion that a predicate is sometimes true is an existential quantification. Some-
times the English sentences are unclear with respect to quantification:

If you can solve any problem we come up with,
then you get an A for the course. (3.19)

The phrase “you can solve any problem we can come up with” could reasonably be
interpreted as either a universal or existential quantification:

you can solve every problem we come up with, (3.20)

or maybe
you can solve at least one problem we come up with. (3.21)
To be precise, let Probs be the set of problems we come up with, Solves.x/ be
the predicate “You can solve problem x,” and G be the proposition, “You get an A
for the course.” Then the two different interpretations of (3.19) can be written as
follows:

.8x 2 Probs: Solves.x// IMPLIES G; for (3.20);
.9x 2 Probs: Solves.x// IMPLIES G: for (3.21):

3.6.2 Mixing Quantifiers
Many mathematical statements involve several quantifiers. For example, we al-
ready described

Goldbach’s Conjecture 1.1.6: Every even integer greater than 2 is the
sum of two primes.

Let’s write this out in more detail to be precise about the quantification:
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3.6. Predicate Formulas 65

For every even integer n greater than 2, there exist primes p and q such
that n D p C q.

Let Evens be the set of even integers greater than 2, and let Primes be the set of
primes. Then we can write Goldbach’s Conjecture in logic notation as follows:

8n
„ 2ƒ‚ … 9p
Evens 2 Primes 9q 2 Primes: n D p C q:
„ ƒ‚ …
for every even there exist primes
integer n > 2 p and q such that

3.6.3 Order of Quantifiers
Swapping the order of different kinds of quantifiers (existential or universal) usually
changes the meaning of a proposition. For example, let’s return to one of our initial,
confusing statements:

“Every American has a dream.”

This sentence is ambiguous because the order of quantifiers is unclear. Let A be
the set of Americans, let D be the set of dreams, and define the predicate H.a; d /
to be “American a has dream d .” Now the sentence could mean there is a single
dream that every American shares—such as the dream of owning their own home:

9 d 2 D 8a 2 A: H.a; d /

Or it could mean that every American has a personal dream:

8a 2 A 9 d 2 D: H.a; d /

For example, some Americans may dream of a peaceful retirement, while others
dream of continuing practicing their profession as long as they live, and still others
may dream of being so rich they needn’t think about work at all.
Swapping quantifiers in Goldbach’s Conjecture creates a patently false statement
that every even number 2 is the sum of the same two primes:

9 p 2 Primes 9 q 2 Primes: 8n 2 Evens n D p C q:
„ ƒ‚ … „ ƒ‚ …
there exist primes for every even
p and q such that integer n > 2

3.6.4 Variables Over One Domain
When all the variables in a formula are understood to take values from the same
nonempty set D it’s conventional to omit mention of D. For example, instead of
8x 2 D 9y 2 D: Q.x; y/ we’d write 8x9y: Q.x; y/. The unnamed nonempty set
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66 Chapter 3 Logical Formulas

that x and y range over is called the domain of discourse, or just plain domain, of
the formula.
It’s easy to arrange for all the variables to range over one domain. For exam-
ple, Goldbach’s Conjecture could be expressed with all variables ranging over the
domain N as

8n: n 2 Evens IMPLIES .9 p 9 q: p 2 Primes AND q 2 Primes AND n D p C q/:

3.6.5 Negating Quantifiers
There is a simple relationship between the two kinds of quantifiers. The following
two sentences mean the same thing:
Not everyone likes ice cream.
There is someone who does not like ice cream.
The equivalence of these sentences is an instance of a general equivalence that
holds between predicate formulas:

NOT .8x: P .x// is equivalent to 9x: NOT.P .x//: (3.22)

Similarly, these sentences mean the same thing:
There is no one who likes being mocked.
Everyone dislikes being mocked.
The corresponding predicate formula equivalence is

NOT .9x: P .x// is equivalent to 8x: NOT.P .x//: (3.23)

Note that the equivalence (3.23) follows directly by negating both sides the equiv-
alence (3.22).
The general principle is that moving a NOT to the other side of an “9” changes
it into “8,” and vice versa.
These equivalences are called De Morgan’s Laws for Quantifiers because they
can be understood as applying De Morgan’s Laws for propositional formulas to
an infinite sequence of AND’s and OR’s. For example, we can explain (3.22) by
supposing the domain of discourse is fd0 ; d1 ; : : : ; dn ; : : :g. Then 9x: NOT.P .x//
means the same thing as the infinite OR:

NOT.P .d0 // OR NOT .P .d1 // OR OR NOT.P .dn // OR : : : : (3.24)

Applying De Morgan’s rule to this infinite OR yields the equivalent formula

NOT ŒP .d0 / AND P .d1 / AND AND P .dn / AND : : : : (3.25)
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3.6. Predicate Formulas 67

But (3.25) means the same thing as
NOTŒ8x: P .x/:

This explains why 9x: NOT.P .x// means the same thing as NOTŒ8x: P .x/, which
confirms(3.22).

3.6.6 Validity for Predicate Formulas
The idea of validity extends to predicate formulas, but to be valid, a formula now
must evaluate to true no matter what the domain of discourse may be, no matter
what values its variables may take over the domain, and no matter what interpreta-
tions its predicate variables may be given. For example, the equivalence (3.22) that
gives the rule for negating a universal quantifier means that the following formula
is valid:
NOT.8x: P .x// IFF 9x: NOT .P .x//: (3.26)
Another useful example of a valid assertion is
9x8y: P .x; y/ IMPLIES 8y9x: P .x; y/: (3.27)
Here’s an explanation why this is valid:

Let D be the domain for the variables and P0 be some binary predi-
cate3 on D. We need to show that if
9x 2 D: 8y 2 D: P0 .x; y/ (3.28)
holds under this interpretation, then so does
8y 2 D 9x 2 D: P0 .x; y/: (3.29)
So suppose (3.28) is true. Then by definition of 9, this means that some
element d0 2 D has the property that
8y 2 D: P0 .d0 ; y/:
By definition of 8, this means that
P0 .d0 ; d /
is true for all d 2 D. So given any d 2 D, there is an element in D,
namely d0 , such that P0 .d0 ; d / is true. But that’s exactly what (3.29)
means, so we’ve proved that (3.29) holds under this interpretation, as
required.
3 That is, a predicate that depends on two variables.
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68 Chapter 3 Logical Formulas

We hope this is helpful as an explanation, but we don’t really want to call it a
“proof.” The problem is that with something as basic as (3.27), it’s hard to see
what more elementary axioms are ok to use in proving it. What the explanation
above did was translate the logical formula (3.27) into English and then appeal to
the meaning, in English, of “for all” and “there exists” as justification.
In contrast to (3.27), the formula

8y9x: P .x; y/ IMPLIES 9x8y: P .x; y/: (3.30)

is not valid. We can prove this just by describing an interpretation where the hy-
pothesis 8y9x: P .x; y/ is true but the conclusion 9x8y: P .x; y/ is not true. For
example, let the domain be the integers and P .x; y/ mean x > y. Then the hy-
pothesis would be true because, given a value n for y we could choose the value
of x to be n C 1, for example. But under this interpretation the conclusion asserts
that there is an integer that is bigger than all integers, which is certainly false. An
interpretation like this that falsifies an assertion is called a counter-model to that
assertion.

3.7 References
[19]

Problems for Section 3.1
Practice Problems
Problem 3.1.
Some people are uncomfortable with the idea that from a false hypothesis you can
prove everything, and instead of having P IMPLIES Q be true when P is false,
they want P IMPLIES Q to be false when P is false. This would lead to IMPLIES
having the same truth table as what propositional connective?

Problem 3.2.
Your class has a textbook and a final exam. Let P , Q and R be the following
propositions:
P WWD You get an A on the final exam.
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3.7. References 69

QWWD You do every exercise in the book.

RWWD You get an A in the class.
Translate following assertions into propositional formulas using P , Q, R and
the propositional connectives AND; NOT; IMPLIES.
(a) You get an A in the class, but you do not do every exercise in the book.

(b) You get an A on the final, you do every exercise in the book, and you get an A
in the class.

(d) You get an A on the final, but you don’t do every exercise in this book; never-
theless, you get an A in this class.

Class Problems
Problem 3.3.
When the mathematician says to his student, “If a function is not continuous, then it
is not differentiable,” then letting D stand for “differentiable” and C for continuous,
the only proper translation of the mathematician’s statement would be

NOT.C / IMPLIES NOT .D/;

or equivalently,
D IMPLIES C:
But when a mother says to her son, “If you don’t do your homework, then you
can’t watch TV,” then letting T stand for “can watch TV” and H for “do your
homework,” a reasonable translation of the mother’s statement would be

NOT .H / IFF NOT .T /;
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70 Chapter 3 Logical Formulas

or equivalently,
H IFF T:
Explain why it is reasonable to translate these two IF-THEN statements in dif-
ferent ways into propositional formulas.

Homework Problems
Problem 3.4.
Describe a simple procedure which, given a positive integer argument, n, produces
a width n array of truth-values whose rows would be all the possible truth-value
assignments for n propositional variables. For example, for n D 2, the array would
be:

T T
T F
F T
F F
Your description can be in English, or a simple program in some familiar lan-
guage such as Python or Java. If you do write a program, be sure to include some
sample output.

Problem 3.5.
Sloppy Sam is trying to prove a certain proposition P . He defines two related
propositions Q and R, and then proceeds to prove three implications:
P IMPLIES Q; Q IMPLIES R; R IMPLIES P:
He then reasons as follows:
If Q is true, then since I proved .Q IMPLIES R/, I can conclude that R
is true. Now, since I proved .R IMPLIES P /, I can conclude that P is
true. Similarly, if R is true, then P is true and so Q is true. Likewise,
if P is true, then so are Q and R. So any way you look at it, all three
of P; Q and R are true.
(a) Exhibit truth tables for
.P IMPLIES Q/ AND .Q IMPLIES R/ AND .R IMPLIES P / (*)
and for
P AND Q AND R: (**)
Use these tables to find a truth assignment for P; Q; R so that (*) is T and (**) is F.
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3.7. References 71

(b) You show these truth tables to Sloppy Sam and he says “OK, I’m wrong that
P; Q and R all have to be true, but I still don’t see the mistake in my reasoning.
Can you help me understand my mistake?” How would you explain to Sammy
where the flaw lies in his reasoning?

Problems for Section 3.2
Class Problems
Problem 3.6.
Propositional logic comes up in digital circuit design using the convention that T
corresponds to 1 and F to 0. A simple example is a 2-bit half-adder circuit. This
circuit has 3 binary inputs, a1 ; a0 and b, and 3 binary outputs, c; s1 ; s0 . The 2-bit
word a1 a0 gives the binary representation of an integer k between 0 and 3. The
3-bit word cs1 s0 gives the binary representation of k C b. The third output bit c is
called the final carry bit.
So if k and b were both 1, then the value of a1 a0 would be 01 and the value of
the output cs1 s0 would 010, namely, the 3-bit binary representation of 1 C 1.
In fact, the final carry bit equals 1 only when all three binary inputs are 1, that is,
when k D 3 and b D 1. In that case, the value of cs1 s0 is 100, namely, the binary
representation of 3 C 1.
This 2-bit half-adder could be described by the following formulas:
c0 D b
s0 D a0 XOR c0
c1 D a0 AND c0 the carry into column 1
s1 D a1 XOR c1
c2 D a1 AND c1 the carry into column 2
c D c2 :
(a) Generalize the above construction of a 2-bit half-adder to an n C 1 bit half-
adder with inputs an ; : : : ; a1 ; a0 and b and outputs c; sn ; : : : ; s1 ; s0 . That is, give
simple formulas for si and ci for 0 i n C 1, where ci is the carry into column
i C 1, and c D cnC1 .

(b) Write similar definitions for the digits and carries in the sum of two n C 1-bit
binary numbers an : : : a1 a0 and bn : : : b1 b0 .
Visualized as digital circuits, the above adders consist of a sequence of single-
digit half-adders or adders strung together in series. These circuits mimic ordinary
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72 Chapter 3 Logical Formulas

pencil-and-paper addition, where a carry into a column is calculated directly from
the carry into the previous column, and the carries have to ripple across all the
columns before the carry into the final column is determined. Circuits with this
design are called ripple-carry adders. Ripple-carry adders are easy to understand
and remember and require a nearly minimal number of operations. But the higher-
order output bits and the final carry take time proportional to n to reach their final
values.
(c) How many of each of the propositional operations does your adder from part (b)
use to calculate the sum?

Homework Problems
Problem 3.7.
As in Problem 3.6, a digital circuit is called an .n C 1/-bit half-adder when it has
with n C 2 inputs
an ; : : : ; a1 ; a0 ; b
and n C 2 outputs
c; sn ; : : : ; s1 ; s0 :
The input-output specification of the half-adder is that, if the 0-1 values of inputs
an ; : : : ; a1 ; a0 are taken to be the .n C 1/-bit binary representation of an integer k
then the 0-1 values of the outputs c; sn ; : : : ; s1 ; s0 are supposed to be the .nC2/-bit
binary representation of k C b.
For example suppose n D 2 and the values of a2 a1 a0 were 101. This is the
binary representation of k D 5. Now if the value of b was 1, then the output should
be the 4-bit representation of 5 C 1 D 6. Namely, the values of cs2 s1 s0 would be
0110.
There are many different circuit designs for half adders. The most straighforward
one is the “ripple carry” design described in Problem 3.6. We will now develop
a different design for a half-adder circuit called a parallel-design or “look-ahead
carry” half-adder. This design works by computing the values of higher-order digits
for both a carry of 0 and a carry of 1, in parallel. Then, when the carry from the
low-order digits finally arrives, the pre-computed answer can be quickly selected.
We’ll illustrate this idea by working out a parallel design for an .n C 1/-bit half-
adder.
Parallel-design half-adders are built out of parallel-design circuits called add1-
modules. The input-output behavior of an add1-module is just a special case of a
half-adder, where instead of an adding an input b to the input, the add1-module
always adds 1. That is, an .n C 1/-bit add1-module has .n C 1/ binary inputs
an ; : : : ; a1 ; a0 ;
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3.7. References 73

and n C 2 binary outputs
c pn ; : : : ; p1 ; p0 :
If an : : : a1 a0 are taken to be the .n C 1/-bit representation of an integer k then
cpn : : : p1 p0 is supposed to be the .n C 2/-bit binary representation of k C 1.
So a 1-bit add1-module just has input a0 and outputs c; p0 where

p0 WWD a0 XOR 1; .or more simply, p0 WWD NOT.a0 //;
c WWD a0 :

In the ripple-carry design, a double-size half-adder with 2.n C 1/ inputs takes
twice as long to produce its output values as an .n C 1/-input ripple-carry circuit.
With parallel-design add1-modules, a double-size add1-module produces its output
values nearly as fast as a single-size add1-modules. To see how this works, suppose
the inputs of the double-size module are

a2nC1 ; : : : ; a1 ; a0

and the outputs are
c; p2nC1 ; : : : ; p1 ; p0 :
We will build the double-size add1-module by having two single-size add1-modules
work in parallel. The setup is illustrated in Figure 3.1.
Namely, the first single-size add1-module handles the first n C 1 inputs. The in-
puts to this module are the low-order n C 1 input bits an ; : : : ; a1 ; a0 , and its outputs
will serve as the first n C 1 outputs pn ; : : : ; p1 ; p0 of the double-size module. Let
c.1/ be the remaining carry output from this module.
The inputs to the second single-size module are the higher-order n C 1 input bits
a2nC1 ; : : : ; anC2 ; anC1 . Call its first n C 1 outputs rn ; : : : ; r1 ; r0 and let c.2/ be its
carry.
(a) Write a formula for the carry c of the double-size add1-module solely in terms
of carries c.1/ and c.2/ of the single-size add1-modules.

(b) Complete the specification of the double-size add1-module by writing propo-
sitional formulas for the remaining outputs pnCi for 1 i n C 1. The formula
for pnCi should only involve the variables anCi , ri 1 and c.1/ .

(c) Explain how to build an .nC1/-bit parallel-design half-adder from an .nC1/-
bit add1-module by writing a propositional formula for the half-adder output si
using only the variables ai , pi and b.
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74 Chapter 3 Logical Formulas

(d) The speed or latency of a circuit is determined by the largest number of gates
on any path from an input to an output. In an n-bit ripple carry circuit(Problem 3.6),
there is a path from an input to the final carry output that goes through about 2n
gates. In contrast, parallel half-adders are exponentially faster than ripple-carry
half-adders. Confirm this by determining the largest number of propositional opera-
tions, that is, gates, on any path from an input to an output of an n-bit add1-module.
(You may assume n is a power of 2.)

a2nC1 anC2 anC1 an a1 a0

c.2/ .nC1/-bit add1 c.1/ .nC1/-bit add1

rn r1 r0
c
2.nC2/-bit add1 module

p2nC1 p nC2 pnC1 pn p1 p0

Figure 3.1 Structure of a Double-size add1 Module.

Exam Problems
Problem 3.8.

Claim. There are exactly two truth environments (assignments) for the variables
M; N; P; Q; R; S that satisfy the following formula:

.P OR Q/ AND .Q OR R/ AND .R OR S / AND .S OR P / AND M AND N
„ ƒ‚ … „ ƒ‚ … „ ƒ‚ … „ ƒ‚ …
clause (1) clause (2) clause (3) clause (4)
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3.7. References 75

(a) This claim could be proved by truth-table. How many rows would the truth
table have?

(b) Instead of a truth-table, prove this claim with an argument by cases according
to the truth value of P .

Problem 3.9.
An n-bit AND-circuit has 0-1 valued inputs a0 ; a1 ; : : : ; an 1 and one output c
whose value will be

c D a0 AND a1 AND AND an 1:

There are various ways to design an n-bit AND-circuit. A serial design is simply
a series of AND-gates, each with one input being a circuit input ai and the other
input being the output of the previous gate as shown in Figure 3.2.
We can also use a tree design. A 1-bit tree design is just a wire, that is c WWD a1 .
Assuming for simplicity that n is a power of two, an n-input tree circuit for n > 1
simply consists of two n=2-input tree circuits whose outputs are AND’d to produce
output c, as in Figure 3.3. For example, a 4-bit tree design circut is shown in
Figure 3.4.
(a) How many AND-gates are in the n-input serial circuit?

(b) The “speed” or latency of a circuit is the largest number of gates on any path
from an input to an output. Briefly explain why the tree circuit is exponentially
faster than the serial circuit.

Problems for Section 3.3
Practice Problems
Problem 3.10.
Indicate whether each of the following propositional formulas is valid (V), satis-
fiable but not valid (S), or not satisfiable (N). For the satisfiable ones, indicate a
satisfying truth assignment.
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76 Chapter 3 Logical Formulas

Figure 3.2 A serial AND-circuit.
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3.7. References 77

Figure 3.3 An n-bit AND-tree circuit.

Figure 3.4 A 4-bit AND-tree circuit.
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78 Chapter 3 Logical Formulas

M IMPLIES Q
M IMPLIES .P OR Q/
M IMPLIES ŒM AND .P IMPLIES M /
.P OR Q/ IMPLIES Q
.P OR Q/ IMPLIES .P AND Q/
.P OR Q/ IMPLIES ŒM AND .P IMPLIES M /
.P XOR Q/ IMPLIES Q
.P XOR Q/ IMPLIES .P OR Q/
.P XOR Q/ IMPLIES ŒM AND .P IMPLIES M /

Problem 3.11.
Show truth tables that verify the equivalence of the following two propositional
formulas

.P XOR Q/;
NOT .P IFF Q/:

Problem 3.12.
Prove that the propositional formulas

P OR Q OR R

and

.P AND NOT.Q// OR .Q AND NOT.R// OR .R AND NOT.P // OR .P AND Q AND R/:

are equivalent.

Problem 3.13.
Prove by truth table that OR distributes over AND, namely,

P OR .Q AND R/ is equivalent to .P OR Q/ AND .P OR R/ (3.31)
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3.7. References 79

Exam Problems
Problem 3.14.
The formula

NOT.A IMPLIES B/ AND A AND C
IMPLIES
D AND E AND F AND G AND H AND I AND J AND K AND L AND M

turns out to be valid.
(a) Explain why verifying the validity of this formula by truth table would be very
hard for one person to do with pencil and paper (no computers).

(b) Verify that the formula is valid, reasoning by cases according to the truth value
of A.

Proof. Case: (A is True).

Case: (A is False).

Class Problems
Problem 3.15. (a) Verify by truth table that

.P IMPLIES Q/ OR .Q IMPLIES P /

is valid.

(b) Let P and Q be propositional formulas. Describe a single formula R using
only AND’s, OR’s, NOT’s, and copies of P and Q, such that R is valid iff P and Q
are equivalent.

(c) A propositional formula is satisfiable iff there is an assignment of truth values
to its variables—an environment—that makes it true. Explain why

P is valid iff NOT .P / is not satisfiable.

(d) A set of propositional formulas P1 ; : : : ; Pk is consistent iff there is an envi-
ronment in which they are all true. Write a formula S such that the set P1 ; : : : ; Pk
is not consistent iff S is valid.
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80 Chapter 3 Logical Formulas

Problem 3.16.
This problem4 examines whether the following specifications are satisfiable:
1. If the file system is not locked, then

(a) new messages will be queued.
(b) new messages will be sent to the messages buffer.
(c) the system is functioning normally, and conversely, if the system is
functioning normally, then the file system is not locked.

2. If new messages are not queued, then they will be sent to the messages buffer.
3. New messages will not be sent to the message buffer.
(a) Begin by translating the five specifications into propositional formulas using
four propositional variables:
L WWD file system locked;
Q WWD new messages are queued;
B WWD new messages are sent to the message buffer;
N WWD system functioning normally:

(b) Demonstrate that this set of specifications is satisfiable by describing a single
truth assignment for the variables L; Q; B; N and verifying that under this assign-
ment, all the specifications are true.

Problems for Section 3.4
Practice Problems
Problem 3.17.
A half dozen different operators may appear in propositional formulas, but just
AND , OR , and NOT are enough to do the job. That is because each of the operators
is equivalent to a simple formula using only these three operators. For example,
A IMPLIES B is equivalent to NOT.A/ OR B. So all occurences of IMPLIES in a
formula can be replaced using just NOT and OR.
4 Revised from Rosen, 5th edition, Exercise 1.1.36
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3.7. References 81

(a) Write formulas using only AND, OR, NOT that are equivalent to each of A IFF B
and A XOR B. Conclude that every propositional formula is equivalent to an AND-
OR - NOT formula.

(b) Explain why you don’t even need AND.

Class Problems
Problem 3.18.
The propositional connective NOR is defined by the rule

P NOR Q WWD .NOT.P / AND NOT.Q//:

Explain why every propositional formula—possibly involving any of the usual op-
erators such as IMPLIES, XOR, . . . —is equivalent to one whose only connective is
NOR .

Problem 3.19.
Explain how to read off a conjunctive form for a propositional formula directly
from a disjunctive form for its complement.

Problem 3.20.
Let P be the proposition depending on propositional variable A; B; C; D whose
truth values for each truth assignment to A; B; C; D are given in the table below.
Write out both a disjunctive and a conjunctive normal form for P .
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82 Chapter 3 Logical Formulas

A B C D P
T T T T T
T T T F F
T T F T T
T T F F F
T F T T T
T F T F T
T F F T T
T F F F T
F T T T T
F T T F F
F T F T T
F T F F F
F F T T F
F F T F F
F F F T T
F F F F T

Homework Problems

Problem 3.21.
Use the equivalence axioms of Section 3.4.2 to convert the formula

A XOR B XOR C

(a) . . . to disjunctive (OR of AND’s) form,

(b) . . . to conjunctive (AND of OR’s) form.

Problems for Section 3.5
Class Problems
Problem 3.22.
The circuit-SAT problem is the problem of determining, for any given digital circuit
with one output wire, whether there are truth values that can be fed into the circuit
input wires which will lead the circuit to give output T.
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3.7. References 83

It’s easy to see that any efficient way of solving the circuit-SAT problem would
yield an efficient way to solve the usual SAT problem for propositional formulas
(Section 3.5). Namely, for any formula F , just construct a circuit CF using that
computes the values of the formula. Then there are inputs for which CF gives
output true iff F is satisfiable. Constructing CF from F is easy, using a binary
gate in CF for each propositional connective in F . So an efficient circuit-SAT
procedure leads to an efficient SAT procedure.
Conversely, there is a simple recursive procedure that will construct, given C , a
formula EC that is equivalent to C in the sense that the truth value EC and the out-
put of C are the same for every truth assignment of the variables. The difficulty is
that, in general, the “equivalent” formula EC , will be exponentially larger than C .
For the purposes of showing that satifiability of circuits and satisfiability of formu-
las take roughly the same effort to solve, spending an exponential time translating
one problem to the other swamps any benefit in switching from one problem to the
other.
So instead of a formula EC that is equivalent to C , we aim instead for a formula
FC that is “equisatisfiable” with C . That is, there will be input values that make
C output True iff there is a truth assignment that satisfies FC . (In fact, FC and C
need not even use the same variables.) But now we make sure that the amount of
computation needed to construct FC is not much larger than the size of the circuit
C . In particular, the size of FC will also not be much larger than C .
The idea behind the construction of FC is that, given any digital circuit C with
binary gates and one output, we can assign a distinct variable to each wire of C .
Then for each gate of C , we can set up a propositional formula that represents the
constraints that the gate places on the values of its input and output wires. For
example, for an AND gate with input wire variables P and Q and output wire
variable R, the constraint proposition would be

.P AND Q/ IFF R: (3.32)

(a) Given a circuit C , explain how to easily find a formula FC of size proportional
to the number of wires in C such that FC is satisfiable iff C gives output T for some
set of input values.

(b) Conclude that any efficient way of solving SAT would yield an efficient way
to solve circuit-SAT.

Homework Problems
Problem 3.23.
A 3-conjunctive form (3CF) formula is a conjunctive form formula in which each
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84 Chapter 3 Logical Formulas

OR -term is an OR of at most 3 variables or negations of variables. Although it
may be hard to tell if a propositional formula F is satisfiable, it is always easy to
construct a formula C.F / that is

in 3-conjunctive form,

has at most 24 times as many occurrences of variables as F , and

is satisfiable iff F is satisfiable.

To construct C.F /, introduce a different new variables for each operator that
occurs in F . For example, if F was

..P XOR Q/ XOR R/ OR .P AND S / (3.33)

we might use new variables X1 , X2 O and A corresponding to the operator occur-
rences as follows:

..P „ƒ‚…
XOR Q/ XOR R/ OR .P „ƒ‚…
„ƒ‚… „ƒ‚… AND S /:
X1 X2 O A

Next we write a formula that constrains each new variable to have the same truth
value as the subformula determined by its corresponding operator. For the example
above, these constraining formulas would be

X1 IFF .P XOR Q/;
X2 IFF .X1 XOR R/;
A IFF .P AND S /;
O IFF .X2 OR A/

(a) Explain why the AND of the four constraining formulas above along with a
fifth formula consisting of just the variable O will be satisfiable iff (3.33) is satisfi-
able.

(b) Explain why each constraining formula will be equivalent to a 3CF formula
with at most 24 occurrences of variables.

(c) Using the ideas illustrated in the previous parts, explain how to construct C.F /
for an arbitrary propositional formula F .
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3.7. References 85

Problems for Section 3.6
Practice Problems
Problem 3.24.
For each of the following propositions:

1. 8x 9y: 2x yD0

2. 8x 9y: x 2y D 0

3. 8x: x < 10 IMPLIES .8y: y < x IMPLIES y < 9/

4. 8x 9y: Œy > x ^ 9z: y C z D 100

determine which propositions are true when the variables range over:
(a) the nonnegative integers.

(b) the integers.

Problem 3.25.
Let Q.x; y/ be the statement

“x has been a contestant on television show y.”

The universe of discourse for x is the set of all students at your school and for y is
the set of all quiz shows that have ever been on television.
Determine whether or not each of the following expressions is logically equiva-
lent to the sentence:

“No student at your school has ever been a contestant on a television quiz show.”

(a) 8x 8y: NOT.Q.x; y//

(b) 9x 9y: NOT.Q.x; y//

(d) NOT.9x 9y: Q.x; y//
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86 Chapter 3 Logical Formulas

Problem 3.26.
Find a counter-model showing the following is not valid.

9x:P .x/ IMPLIES 8x:P .x/

(Just define your counter-model. You do not need to verify that it is correct.)

Problem 3.27.
Find a counter-model showing the following is not valid.

Œ9x: P .x/ AND 9x:Q.x/ IMPLIES 9x:ŒP .x/ AND Q.x/

(Just define your counter-model. You do not need to verify that it is correct.)

Problem 3.28.
Which of the following are valid? For those that are not valid, desribe a counter-
model.
(a) 9x9y: P .x; y/ IMPLIES 9y9x: P .x; y/

(b) 8x9y: Q.x; y/ IMPLIES 9y8x: Q.x; y/

(d) NOT.9x S.x// IFF 8x NOT.S.x//

Problem 3.29. (a) Verify that the propositional formula

.P IMPLIES Q/ OR .Q IMPLIES P /

is valid.

(b) The valid formula of part (a) leads to sound proof method: to prove that an im-
plication is true, just prove that its converse is false.5 For example, from elementary
calculus we know that the assertion

If a function is continuous, then it is differentiable

is false. This allows us to reach at the correct conclusion that its converse,
5 This problem was stimulated by the discussion of the fallacy in [3].
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3.7. References 87

If a function is differentiable, then it is continuous

is true, as indeed it is.
But wait a minute! The implication

If a function is differentiable, then it is not continuous

is completely false. So we could conclude that its converse

If a function is not continuous, then it is differentiable,

should be true, but in fact the converse is also completely false.
So something has gone wrong here. Explain what.

Class Problems
Problem 3.30.
A media tycoon has an idea for an all-news television network called LNN: The
Logic News Network. Each segment will begin with a definition of the domain of
discourse and a few predicates. The day’s happenings can then be communicated
concisely in logic notation. For example, a broadcast might begin as follows:

THIS IS LNN. The domain of discourse is

fAlbert; Ben; Claire; David; Emilyg:

Let D.x/ be a predicate that is true if x is deceitful. Let L.x; y/
be a predicate that is true if x likes y. Let G.x; y/ be a predicate that
is true if x gave gifts to y.

Translate the following broadcasts in logic notation into (English) statements.
(a)

NOT.D.Ben/ OR D.David// IMPLIES
.L.Albert; Ben/ AND L.Ben; Albert//:

(b)

8x: ..x D Claire AND NOT.L.x; Emily/// OR .x ¤ Claire AND L.x; Emily///
AND
8x: ..x D David AND L.x; Claire// OR .x ¤ David AND NOT.L.x; Claire////
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88 Chapter 3 Logical Formulas

(c)

NOT .D.Claire// IMPLIES .G.Albert; Ben/ AND 9x: G.Ben; x//

(d)
8x9y9z .y ¤ z/ AND L.x; y/ AND NOT.L.x; z//:

(e) How could you express “Everyone except for Claire likes Emily” using just
propositional connectives without using any quantifiers (8; 9)? Can you generalize
to explain how any logical formula over this domain of discourse can be expressed
without quantifiers? How big would the formula in the previous part be if it was
expressed this way?

Problem 3.31.
The goal of this problem is to translate some assertions about binary strings into
logic notation. The domain of discourse is the set of all finite-length binary strings:
, 0, 1, 00, 01, 10, 11, 000, 001, . . . . (Here denotes the empty string.) In your
translations, you may use all the ordinary logic symbols (including =), variables,
and the binary symbols 0, 1 denoting 0, 1.
A string like 01x0y of binary symbols and variables denotes the concatenation
of the symbols and the binary strings represented by the variables. For example, if
the value of x is 011 and the value of y is 1111, then the value of 01x0y is the
binary string 0101101111.
Here are some examples of formulas and their English translations. Names for
these predicates are listed in the third column so that you can reuse them in your
solutions (as we do in the definition of the predicate NO -1 S below).

Meaning Formula Name
x is a prefix of y 9z .xz D y/ PREFIX (x; y)
x is a substring of y 9u9v .uxv D y/ SUBSTRING (x; y)
x is empty or a string of 0’s NOT . SUBSTRING .1; x// NO -1 S (x)

(a) x consists of three copies of some string.

(b) x is an even-length string of 0’s.

(d) x is the binary representation of 2k C 1 for some integer k 0.
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3.7. References 89

(e) An elegant, slightly trickier way to define NO -1 S.x/ is:

PREFIX .x; 0x/: (*)

Explain why (*) is true only when x is a string of 0’s.

Problem 3.32.
For each of the logical formulas, indicate whether or not it is true when the do-
main of discourse is N, (the nonnegative integers 0, 1, 2, . . . ), Z (the integers), Q
(the rationals), R (the real numbers), and C (the complex numbers). Add a brief
explanation to the few cases that merit one.

9x: x 2 D 2
8x:9y: x 2 D y
8y:9x: x 2 D y
8x ¤ 0:9y: xy D 1
9x:9y: x C 2y D 2 AND 2x C 4y D 5

Problem 3.33.
Show that
.8x9y: P .x; y// ! 8z: P .z; z/
is not valid by describing a counter-model.

Homework Problems
Problem 3.34.
Express each of the following predicates and propositions in formal logic notation.
The domain of discourse is the nonnegative integers, N. Moreover, in addition to
the propositional operators, variables and quantifiers, you may define predicates
using addition, multiplication, and equality symbols, and nonnegative integer con-
stants (0, 1,. . . ), but no exponentiation (like x y ). For example, the predicate “n is
an even number” could be defined by either of the following formulas:

9m: .2m D n/; 9m: .m C m D n/:

(a) m is a divisor of n.
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90 Chapter 3 Logical Formulas

(b) n is a prime number.

Problem 3.35.
Translate the following sentence into a predicate formula:
There is a student who has e-mailed at most two other people in the
class, besides possibly himself.
The domain of discourse should be the set of students in the class; in addition,
the only predicates that you may use are
equality, and
E.x; y/, meaning that “x has sent e-mail to y.”

Problem 3.36. (a) Translate the following sentence into a predicate formula:
There is a student who has e-mailed at most n other people in the class,
besides possibly himself.
The domain of discourse should be the set of students in the class; in addition, the
only predicates that you may use are
equality,
E.x; y/, meaning that “x has sent e-mail to y.”

(b) Explain how you would use your predicate formula (or some variant of it) to
express the following two sentences.
1. There is a student who has emailed at least n other people in the class, besides
possibly himself.
2. There is a student who has emailed exactly n other people in the class, besides
possibly himself.

Exam Problems
Problem 3.37.
For each of the logic formulas below, indicate the smallest domain in which it is
true, among
N(nonnegative integers); Z(integers); Q(rationals); R(reals); C(complex numbers);
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3.7. References 91

or state “none” if it is not true in any of them.

i. 8x9y: y D 3x

ii. 8x9y: 3y D x

iii. 8x9y: y 2 D x

iv. 8x9y: y < x

v. 8x9y: y 3 D x

vi. 8x ¤ 0: 9y; z: y ¤ z AND y 2 D x D z 2

Problem 3.38.

The following predicate logic formula is invalid:

8x; 9y:P .x; y/ ! 9y; 8x:P .x; y/
Which of the following are counter models for it?

1. The predicate P .x; y/ D ‘y x D 1’ where the domain of discourse is Q.

2. The predicate P .x; y/ D ‘y < x’ where the domain of discourse is R.

3. The predicate P .x; y/ D ‘y x D 2’ where the domain of discourse is R
without 0.

4. The predicate P .x; y/ D ‘yxy D x’ where the domain of discourse is the
set of all binary strings, including the empty string.

Problem 3.39.
Some students from a large class will be lined up left to right. There will be at least
two students in the line. Translate each of the following assertions into predicate
formulas with the set of students in the class as the domain of discourse. The only
predicates you may use are

equality and,

F .x; y/, meaning that “x is somewhere to the left of y in the line.” For
example, in the line “CDA”, both F .C; A/ and F .C; D/ are true.
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92 Chapter 3 Logical Formulas

Once you have defined a formula for a predicate P you may use the abbreviation
“P ” in further formulas.
(a) Student x is in the line.

(b) Student x is first in line.

(d) Student x is second.

Problem 3.40.
We want to find predicate formulas about the nonnegative integers N in which is
the only predicate that appears, and no constants appear.
For example, there is such a formula defining the equality predicate:

Œx D y WWD Œx y AND y x:

Once predicate is shown to be expressible solely in terms of , it may then be used
in subsequent translations. For example,

Œx > 0 WWD 9y: NOT.x D y/ AND y x:

(a) Œx D 0.

(b) Œx D y C 1.
Hint: If an integer is bigger than y, then it must be x.

Problem 3.41.
Predicate Formulas whose only predicate symbol is equality are called “pure equal-
ity” formulas. For example,
8x 8y: x D y (1-element)
is a pure equality formula. Its meaning is that there is exactly one element in the
domain of discourse.6 Another such formula is

9a 9b 8x: x D a OR x D b: ( 2-elements)
6 Remember, a domain of discourse is not allowed to be empty.
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3.7. References 93

Its meaning is that there are at most two elements in the domain of discourse.
A formula that is not a pure equality formula is

x y: (not-pure)

Formula (not-pure) uses the less-than-or-equal predicate which is not allowed.7
(a) Describe a pure equality formula that means that there are exactly two ele-
ments in the domain of discourse.

(b) Describe a pure equality formula that means that there are exactly three ele-
ments in the domain of discourse.

7 Infact, formula (not-pure) only makes sense when the domain elements are ordered, while pure
equality formulas make sense over every domain.
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4 Mathematical Data Types
We have assumed that you’ve already been introduced to the concepts of sets, se-
quences, and functions, and we’ve used them informally several times in previous
sections. In this chapter, we’ll now take a more careful look at these mathemati-
cal data types. We’ll quickly review the basic definitions, add a few more such as
“images” and “inverse images” that may not be familiar, and end the chapter with
some methods for comparing the sizes of sets.

4.1 Sets
Informally, a set is a bunch of objects, which are called the elements of the set.
The elements of a set can be just about anything: numbers, points in space, or even
other sets. The conventional way to write down a set is to list the elements inside
curly-braces. For example, here are some sets:

A D fAlex; Tippy; Shells; Shadowg dead pets
B D fred; blue; yellowg primary colors
C D ffa; bg; fa; cg; fb; cgg a set of sets

This works fine for small finite sets. Other sets might be defined by indicating how
to generate a list of them:

D WWD f1; 2; 4; 8; 16; : : :g the powers of 2

The order of elements is not significant, so fx; yg and fy; xg are the same set
written two different ways. Also, any object is, or is not, an element of a given set—
there is no notion of an element appearing more than once in a set.1 So, writing
fx; xg is just indicating the same thing twice: that x is in the set. In particular,
fx; xg D fxg.
The expression “e 2 S” asserts that e is an element of set S. For example,
32 2 D and blue 2 B, but Tailspin 62 A—yet.
Sets are simple, flexible, and everywhere. You’ll find some set mentioned in
nearly every section of this text.
1 It’s
not hard to develop a notion of multisets in which elements can occur more than once, but
multisets are not ordinary sets and are not covered in this text.
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96 Chapter 4 Mathematical Data Types

4.1.1 Some Popular Sets
Mathematicians have devised special symbols to represent some common sets.

symbol set elements
; the empty set none
N nonnegative integers f0; 1; 2; 3; : : :g
Z integers f: : : ; 3; 2; 1; 0; 1; 2; 3; : : :g
1 5
Q rational numbers 2; 3 ; 16;petc.
R real numbers ; e; p9; 2; etc.
C complex numbers i; 192 ; 2 2i; etc.

A superscript “C ” restricts a set to its positive elements; for example, RC denotes
the set of positive real numbers. Similarly, Z denotes the set of negative integers.

4.1.2 Comparing and Combining Sets
The expression S T indicates that set S is a subset of set T , which means that
every element of S is also an element of T . For example, N Z because every
nonnegative integer is an integer; Q R because every rational number is a real
number, but C 6 R because not every complex number is a real number.
As a memory trick, think of the “” symbol as like the “” sign with the smaller
set or number on the left-hand side. Notice that just as n n for any number n,
also S S for any set S .
There is also a relation on sets like the “less than” relation < on numbers.
S T means that S is a subset of T , but the two are not equal. So just as n 6< n
for every number n, also A 6 A, for every set A. “S T ” is read as “S is a strict
subset of T .”
There are several basic ways to combine sets. For example, suppose

X WWD f1; 2; 3g;
Y WWD f2; 3; 4g:

Definition 4.1.1.

The union of sets A and B, denoted A [ B, includes exactly the elements
appearing in A or B or both. That is,

x 2A[B IFF x 2 A OR x 2 B:

So X [ Y D f1; 2; 3; 4g.
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4.1. Sets 97

The intersection of A and B, denoted A \ B, consists of all elements that
appear in both A and B. That is,

x 2A\B IFF x 2 A AND x 2 B:

So, X \ Y D f2; 3g.

The set difference of A and B, denoted A B, consists of all elements that
are in A, but not in B. That is,

x2A B IFF x 2 A AND x … B:

So, X Y D f1g and Y X D f4g.

Often all the sets being considered are subsets of a known domain of discourse
D. Then for any subset A of D, we define A to be the set of all elements of D not
in A. That is,
A WWD D A:
The set A is called the complement of A. So

A D ; IFF A D D:

For example, if the domain we’re working with is the integers, the complement
of the nonnegative integers is the set of negative integers:

NDZ :

We can use complement to rephrase subset in terms of equality

A B is equivalent to A \ B D ;:

4.1.3 Power Set
The set of all the subsets of a set A is called the power set pow.A/ of A. So

B 2 pow.A/ IFF B A:

For example, the elements of pow.f1; 2g/ are ;; f1g; f2g and f1; 2g.
More generally, if A has n elements, then there are 2n sets in pow.A/—see The-
orem 4.5.5. For this reason, some authors use the notation 2A instead of pow.A/.
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98 Chapter 4 Mathematical Data Types

4.1.4 Set Builder Notation
An important use of predicates is in set builder notation. We’ll often want to talk
about sets that cannot be described very well by listing the elements explicitly or
by taking unions, intersections, etc., of easily described sets. Set builder notation
often comes to the rescue. The idea is to define a set using a predicate; in particular,
the set consists of all values that make the predicate true. Here are some examples
of set builder notation:

A WWD fn 2 N j n is a prime and n D 4k C 1 for some integer kg;
B WWD fx 2 R j x 3 3x C 1 > 0g;
C WWD fa C bi 2 C j a2 C 2b 2 1g;
D WWD fL 2 books j L is cited in this textg:
The set A consists of all nonnegative integers n for which the predicate
“n is a prime and n D 4k C 1 for some integer k”
is true. Thus, the smallest elements of A are:
5; 13; 17; 29; 37; 41; 53; 61; 73; : : : :
Trying to indicate the set A by listing these first few elements wouldn’t work very
well; even after ten terms, the pattern is not obvious. Similarly, the set B consists
of all real numbers x for which the predicate
x3 3x C 1 > 0
is true. In this case, an explicit description of the set B in terms of intervals would
require solving a cubic equation. Set C consists of all complex numbers a C bi
such that:
a2 C 2b 2 1
This is an oval-shaped region around the origin in the complex plane. Finally, the
members of set D can be determined by filtering out journal articles in from the list
of references in the Bibliography 22.5.

4.1.5 Proving Set Equalities
Two sets are defined to be equal if they have exactly the same elements. That is,
X D Y means that z 2 X if and only if z 2 Y , for all elements z.2 So, set
equalities can be formulated and proved as “iff” theorems. For example:
2 Thisis actually the first of the ZFC axioms for set theory mentioned at the end of Section 1.3
and discussed further in Section 8.3.2.
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4.1. Sets 99

Theorem 4.1.2. [Distributive Law for Sets] Let A, B and C be sets. Then:

A \ .B [ C / D .A \ B/ [ .A \ C / (4.1)

Proof. The equality (4.1) is equivalent to the assertion that

z 2 A \ .B [ C / iff z 2 .A \ B/ [ .A \ C / (4.2)

for all z. Now we’ll prove (4.2) by a chain of iff’s.
Now we have

z 2 A \ .B [ C /
iff .z 2 A/ AND .z 2 B [ C / (def of \)
iff .z 2 A/ AND .z 2 B OR z 2 C / (def of [)
iff .z 2 A AND z 2 B/ OR .z 2 A AND z 2 C / (AND distributivity (3.9))
iff .z 2 A \ B/ OR .z 2 A \ C / (def of \)
iff z 2 .A \ B/ [ .A \ C / (def of [)

The proof of Theorem 4.1.2 illustrates a general method for proving a set equality
involving the basic set operations by checking that a corresponding propositional
formula is valid. As a further example, from De Morgan’s Law (3.14) for proposi-
tions
NOT .P AND Q/ is equivalent to P OR Q

we can derive (Problem 4.5) a corresponding De Morgan’s Law for set equality:

A \ B D A [ B: (4.3)

Despite this correspondence between two kinds of operations, it’s important not
to confuse propositional operations with set operations. For example, if X and Y
are sets, then it is wrong to write “X AND Y ” instead of “X \ Y .” Applying AND
to sets will cause your compiler—or your grader—to throw a type error, because
an operation that is only supposed to be applied to truth values has been applied to
sets. Likewise, if P and Q are propositions, then it is a type error to write “P [ Q”
instead of “P OR Q.”
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100 Chapter 4 Mathematical Data Types

4.2 Sequences
Sets provide one way to group a collection of objects. Another way is in a sequence,
which is a list of objects called its components, members, or elements. Short se-
quences are commonly described by listing the elements between parentheses; for
example, the sequence .a; b; c/ has three components. It would also be referred to
as a three element sequence or a sequence of length three. These phrases are all
synonyms—sequences are so basic that they appear everywhere and there are a lot
of ways to talk about them.
While both sets and sequences perform a gathering role, there are several differ-
ences.

The elements of a set are required to be distinct, but elements in a sequence
can be the same. Thus, .a; b; a/ is a valid sequence of length three, but
fa; b; ag is a set with two elements, not three.

The elements in a sequence have a specified order, but the elements of a
set do not. For example, .a; b; c/ and .a; c; b/ are different sequences, but
fa; b; cg and fa; c; bg are the same set.

Texts differ on notation for the empty sequence; we use for the empty
sequence.

The product operation is one link between sets and sequences. A Cartesian
product of sets, S1 S2 Sn , is a new set consisting of all sequences where
the first component is drawn from S1 , the second from S2 , and so forth. Length two
sequences are called pairs.3 For example, N fa; bg is the set of all pairs whose
first element is a nonnegative integer and whose second element is an a or a b:

N fa; bg D f.0; a/; .0; b/; .1; a/; .1; b/; .2; a/; .2; b/; : : :g

A product of n copies of a set S is denoted S n . For example, f0; 1g3 is the set of
all 3-bit sequences:

f0; 1g3 D f.0; 0; 0/; .0; 0; 1/; .0; 1; 0/; .0; 1; 1/; .1; 0; 0/; .1; 0; 1/; .1; 1; 0/; .1; 1; 1/g
3 Some texts call them ordered pairs.
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4.3. Functions 101

4.3 Functions
4.3.1 Domains and Images
A function assigns an element of one set, called the domain, to an element of an-
other set, called the codomain. The notation

f WA!B

indicates that f is a function with domain A and codomain B. The familiar notation
“f .a/ D b” indicates that f assigns the element b 2 B to a. Here b would be
called the value of f at argument a.
Functions are often defined by formulas, as in:
1
f1 .x/ WWD
x2
where x is a real-valued variable, or

f2 .y; z/ WWD y10yz

where y and z range over binary strings, or

f3 .x; n/ WWD the length n sequence .x; : : : ; x/
„ ƒ‚ …
n x’s

where n ranges over the nonnegative integers.
A function with a finite domain could be specified by a table that shows the value
of the function at each element of the domain. For example, a function f4 .P; Q/
where P and Q are propositional variables is specified by:

P Q f4 .P; Q/
T T T
T F F
F T T
F F T

Notice that f4 could also have been described by a formula:

f4 .P; Q/ WWD ŒP IMPLIES Q:

A function might also be defined by a procedure for computing its value at any
element of its domain, or by some other kind of specification. For example, define
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102 Chapter 4 Mathematical Data Types

f5 .y/ to be the length of a left to right search of the bits in the binary string y until
a 1 appears, so
f5 .0010/ D 3;
f5 .100/ D 1;
f5 .0000/ is undefined:
Notice that f5 does not assign a value to any string of just 0’s. This illustrates an
important fact about functions: they need not assign a value to every element in the
domain. In fact this came up in our first example f1 .x/ D 1=x 2 , which does not
assign a value to 0. So in general, functions may be partial functions, meaning that
there may be domain elements for which the function is not defined. If a function
is defined on every element of its domain, it is called a total function.
It’s often useful to find the set of values a function takes when applied to the
elements in a set of arguments. So if f W A ! B, and S is a subset of A, we define
f .S / to be the set of all the values that f takes when it is applied to elements of S .
That is,
f .S / WWD fb 2 B j f .s/ D b for some s 2 Sg:
For example, if we let Œr; s denote set of numbers in the interval from r to s on the
real line, then f1 .Œ1; 2/ D Œ1=4; 1.
For another example, let’s take the “search for a 1” function f5 . If we let X be
the set of binary words which start with an even number of 0’s followed by a 1,
then f5 .X / would be the odd nonnegative integers.
Applying f to a set S of arguments is referred to as “applying f pointwise to
S ”, and the set f .S / is referred to as the image of S under f .4 The set of values
that arise from applying f to all possible arguments is called the range of f . That
is,
range.f / WWD f .domain.f //:
Some authors refer to the codomain as the range of a function, but they shouldn’t.
The distinction between the range and codomain will be important later in Sec-
tions 4.5 when we relate sizes of sets to properties of functions between them.

4.3.2 Function Composition
Doing things step by step is a universal idea. Taking a walk is a literal example, but
so is cooking from a recipe, executing a computer program, evaluating a formula,
and recovering from substance abuse.
4 There is a picky distinction between the function f which applies to elements of A and the
function which applies f pointwise to subsets of A, because the domain of f is A, while the domain
of pointwise-f is pow.A/. It is usually clear from context whether f or pointwise-f is meant, so
there is no harm in overloading the symbol f in this way.
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4.4. Binary Relations 103

Abstractly, taking a step amounts to applying a function, and going step by step
corresponds to applying functions one after the other. This is captured by the op-
eration of composing functions. Composing the functions f and g means that first
f is applied to some argument, x, to produce f .x/, and then g is applied to that
result to produce g.f .x//.

Definition 4.3.1. For functions f W A ! B and g W B ! C , the composition,
g ı f , of g with f is defined to be the function from A to C defined by the rule:

.g ı f /.x/ WWD g.f .x//;

for all x 2 A.

Function composition is familiar as a basic concept from elementary calculus,
and it plays an equally basic role in discrete mathematics.

4.4 Binary Relations
Binary relations define relations between two objects. For example, “less-than” on
the real numbers relates every real number a to a real number b, precisely when
a < b. Similarly, the subset relation relates a set A to another set B precisely when
A B. A function f W A ! B is a special case of binary relation in which an
element a 2 A is related to an element b 2 B precisely when b D f .a/.
In this section we’ll define some basic vocabulary and properties of binary rela-
tions.

Definition 4.4.1. A binary relation R consists of a set A, called the domain of R,
a set B called the codomain of R, and a subset of A B called the graph of R.

A relation whose domain is A and codomain is B is said to be “between A and
B”, or “from A to B.” As with functions, we write R W A ! B to indicate that R
is a relation from A to B. When the domain and codomain are the same set A we
simply say the relation is “on A.” It’s common to use “a R b” to mean that the pair
.a; b/ is in the graph of R.5
Notice that Definition 4.4.1 is exactly the same as the definition in Section 4.3
of a function, except that it doesn’t require the functional condition that, for each
5 Writing the relation or operator symbol between its arguments is called infix notation. Infix
expressions like “m < n” or “m C n” are the usual notation used for things like the less-then relation
or the addition operation rather than prefix notation like “< .m; n/” or “C.m; n/.”
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104 Chapter 4 Mathematical Data Types

domain element a, there is at most one pair in the graph whose first coordinate is
a. As we said, a function is a special case of a binary relation.
The “in-charge of” relation Chrg for MIT in Spring ’10 subjects and instructors
is a handy example of a binary relation. Its domain Fac is the names of all the
MIT faculty and instructional staff, and its codomain is the set SubNums of subject
numbers in the Fall ’09–Spring ’10 MIT subject listing. The graph of Chrg contains
precisely the pairs of the form

.hinstructor-namei ; hsubject-numi/

such that the faculty member named hinstructor-namei is in charge of the subject
with number hsubject-numi that was offered in Spring ’10. So graph.Chrg/ con-
tains pairs like
.T. Eng; 6.UAT/
.G. Freeman; 6.011/
.G. Freeman; 6.UAT/
.G. Freeman; 6.881/
.G. Freeman; 6.882/
.J. Guttag; 6.00/
.A. R. Meyer; 6.042/ (4.4)
.A. R. Meyer; 18.062/
.A. R. Meyer; 6.844/
.T. Leighton; 6.042/
.T. Leighton; 18.062/
::
:
Some subjects in the codomain SubNums do not appear among this list of pairs—
that is, they are not in range.Chrg/. These are the Fall term-only subjects. Simi-
larly, there are instructors in the domain Fac who do not appear in the list because
they are not in charge of any Spring term subjects.

4.4.1 Relation Diagrams
Some standard properties of a relation can be visualized in terms of a diagram. The
diagram for a binary relation R has points corresponding to the elements of the
domain appearing in one column (a very long column if domain.R/ is infinite). All
the elements of the codomain appear in another column which we’ll usually picture
as being to the right of the domain column. There is an arrow going from a point a
in the left-hand, domain column to a point b in the right-hand, codomain column,
precisely when the corresponding elements are related by R. For example, here are
diagrams for two functions:
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4.4. Binary Relations 105

A B A B
a - 1 a - 1
b PP 2 b PP 2
PP3 PP
3
c PP P
Pq
c Q P
Pq
3 3
PP 3
Q

PP
d

q
4 d QQ 4
QQ
e s
5

Being a function is certainly an important property of a binary relation. What it
means is that every point in the domain column has at most one arrow coming out
of it. So we can describe being a function as the “ 1 arrow out” property. There
are four more standard properties of relations that come up all the time. Here are
all five properties defined in terms of arrows:
Definition 4.4.2. A binary relation R is:
a function when it has the Œ 1 arrow out property.
surjective when it has the Œ 1 arrows in property. That is, every point in
the right-hand, codomain column has at least one arrow pointing to it.
total when it has the Œ 1 arrows out property.
injective when it has the Œ 1 arrow in property.
bijective when it has both the ŒD 1 arrow out and the ŒD 1 arrow in prop-
erty.
From here on, we’ll stop mentioning the arrows in these properties and for ex-
ample, just write Œ 1 in instead of Œ 1 arrows in.
So in the diagrams above, the relation on the left has the ŒD 1 out and Œ 1 in
properties, which means it is a total, surjective function. But it does not have the
Œ 1 in property because element 3 has two arrows going into it; it is not injective.
The relation on the right has the ŒD 1 out and Œ 1 in properties, which means
it is a total, injective function. But it does not have the Œ 1 in property because
element 4 has no arrow going into it; it is not surjective.
The arrows in a diagram for R correspond, of course, exactly to the pairs in the
graph of R. Notice that the arrows alone are not enough to determine, for example,
if R has the Œ 1 out, total, property. If all we knew were the arrows, we wouldn’t
know about any points in the domain column that had no arrows out. In other
words, graph.R/ alone does not determine whether R is total: we also need to
know what domain.R/ is.
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106 Chapter 4 Mathematical Data Types

Example 4.4.3. The function defined by the formula 1=x 2 has the Œ 1 out prop-
erty if its domain is RC , but not if its domain is some set of real numbers including
0. It has the ŒD 1 in and ŒD 1 out property if its domain and codomain are both
RC , but it has neither the Œ 1 in nor the Œ 1 out property if its domain and
codomain are both R.

4.4.2 Relational Images
The idea of the image of a set under a function extends directly to relations.

Definition 4.4.4. The image of a set Y under a relation R written R.Y /, is the set
of elements of the codomain B of R that are related to some element in Y . In terms
of the relation diagram, R.Y / is the set of points with an arrow coming in that starts
from some point in Y .

For example, the set of subject numbers that Meyer is in charge of in Spring ’10
is exactly Chrg.A. Meyer/. To figure out what this is, we look for all the arrows
in the Chrg diagram that start at “A. Meyer,” and see which subject-numbers are
at the other end of these arrows. Looking at the list (4.4) of pairs in graph.Chrg/,
we see that these subject-numbers are f6.042, 18.062, 6.844g. Similarly, to find the
subject numbers that either Freeman or Eng are in charge of, we can collect all the
arrows that start at either “G. Freeman,” or “T. Eng” and, again, see which subject-
numbers are at the other end of these arrows. This is Chrg.fG. Freeman; T. Engg/.
Looking again at the list (4.4), we see that

Chrg.fG. Freeman; T. Engg/ D f6.011, 6.881, 6.882, 6.UATg

Finally, Fac is the set of all in-charge instructors, so Chrg.Fac/ is the set of all the
subjects listed for Spring ’10.

Inverse Relations and Images
Definition 4.4.5. The inverse, R 1 of a relation R W A ! B is the relation from B
to A defined by the rule
b R 1 a IFF a R b:

In other words, R 1 is the relation you get by reversing the direction of the
arrows in the diagram of R.

Definition 4.4.6. The image of a set under the relation R 1 is called the inverse
image of the set. That is, the inverse image of a set X under the relation R is defined
to be R 1 .X /.
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4.5. Finite Cardinality 107

Continuing with the in-charge example above, the set of instructors in charge
of 6.UAT in Spring ’10 is exactly the inverse image of f6.UATg under the Chrg
relation. From the list (4.4), we see that Eng and Freeman are both in charge of
6.UAT, that is,
1
fT. Eng; D. Freemang Chrg .f6.UATg/:

We can’t assert equality here because there may be additional pairs further down
the list showing that additional instructors are co-incharge of 6.UAT.
Now let Intro be the set of introductory course 6 subject numbers. These are the
subject numbers that start with “6.0.” So the set of names of the instructors who
were in-charge of introductory course 6 subjects in Spring ’10, is Chrg 1 .Intro/.
From the part of the Chrg list shown in (4.4), we see that Meyer, Leighton, Free-
man, and Guttag were among the instructors in charge of introductory subjects in
Spring ’10. That is,
1
fMeyer, Leighton, Freeman, Guttagg Chrg .Intro/:

Finally, Chrg 1 .SubNums/ is the set of all instructors who were in charge of a
subject listed for Spring ’10.

4.5 Finite Cardinality
A finite set is one that has only a finite number of elements. This number of ele-
ments is the “size” or cardinality of the set:
Definition 4.5.1. If A is a finite set, the cardinality jAj of A is the number of
elements in A.
A finite set may have no elements (the empty set), or one element, or two ele-
ments,. . . , so the cardinality of finite sets is always a nonnegative integer.
Now suppose R W A ! B is a function. This means that every element of A
contributes at most one arrow to the diagram for R, so the number of arrows is at
most the number of elements in A. That is, if R is a function, then

jAj #arrows:

If R is also surjective, then every element of B has an arrow into it, so there must
be at least as many arrows in the diagram as the size of B. That is,

#arrows jBj:
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108 Chapter 4 Mathematical Data Types

Combining these inequalities implies that if R is a surjective function, then jAj
jBj.
In short, if we write A surj B to mean that there is a surjective function from
A to B, then we’ve just proved a lemma: if A surj B for finite sets A; B, then
jAj jBj. The following definition and lemma lists this statement and three similar
rules relating domain and codomain size to relational properties.

Definition 4.5.2. Let A; B be (not necessarily finite) sets. Then

1. A surj B iff there is a surjective function from A to B.

2. A inj B iff there is an injective total relation from A to B.

3. A bij B iff there is a bijection from A to B.

Lemma 4.5.3. For finite sets A; B:

1. If A surj B, then jAj jBj.

2. If A inj B, then jAj jBj.

3. If A bij B, then jAj D jBj.

Proof. We’ve already given an “arrow” proof of implication 1. Implication 2. fol-
lows immediately from the fact that if R has the Œ 1 out, function property, and
the Œ 1 in, surjective property, then R 1 is total and injective, so A surj B iff
B inj A. Finally, since a bijection is both a surjective function and a total injective
relation, implication 3. is an immediate consequence of the first two.

Lemma 4.5.3.1. has a converse: if the size of a finite set A is greater than or equal
to the size of another finite set B then it’s always possible to define a surjective
function from A to B. In fact, the surjection can be a total function. To see how
this works, suppose for example that

A D fa0 ; a1 ; a2 ; a3 ; a4 ; a5 g
B D fb0 ; b1 ; b2 ; b3 g:

Then define a total function f W A ! B by the rules

f .a0 / WWD b0 ; f .a1 / WWD b1 ; f .a2 / WWD b2 ; f .a3 / D f .a4 / D f .a5 / WWD b3 :

More concisely,
f .ai / WWD bmin.i;3/ ;
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4.5. Finite Cardinality 109

for 0 i 5. Since 5 3, this f is a surjection.
So we have figured out that if A and B are finite sets, then jAj jBj if and only if
A surj B. All told, this argument wraps up the proof of a theorem that summarizes
the whole finite cardinality story:
Theorem 4.5.4. [Mapping Rules] For finite sets A; B,

jAj jBj iff A surj B; (4.5)
jAj jBj iff A inj B; (4.6)
jAj D jBj iff A bij B; (4.7)

4.5.1 How Many Subsets of a Finite Set?
As an application of the bijection mapping rule (4.7), we can give an easy proof of:
Theorem 4.5.5. There are 2n subsets of an n-element set. That is,

jAj D n implies j pow.A/j D 2n :

For example, the three-element set fa1 ; a2 ; a3 g has eight different subsets:

; fa1 g fa2 g fa1 ; a2 g
fa3 g fa1 ; a3 g fa2 ; a3 g fa1 ; a2 ; a3 g
Theorem 4.5.5 follows from the fact that there is a simple bijection from subsets
of A to f0; 1gn , the n-bit sequences. Namely, let a1 ; a2 ; : : : ; an be the elements
of A. The bijection maps each subset of S A to the bit sequence .b1 ; : : : ; bn /
defined by the rule that
bi D 1 iff ai 2 S:
For example, if n D 10, then the subset fa2 ; a3 ; a5 ; a7 ; a10 g maps to a 10-bit
sequence as follows:

subset: f a2 ; a 3 ; a5 ; a7 ; a10 g
sequence: . 0; 1; 1; 0; 1; 0; 1; 0; 0; 1 /
Now by bijection case of the Mapping Rules 4.5.4.(4.7),

j pow.A/j D jf0; 1gn j:

But every computer scientist knows6 that there are 2n n-bit sequences! So we’ve
proved Theorem 4.5.5!
6 Incase you’re someone who doesn’t know how many n-bit sequences there are, you’ll find the
2n explained in Section 15.2.2.
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110 Chapter 4 Mathematical Data Types

Problems for Section 4.1
Practice Problems
Problem 4.1.
For any set A, let pow.A/ be its power set, the set of all its subsets; note that A is
itself a member of pow.A/. Let ; denote the empty set.
(a) The elements of pow.f1; 2g/ are:

(b) The elements of pow.f;; f;gg/ are:

Problem 4.2.
Express each of the following assertions about sets by a formula of set theory.7
Expressions may use abbreviations introduced earlier (so it is now legal to use “D”
because we just defined it).
(a) x D ;.

(b) x D fy; zg.

(c) x y. (x is a subset of y that might equal y.)
Now we can explain how to express “x is a proper subset of y” as a set theory
formula using things we already know how to express. Namely, letting “x ¤ y”
abbreviate NOT.x D y/, the expression

.x y AND x ¤ y/;

describes a formula of set theory that means x y.
From here on, feel free to use any previously expressed property in describing
formulas for the following:
(d) x D y [ z.

(e) x D y z.

(f) x D pow.y/.
7 See Section 8.3.2.
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4.5. Finite Cardinality 111

S
(g) x D z2y z.
This means that y is supposed to be S S x is the union of all of
a collection of sets, and
them. A more concise notation for “ z2y z’ is simply “ y.”

Class Problems
Problem 4.3.
Set Formulas and Propositional Formulas.
(a) Verify that the propositional formula .P AND Q/ OR .P AND Q/ is equivalent
to P .

(b) Prove that
A D .A B/ [ .A \ B/
for all sets, A; B, by showing

x 2 A IFF x 2 .A B/ [ .A \ B/

for all elements x using the equivalence of part (a) in a chain of IFF’s.

Problem 4.4.
Prove

Theorem (Distributivity of union over intersection).

A [ .B \ C / D .A [ B/ \ .A [ C / (4.8)

for all sets, A; B; C , by using a chain of iff’s to show that

x 2 A [ .B \ C / IFF x 2 .A [ B/ \ .A [ C /

for all elements x. You may assume the corresponding propositional equivalence 3.10.

Problem 4.5.
Prove De Morgan’s Law for set equality

A \ B D A [ B: (4.9)

by showing with a chain of IFF’s that x 2 the left-hand side of (4.9) iff x 2 the
right-hand side. You may assume the propositional version (3.14) of De Morgan’s
Law.
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112 Chapter 4 Mathematical Data Types

Problem 4.6.
Powerset Properties.
Let A and B be sets.
(a) Prove that
pow.A \ B/ D pow.A/ \ pow.B/:

(b) Prove that
.pow.A/ [ pow.B// pow.A [ B/;
with equality holding iff one of A or B is a subset of the other.

Problem 4.7.
Subset take-away8 is a two player game played with a finite set A of numbers.
Players alternately choose nonempty subsets of A with the conditions that a player
may not choose

the whole set A, or

any set containing a set that was named earlier.

The first player who is unable to move loses the game.
For example, if the size of A is one, then there are no legal moves and the second
player wins. If A has exactly two elements, then the only legal moves are the two
one-element subsets of A. Each is a good reply to the other, and so once again the
second player wins.
The first interesting case is when A has three elements. This time, if the first
player picks a subset with one element, the second player picks the subset with the
other two elements. If the first player picks a subset with two elements, the second
player picks the subset whose sole member is the third element. In both cases, these
moves lead to a situation that is the same as the start of a game on a set with two
elements, and thus leads to a win for the second player.
Verify that when A has four elements, the second player still has a winning strat-
egy.9
8 From Christenson & Tilford, David Gale’s Subset Takeaway Game, American Mathematical
Monthly, Oct. 1997
9 David Gale worked out some of the properties of this game and conjectured that the second

player wins the game for any set A. This remains an open problem.
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4.5. Finite Cardinality 113

Homework Problems
Problem 4.8.
Let A, B and C be sets. Prove that

A [ B [ C D .A B/ [ .B C / [ .C A/ [ .A \ B \ C / (4.10)

using a chain of IFF’s as Section 4.1.5.

Problem 4.9.
Union distributes over the intersection of two sets:

A [ .B \ C / D .A [ B/ \ .A [ C / (4.11)

(see Problem 4.4).
Use (4.11) and the Well Ordering Principle to prove the Distributive Law of
union over the intersection of n sets:

A [ .B1 \ \ Bn 1 \ Bn /
D .A [ B1 / \ \ .A [ Bn 1/ \ .A [ Bn / (4.12)

Extending formulas to an arbitrary number of terms is a common (if mundane)
application of the WOP.

Exam Problems
Problem 4.10.
You’ve seen how certain set identities follow from corresponding propositional
equivalences. For example, you proved by a chain of iff’s that

.A B/ [ .A \ B/ D A

using the fact that the propositional formula .P AND Q/ OR .P AND Q/ is equivalent
to P .
State a similar propositional equivalence that would justify the key step in a proof
for the following set equality organized as a chain of iff’s:

A B D A C [ .B \ C / [ A [ B \ C (4.13)

(You are not being asked to write out an iff-proof of the equality or to write out
a proof of the propositional equivalence. Just state the equivalence.)
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114 Chapter 4 Mathematical Data Types

Problem 4.11.
You’ve seen how certain set identities follow from corresponding propositional
equivalences. For example, you proved by a chain of iff’s that
.A B/ [ .A \ B/ D A
using the fact that the propositional formula .P AND Q/ OR .P AND Q/ is equivalent
to P .
State a similar propositional equivalence that would justify the key step in a proof
for the following set equality organized as a chain of iff’s:
A \ B \ C D A [ .B A/ [ C :
(You are not being asked to write out an iff-proof of the equality or to write out
a proof of the propositional equivalence. Just state the equivalence.)

Problem 4.12.
The set equation
A\B DA[B
follows from a certain equivalence between propositional formulas.
(a) What is the equivalence?

(b) Show how to derive the equation from this equivalence.

Problems for Section 4.2
Homework Problems
Problem 4.13.
Prove that for any sets A, B, C and D, if the Cartesian products A B and C D
are disjoint, then either A and C are disjoint or B and D are disjoint.

Problem 4.14. (a) Give a simple example where the following result fails, and
briefly explain why:
False Theorem. For sets A, B, C and D, let
L WWD .A [ B/ .C [ D/;
R WWD .A C / [ .B D/:
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4.5. Finite Cardinality 115

Then L D R.

(b) Identify the mistake in the following proof of the False Theorem.

Bogus proof. Since L and R are both sets of pairs, it’s sufficient to prove that
.x; y/ 2 L ! .x; y/ 2 R for all x; y.
The proof will be a chain of iff implications:

.x; y/ 2 R
iff .x; y/ 2 .A C / [ .B D/
iff .x; y/ 2 A C , or .x; y/ 2 B D
iff (x 2 A and y 2 C ) or else (x 2 B and y 2 D)
iff either x 2 A or x 2 B, and either y 2 C or y 2 D
iff x 2 A [ B and y 2 C [ D
iff .x; y/ 2 L.

Problems for Section 4.4
Practice Problems
Problem 4.15.
The inverse R 1 of a binary relation R from A to B is the relation from B to A
defined by:
1
bR a iff a R b:
In other words, you get the diagram for R 1 from R by “reversing the arrows” in
the diagram describing R. Now many of the relational properties of R correspond
to different properties of R 1 . For example, R is total iff R 1 is a surjection.
Fill in the remaining entries is this table:
R is iff R 1 is
total a surjection
a function
a surjection
an injection
a bijection
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116 Chapter 4 Mathematical Data Types

Hint: Explain what’s going on in terms of “arrows” from A to B in the diagram
for R.

Problem 4.16.
Describe a total injective function ŒD 1 out, Œ 1 in; from R ! R that is not a
bijection.

Problem 4.17.
For a binary relation R W A ! B, some properties of R can be determined from
just the arrows of R, that is, from graph.R/, and others require knowing if there
are elements in the domain A or the codomain B that don’t show up in graph.R/.
For each of the following possible properties of R, indicate whether it is always
determined by
1. graph.R/ alone,

2. graph.R/ and A alone,

3. graph.R/ and B alone,

4. all three parts of R.
Properties:
(a) surjective

(b) injective

(d) function

(e) bijection

Problem 4.18.
For each of the following real-valued functions on the real numbers, indicate whether
it is a bijection, a surjection but not a bijection, an injection but not a bijection, or
neither an injection nor a surjection.
(a) x ! x C 2

(b) x ! 2x
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4.5. Finite Cardinality 117

(d) x ! x 3

(e) x ! sin x

(f) x ! x sin x

(g) x ! e x

Problem 4.19.
Let f W A ! B and g W B ! C be functions and h W A ! C be their composition,
namely, h.a/ WWD g.f .a// for all a 2 A.
(a) Prove that if f and g are surjections, then so is h.

(b) Prove that if f and g are bijections, then so is h.

Problem 4.20.
Give an example of a relation R that is a total injective function from a set A to
itself but is not a bijection.

Problem 4.21.
Let R W A ! B be a binary relation. Each of the following formulas expresses
the fact that R has a familiar relational “arrow” property such as being surjective
or being a function.
Identify the relational property expressed by each of the following relational
expressions. Explain your reasoning.
(a) R ı R 1 IdB

(b) R 1 ı R IdA

(d) R ı R 1 IdB

Class Problems
Problem 4.22. (a) Prove that if A surj B and B surj C , then A surj C .
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118 Chapter 4 Mathematical Data Types

(b) Explain why A surj B iff B inj A.

(d) Explain why A inj B iff there is a total injective function (ŒD 1 out; 1 in)
from A to B. 10

Problem 4.23.
Five basic properties of binary relations R W A ! B are:

1. R is a surjection Œ 1 in

2. R is an injection Œ 1 in

3. R is a function Œ 1 out

4. R is total Œ 1 out

5. R is empty ŒD 0 out

Below are some assertions about a relation R. For each assertion, write the
numbers of all the properties above that the relation R must have; write “none” if
R might not have any of these properties. For example, you should write “(1), (4)”
next to the first assertion.
Variables a; a1 ; : : : range over A and b; b1 ; : : : range over B.
(a) 8a 8b: a R b. (1), (4)

(b) NOT.8a 8b: a R b/.

(d) 8a 9b: a R b.

(e) 8b 9a: a R b.

(f) R is a bijection.
V
(g) 8a 9b1 a R b1 8b: a R b IMPLIES b D b1 .

(h) 8a; b: a R b OR a ¤ b.

(i) 8b1 ; b2 ; a: .a R b1 AND a R b2 / IMPLIES b1 D b2 .
10 The official definition of inj is with a total injective relation (Œ 1 out; 1 in)
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4.5. Finite Cardinality 119

(j) 8a1 ; a2 ; b: .a1 R b AND a2 R b/ IMPLIES a1 D a2 .

(k) 8a1 ; a2 ; b1 ; b2 : .a1 R b1 AND a2 R b2 AND a1 ¤ a2 / IMPLIES b1 ¤ b2 .

(l) 8a1 ; a2 ; b1 ; b2 : .a1 R b1 AND a2 R b2 AND b1 ¤ b2 / IMPLIES a1 ¤ a2 .

Homework Problems
Problem 4.24.
Let f W A ! B and g W B ! C be functions.
(a) Prove that if the composition g ı f is a bijection, then f is a total injection
and g is a surjection.

(b) Show there is a total injection f and a bijection, g, such that g ı f is not a
bijection.

Problem 4.25.
Let A, B and C be nonempty sets, and let f W B ! C and g W A ! B be
functions. Let h WWD f ı g be the composition function of f and g, namely, the
function with domain A and codomain C such that h.x/ D f .g.x//.
(a) Prove that if h is surjective and f is total and injective, then g must be surjec-
tive.
Hint: contradiction.

(b) Suppose that h is injective and f is total. Prove that g must be injective and
provide a counterexample showing how this claim could fail if f was not total.

Problem 4.26.
Let A, B and C be sets, and let f W B ! C and g W A ! B be functions. Let
h W A ! C be the composition f ı g; that is, h.x/ WWD f .g.x// for x 2 A. Prove
or disprove the following claims:
(a) If h is surjective, then f must be surjective.

(b) If h is surjective, then g must be surjective.

(d) If h is injective and f is total, then g must be injective.
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120 Chapter 4 Mathematical Data Types

Problem 4.27.
Let R be a binary relation on a set D. Let x; y be variables ranging over D. Indicate
the expressions below whose meaning is that R is an injection Œ 1 in. Remember
R is a not necessarily total or a function.

1. R.x/ D R.y/ IMPLIES x D y

2. R.x/ \ R.y/ D ; IMPLIES x ¤ y

3. R.x/ \ R.y/ ¤ ; IMPLIES x ¤ y

4. R.x/ \ R.y/ ¤ ; IMPLIES x D y

5. R 1 .R.x// D fxg

6. R 1 .R.x// fxg

7. R 1 .R.x// fxg

8. R.R 1 .x// Dx

Problem 4.28.
The language of sets and relations may seem remote from the practical world of
programming, but in fact there is a close connection to relational databases, a
very popular software application building block implemented by such software
packages as MySQL. This problem explores the connection by considering how to
manipulate and analyze a large data set using operators over sets and relations. Sys-
tems like MySQL are able to execute very similar high-level instructions efficiently
on standard computer hardware, which helps programmers focus on high-level de-
sign.
Consider a basic Web search engine, which stores information on Web pages and
processes queries to find pages satisfying conditions provided by users. At a high
level, we can formalize the key information as:

A set P of pages that the search engine knows about

A binary relation L (for link) over pages, defined such that p1 L p2 iff page
p1 links to p2

A set E of endorsers, people who have recorded their opinions about which
pages are high-quality
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4.5. Finite Cardinality 121

A binary relation R (for recommends) between endorsers and pages, such
that e R p iff person e has recommended page p

A set W of words that may appear on pages

A binary relation M (for mentions) between pages and words, where p M w
iff word w appears on page p

Each part of this problem describes an intuitive, informal query over the data,
and your job is to produce a single expression using the standard set and relation
operators, such that the expression can be interpreted as answering the query cor-
rectly, for any data set. Your answers should use only the set and relation symbols
given above, in addition to terms standing for constant elements of E or W , plus
the following operators introduced in the text:

set union [.

set intersection \.

set difference .

relational image—for example, R.A/ for some set A, or R.a/ for some spe-
cific element a.

relational inverse 1.

. . . and one extra: relational composition which generalizes composition of
functions
a .R ı S / c WWD 9b 2 B: .a S b/ AND .b R c/:
In other words, a is related to c in R ı S if starting at a you can follow an S
arrow to the start of an R arrow and then follow the R arrow to get to c.11

Here is one worked example to get you started:

Search description: The set of pages containing the word “logic”

Solution expression: M 1 .“logic”/

Find similar solutions for each of the following searches:
(a) The set of pages containing the word “logic” but not the word “predicate”
11 Note the reversal of R and S in the definition; this is to make relational composition work like
function composition. For functions, f ı g means you apply g first. That is, if we let h be f ı g,
then h.x/ D f .g.x//.
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122 Chapter 4 Mathematical Data Types

(b) The set of pages containing the word “set” that have been recommended by
“Meyer”

(d) The relation that relates endorser e and word w iff e has recommended a page
containing w

(e) The set of pages that have at least one incoming or outgoing link

(f) The relation that relates word w and page p iff w appears on a page that links
to p

(g) The relation that relates word w and endorser e iff w appears on a page that
links to a page that e recommends

(h) The relation that relates pages p1 and p2 iff p2 can be reached from p1 by
following a sequence of exactly 3 links

Exam Problems
Problem 4.29.
Let A be the set containing the five sets: fag; fb; cg; fb; d g; fa; eg; fe; f g, and let
B be the set containing the three sets: fa; bg; fb; c; d g; fe; f g. Let R be the “is
subset of” binary relation from A to B defined by the rule:

XRY IFF X Y:

(a) Fill in the arrows so the following figure describes the graph of the relation,
R:
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4.5. Finite Cardinality 123

A arrows B

fag

fa; bg

fb; cg

fb; c; d g

fb; d g

fe; f g

fa; eg

fe; f g

(b) Circle the properties below possessed by the relation R:

function total injective surjective bijective

Problem 4.30. (a) Five assertions about a binary relation R W A ! B are bulleted
below. There are nine predicate formulas that express some of these assertions.
Write the numbers of the formulas next to the assertions they express. For example,
you should write “4” next to the last assertion, since formula (4) expresses the
assertion that R is the identity relation.
Variables a; a1 ; : : : range over the domain A and b; b1 ; : : : range over the codomain
B. More than one formula may express one assertion.

R is a surjection
R is an injection
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124 Chapter 4 Mathematical Data Types

R is a function
R is total
R is the identity relation.

1. 8b: 9a: a R b.
2. 8a: 9b: a R b.
3. 8a: a R a.
4. 8a; b: a R b IFF a D b.
5. 8a; b: a R b OR a ¤ b.
6. 8b1 ; b2 ; a: .a R b1 AND a R b2 / IMPLIES b1 D b2 .
7. 8a1 ; a2 ; b: .a1 R b AND a2 R b/ IMPLIES a1 D a2 .
8. 8a1 ; a2 ; b1 ; b2 : .a1 R b1 AND a2 R b2 AND a1 ¤ a2 / IMPLIES b1 ¤ b2 .
9. 8a1 ; a2 ; b1 ; b2 : .a1 R b1 AND a2 R b2 AND b1 ¤ b2 / IMPLIES a1 ¤ a2 .

(b) Give an example of a relation R that satisfies three of the properties surjection,
injection, total, and function (you indicate which) but is not a bijection.

Problem 4.31.
Prove that if relation R W A ! B is a total injection, Œ 1 out; Œ 1 in, then
1
R ı R D IdA ;

where IdA is the identity function on A.
(A simple argument in terms of ”arrows” will do the job.)

Problem 4.32.
Let R W A ! B be a binary relation.
(a) Prove that R is a function iff R ı R 1 IdB .
Write similar containment formulas involving R 1 ıR, RıR 1 , Ida , IdB equivalent
to the assertion that R has each of the following properties. No proof is required.
(b) total.

(d) a injection.
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4.5. Finite Cardinality 125

Problem 4.33.
Let R W A ! B and S W B ! C be binary relations such that S ı R is a bijection
and jAj D 2.
Give an example of such R; S where neither R nor S is a function. Indicate ex-
actly which properties—total, surjection, function, and injection—your examples
of R and S have.
Hint: Let jBj D 4.

Problem 4.34.
The set f1; 2; 3g! consists of the infinite sequences of the digits 1,2, and 3, and
likewise f4; 5g! is the set of infinite sequences of the digits 4,5. For example
123123123 : : : 2 f1; 2; 3g! ;
222222222222 : : : 2 f1; 2; 3g! ;
4554445554444 : : : 2 f4; 5g! :
(a) Give an example of a total injective function
f W f1; 2; 3g! ! f4; 5g! :

(b) Give an example of a bijection g W .f1; 2; 3g! f1; 2; 3g! / ! f1; 2; 3g! .

(c) Explain why there is a bijection between f1; 2; 3g! f1; 2; 3g! and f4; 5g! .
(You need not explicitly define the bijection.)

Problems for Section 4.5
Practice Problems
Problem 4.35.
Assume f W A ! B is total function, and A is finite. Replace the ? with one of
; D; to produce the strongest correct version of the following statements:
(a) jf .A/j ? jBj.

(b) If f is a surjection, then jAj ? jBj.

(d) If f is an injection, then jf .A/j ? jAj.

(e) If f is a bijection, then jAj ? jBj.
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126 Chapter 4 Mathematical Data Types

Class Problems
Problem 4.36.
Let A D fa0 ; a1 ; : : : ; an 1 g be a set of size n, and B D fb0 ; b1 ; : : : ; bm 1 g a set
of size m. Prove that jA Bj D mn by defining a simple bijection from A B to
the nonnegative integers from 0 to mn 1.

Problem 4.37.
Let R W A ! B be a binary relation. Use an arrow counting argument to prove the
following generalization of the Mapping Rule 1.

Lemma. If R is a function, and X A, then

jXj jR.X /j:
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5 Induction
Induction is a powerful method for showing a property is true for all nonnegative
integers. Induction plays a central role in discrete mathematics and computer sci-
ence. In fact, its use is a defining characteristic of discrete—as opposed to contin-
uous—mathematics. This chapter introduces two versions of induction, Ordinary
and Strong, and explains why they work and how to use them in proofs. It also
introduces the Invariant Principle, which is a version of induction specially adapted
for reasoning about step-by-step processes.

5.1 Ordinary Induction
To understand how induction works, suppose there is a professor who brings a
bottomless bag of assorted miniature candy bars to her large class. She offers to
share the candy in the following way. First, she lines the students up in order. Next
she states two rules:

1. The student at the beginning of the line gets a candy bar.

2. If a student gets a candy bar, then the following student in line also gets a
candy bar.

Let’s number the students by their order in line, starting the count with 0, as usual
in computer science. Now we can understand the second rule as a short description
of a whole sequence of statements:

If student 0 gets a candy bar, then student 1 also gets one.

If student 1 gets a candy bar, then student 2 also gets one.

If student 2 gets a candy bar, then student 3 also gets one.
::
:

Of course, this sequence has a more concise mathematical description:

If student n gets a candy bar, then student n C 1 gets a candy bar, for
all nonnegative integers n.
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128 Chapter 5 Induction

So suppose you are student 17. By these rules, are you entitled to a miniature candy
bar? Well, student 0 gets a candy bar by the first rule. Therefore, by the second
rule, student 1 also gets one, which means student 2 gets one, which means student
3 gets one as well, and so on. By 17 applications of the professor’s second rule,
you get your candy bar! Of course the rules really guarantee a candy bar to every
student, no matter how far back in line they may be.

5.1.1 A Rule for Ordinary Induction
The reasoning that led us to conclude that every student gets a candy bar is essen-
tially all there is to induction.

The Induction Principle.
Let P be a predicate on nonnegative integers. If

P .0/ is true, and

P .n/ IMPLIES P .n C 1/ for all nonnegative integers n

then

P .m/ is true for all nonnegative integers m.

Since we’re going to consider several useful variants of induction in later sec-
tions, we’ll refer to the induction method described above as ordinary induction
when we need to distinguish it. Formulated as a proof rule as in Section 1.4.1, this
would be

Rule. Induction Rule
P .0/; 8n 2 N: P .n/ IMPLIES P .n C 1/
8m 2 N: P .m/

This Induction Rule works for the same intuitive reason that all the students get
candy bars, and we hope the explanation using candy bars makes it clear why the
soundness of ordinary induction can be taken for granted. In fact, the rule is so
obvious that it’s hard to see what more basic principle could be used to justify it.1
What’s not so obvious is how much mileage we get by using it.
1 But see Section 5.3.
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5.1. Ordinary Induction 129

5.1.2 A Familiar Example
Below is the formula (5.1) for the sum of the nonnegative integers up to n. The
formula holds for all nonnegative integers, so it is the kind of statement to which
induction applies directly. We’ve already proved this formula using the Well Or-
dering Principle (Theorem 2.2.1), but now we’ll prove it by induction, that is, using
the Induction Principle.
Theorem 5.1.1. For all n 2 N,
n.n C 1/
1 C 2 C 3 C C n D (5.1)
2
To prove the theorem by induction, define predicate P .n/ to be the equation (5.1).
Now the theorem can be restated as the claim that P .n/ is true for all n 2 N. This
is great, because the Induction Principle lets us reach precisely that conclusion,
provided we establish two simpler facts:
P .0/ is true.
For all n 2 N, P .n/ IMPLIES P .n C 1/.
So now our job is reduced to proving these two statements.
The first statement follows because of the convention that a sum of zero terms
is equal to 0. So P .0/ is the true assertion that a sum of zero terms is equal to
0.0 C 1/=2 D 0.
The second statement is more complicated. But remember the basic plan from
Section 1.5 for proving the validity of any implication: assume the statement on
the left and then prove the statement on the right. In this case, we assume P .n/—
namely, equation (5.1)—in order to prove P .n C 1/, which is the equation
.n C 1/.n C 2/
1 C 2 C 3 C C n C .n C 1/ D : (5.2)
2
These two equations are quite similar; in fact, adding .n C 1/ to both sides of
equation (5.1) and simplifying the right side gives the equation (5.2):
n.n C 1/
1 C 2 C 3 C C n C .n C 1/ D C .n C 1/
2
.n C 2/.n C 1/
D
2
Thus, if P .n/ is true, then so is P .n C 1/. This argument is valid for every non-
negative integer n, so this establishes the second fact required by the induction
proof. Therefore, the Induction Principle says that the predicate P .m/ is true for
all nonnegative integers, m. The theorem is proved.
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130 Chapter 5 Induction

5.1.3 A Template for Induction Proofs
The proof of equation (5.1) was relatively simple, but even the most complicated
induction proof follows exactly the same template. There are five components:

1. State that the proof uses induction. This immediately conveys the overall
structure of the proof, which helps your reader follow your argument.

2. Define an appropriate predicate P .n/. The predicate P .n/ is called the
induction hypothesis. The eventual conclusion of the induction argument
will be that P .n/ is true for all nonnegative n. A clearly stated induction
hypothesis is often the most important part of an induction proof, and its
omission is the largest source of confused proofs by students.
In the simplest cases, the induction hypothesis can be lifted straight from the
proposition you are trying to prove, as we did with equation (5.1). Sometimes
the induction hypothesis will involve several variables, in which case you
should indicate which variable serves as n.

3. Prove that P .0/ is true. This is usually easy, as in the example above. This
part of the proof is called the base case or basis step.

4. Prove that P .n/ implies P .n C 1/ for every nonnegative integer n. This
is called the inductive step. The basic plan is always the same: assume that
P .n/ is true and then use this assumption to prove that P .n C 1/ is true.
These two statements should be fairly similar, but bridging the gap may re-
quire some ingenuity. Whatever argument you give must be valid for every
nonnegative integer n, since the goal is to prove that all the following impli-
cations are true:

P .0/ ! P .1/; P .1/ ! P .2/; P .2/ ! P .3/; : : : :

5. Invoke induction. Given these facts, the induction principle allows you to
conclude that P .n/ is true for all nonnegative n. This is the logical capstone
to the whole argument, but it is so standard that it’s usual not to mention it
explicitly.

Always be sure to explicitly label the base case and the inductive step. Doing
so will make your proofs clearer and will decrease the chance that you forget a key
step—like checking the base case.
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5.1. Ordinary Induction 131

5.1.4 A Clean Writeup
The proof of Theorem 5.1.1 given above is perfectly valid; however, it contains a
lot of extraneous explanation that you won’t usually see in induction proofs. The
writeup below is closer to what you might see in print and should be prepared to
produce yourself.

Revised proof of Theorem 5.1.1. We use induction. The induction hypothesis P .n/
will be equation (5.1).
Base case: P .0/ is true, because both sides of equation (5.1) equal zero when
n D 0.
Inductive step: Assume that P .n/ is true, that is equation (5.1) holds for some
nonnegative integer n. Then adding n C 1 to both sides of the equation implies that
n.n C 1/
1 C 2 C 3 C C n C .n C 1/ D C .n C 1/
2
.n C 1/.n C 2/
D (by simple algebra)
2
which proves P .n C 1/.
So it follows by induction that P .n/ is true for all nonnegative n.

It probably bothers you that induction led to a proof of this summation formula
but did not provide an intuitive way to understand it nor did it explain where the
formula came from in the first place.2 This is both a weakness and a strength. It is a
weakness when a proof does not provide insight. But it is a strength that a proof can
provide a reader with a reliable guarantee of correctness without requiring insight.

5.1.5 A More Challenging Example
During the development of MIT’s famous Stata Center, as costs rose further and
further beyond budget, some radical fundraising ideas were proposed. One rumored
plan was to install a big square courtyard divided into unit squares. The big square
would be 2n units on a side for some undetermined nonnegative integer n, and
one of the unit squares in the center3 occupied by a statue of a wealthy potential
donor—whom the fund raisers privately referred to as “Bill.” The n D 3 case is
shown in Figure 5.1.
A complication was that the building’s unconventional architect, Frank Gehry,
was alleged to require that only special L-shaped tiles (shown in Figure 5.2) be
2 Methods for finding such formulas are covered in Part III of the text.
3 In the special case n D 0, the whole courtyard consists of a single central square; otherwise,
there are four central squares.
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132 Chapter 5 Induction

Figure 5.1 A 2n 2n courtyard for n D 3.

Figure 5.2 The special L-shaped tile.

used for the courtyard. For n D 2, a courtyard meeting these constraints is shown
in Figure 5.3. But what about for larger values of n? Is there a way to tile a 2n 2n
courtyard with L-shaped tiles around a statue in the center? Let’s try to prove that
this is so.

Theorem 5.1.2. For all n 0 there exists a tiling of a 2n 2n courtyard with Bill
in a central square.

Proof. (doomed attempt) The proof is by induction. Let P .n/ be the proposition
that there exists a tiling of a 2n 2n courtyard with Bill in the center.
Base case: P .0/ is true because Bill fills the whole courtyard.
Inductive step: Assume that there is a tiling of a 2n 2n courtyard with Bill in the
center for some n 0. We must prove that there is a way to tile a 2nC1 2nC1
courtyard with Bill in the center . . . .

Now we’re in trouble! The ability to tile a smaller courtyard with Bill in the
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5.1. Ordinary Induction 133

Figure 5.3 A tiling using L-shaped tiles for n D 2 with Bill in a center square.

center isn’t much help in tiling a larger courtyard with Bill in the center. We haven’t
figured out how to bridge the gap between P .n/ and P .n C 1/.
So if we’re going to prove Theorem 5.1.2 by induction, we’re going to need some
other induction hypothesis than simply the statement about n that we’re trying to
prove.
When this happens, your first fallback should be to look for a stronger induction
hypothesis; that is, one which implies your previous hypothesis. For example,
we could make P .n/ the proposition that for every location of Bill in a 2n 2n
courtyard, there exists a tiling of the remainder.
This advice may sound bizarre: “If you can’t prove something, try to prove some-
thing grander!” But for induction arguments, this makes sense. In the inductive
step, where you have to prove P .n/ IMPLIES P .n C 1/, you’re in better shape
because you can assume P .n/, which is now a more powerful statement. Let’s see
how this plays out in the case of courtyard tiling.

Proof (successful attempt). The proof is by induction. Let P .n/ be the proposition
that for every location of Bill in a 2n 2n courtyard, there exists a tiling of the
remainder.
Base case: P .0/ is true because Bill fills the whole courtyard.
Inductive step: Assume that P .n/ is true for some n 0; that is, for every location
of Bill in a 2n 2n courtyard, there exists a tiling of the remainder. Divide the
2nC1 2nC1 courtyard into four quadrants, each 2n 2n . One quadrant contains
Bill (B in the diagram below). Place a temporary Bill (X in the diagram) in each of
the three central squares lying outside this quadrant as shown in Figure 5.4.
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134 Chapter 5 Induction

B
2n

X
X X

2n 2n

Figure 5.4 Using a stronger inductive hypothesis to prove Theorem 5.1.2.

Now we can tile each of the four quadrants by the induction assumption. Replac-
ing the three temporary Bills with a single L-shaped tile completes the job. This
proves that P .n/ implies P .n C 1/ for all n 0. Thus P .m/ is true for all m 2 N,
and the theorem follows as a special case where we put Bill in a central square.

This proof has two nice properties. First, not only does the argument guarantee
that a tiling exists, but also it gives an algorithm for finding such a tiling. Second,
we have a stronger result: if Bill wanted a statue on the edge of the courtyard, away
from the pigeons, we could accommodate him!
Strengthening the induction hypothesis is often a good move when an induction
proof won’t go through. But keep in mind that the stronger assertion must actually
be true; otherwise, there isn’t much hope of constructing a valid proof. Sometimes
finding just the right induction hypothesis requires trial, error, and insight. For
example, mathematicians spent almost twenty years trying to prove or disprove
the conjecture that every planar graph is 5-choosable.4 Then, in 1994, Carsten
Thomassen gave an induction proof simple enough to explain on a napkin. The
key turned out to be finding an extremely clever induction hypothesis; with that in
hand, completing the argument was easy!
4 5-choosabilityis a slight generalization of 5-colorability. Although every planar graph is 4-
colorable and therefore 5-colorable, not every planar graph is 4-choosable. If this all sounds like
nonsense, don’t panic. We’ll discuss graphs, planarity, and coloring in Part II of the text.
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5.1. Ordinary Induction 135

5.1.6 A Faulty Induction Proof
If we have done a good job in writing this text, right about now you should be
thinking, “Hey, this induction stuff isn’t so hard after all—just show P .0/ is true
and that P .n/ implies P .n C 1/ for any number n.” And, you would be right,
although sometimes when you start doing induction proofs on your own, you can
run into trouble. For example, we will now use induction to “prove” that all horses
are the same color—just when you thought it was safe to skip class and work on
your robot program instead. Sorry!
False Theorem. All horses are the same color.
Notice that no n is mentioned in this assertion, so we’re going to have to re-
formulate it in a way that makes an n explicit. In particular, we’ll (falsely) prove
that
False Theorem 5.1.3. In every set of n 1 horses, all the horses are the same
color.
This is a statement about all integers n 1 rather 0, so it’s natural to use a
slight variation on induction: prove P .1/ in the base case and then prove that P .n/
implies P .n C 1/ for all n 1 in the inductive step. This is a perfectly valid variant
of induction and is not the problem with the proof below.

Bogus proof. The proof is by induction on n. The induction hypothesis P .n/ will
be
In every set of n horses, all are the same color. (5.3)

Base case: (n D 1). P .1/ is true, because in a size-1 set of horses, there’s only one
horse, and this horse is definitely the same color as itself.
Inductive step: Assume that P .n/ is true for some n 1. That is, assume that in
every set of n horses, all are the same color. Now suppose we have a set of n C 1
horses:
h1 ; h2 ; : : : ; hn ; hnC1 :
We need to prove these n C 1 horses are all the same color.
By our assumption, the first n horses are the same color:
h ; h ; : : : ; hn ; hnC1
„1 2ƒ‚ …
same color

Also by our assumption, the last n horses are the same color:
h1 ; h2 ; : : : ; hn ; hnC1
„ ƒ‚ …
same color
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136 Chapter 5 Induction

So h1 is the same color as the remaining horses besides hnC1 —that is, h2 ; : : : ; hn .
Likewise, hnC1 is the same color as the remaining horses besides h1 —that is,
h2 ; : : : ; hn , again. Since h1 and hnC1 are the same color as h2 ; : : : ; hn , all n C 1
horses must be the same color, and so P .n C 1/ is true. Thus, P .n/ implies
P .n C 1/.
By the principle of induction, P .n/ is true for all n 1.

We’ve proved something false! Does this mean that math broken and we should
all take up poetry instead? Of course not! It just means that this proof has a mistake.
The mistake in this argument is in the sentence that begins “So h1 is the same
color as the remaining horses besides hnC1 —that is h2 ; : : : ; hn ; : : : .” The ellipis
notation (“: : : ”) in the expression “h1 ; h2 ; : : : ; hn ; hnC1 ” creates the impression
that there are some remaining horses—namely h2 ; : : : ; hn —besides h1 and hnC1 .
However, this is not true when n D 1. In that case, h1 ; h2 ; : : : ; hn ; hnC1 is just
h1 ; h2 and there are no “remaining” horses for h1 to share a color with. And of
course, in this case h1 and h2 really don’t need to be the same color.
This mistake knocks a critical link out of our induction argument. We proved
P .1/ and we correctly proved P .2/ ! P .3/, P .3/ ! P .4/, etc. But we failed
to prove P .1/ ! P .2/, and so everything falls apart: we cannot conclude that
P .2/, P .3/, etc., are true. And naturally, these propositions are all false; there are
sets of n horses of different colors for all n 2.
Students sometimes explain that the mistake in the proof is because P .n/ is false
for n 2, and the proof assumes something false, P .n/, in order to prove P .nC1/.
You should think about how to help such a student understand why this explanation
would get no credit on a Math for Computer Science exam.

5.2 Strong Induction
A useful variant of induction is called strong induction. Strong induction and ordi-
nary induction are used for exactly the same thing: proving that a predicate is true
for all nonnegative integers. Strong induction is useful when a simple proof that
the predicate holds for n C 1 does not follow just from the fact that it holds at n,
but from the fact that it holds for other values n.
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5.2. Strong Induction 137

5.2.1 A Rule for Strong Induction

Principle of Strong Induction.
Let P be a predicate on nonnegative integers. If

P .0/ is true, and

for all n 2 N, P .0/, P .1/, . . . , P .n/ together imply P .n C 1/,

then P .m/ is true for all m 2 N.

The only change from the ordinary induction principle is that strong induction
allows you make more assumptions in the inductive step of your proof! In an
ordinary induction argument, you assume that P .n/ is true and try to prove that
P .n C 1/ is also true. In a strong induction argument, you may assume that P .0/,
P .1/, . . . , and P .n/ are all true when you go to prove P .nC1/. So you can assume
a stronger set of hypotheses which can make your job easier.
Formulated as a proof rule, strong induction is

Rule. Strong Induction Rule

P .0/; 8n 2 N: P .0/ AND P .1/ AND : : : AND P .n/ IMPLIES P .n C 1/
8m 2 N: P .m/

Stated more succintly, the rule is

Rule.
P .0/; Œ8k n 2 N: P .k/ IMPLIES P .n C 1/
8m 2 N: P .m/
The template for strong induction proofs is identical to the template given in
Section 5.1.3 for ordinary induction except for two things:

you should state that your proof is by strong induction, and

you can assume that P .0/, P .1/, . . . , P .n/ are all true instead of only P .n/
during the inductive step.

5.2.2 Fibonacci numbers
The numbers that bear his name arose out of the Italian mathematician Fibonacci’s
models of population growth at the beginning of the thirteenth century. Fibonacci
numbers turn out to describe the growth of lots of interesting biological quantities
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138 Chapter 5 Induction

such as the shape of pineapple sprouts or pine cones, and they also come up regu-
larly in Computer Science where they describe the growth of various data structures
and computation times of algorithms.
To generate the list of successive Fibonacci numbers, you start by writing 0; 1
and then keep adding another element to the list by summing the two previous
ones:
0; 1; 1; 2; 3; 5; 8; 13; 21; : : : :
Another way to describe this process is to define nth Fibonacci number F .n/ by
the equations:

F .0/ WWD 0;
F .1/ WWD 1;
F .n/ WWD F .n 1/ C F .n 2/ for n 2.

Note that because the general rule for finding the Fibonacci F .n/ refers to the two
previous values F .n 1/ and F .n 2/, we needed to know the two values F .0/
and F .1/ in order to get started.
One simple property of Fibonacci numbers is that the even/odd pattern of Fi-
bonacci numbers repeats in a cycle of length three. A nice way to say this is that
for all n 0,
F .n/ is even IFF F .n C 3/ is even: (5.4)
We will verify the equivalence (5.4) by induction, but strong induction is called for
because properties of F .n/ depend not just on F .n 1/ but also on F .n 2/.

Proof. The (strong) induction hypothesis P .n/ will be (5.4).
Base cases:

(n D 0). F .0/ D 0 and F .3/ D 2 are both even.

(n D 1). F .1/ D 1 and F .4/ D 3 are both not even.

Induction step: For n 1, we want to prove P .n C 1/ is true assuming by strong
induction that P .n/ and P .n 1/ are true.
Now it is easy to verify that for all integers k; m,

m C k is even IFF Œm is even IFF k is even: (*)
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5.2. Strong Induction 139

So for n 1,

F .n C 1/ is even
IFF F .n/ C F .n 1/ is even (def of F .n C 1/)
IFF ŒF .n/ is even IFF F .n 1/ is even (by (*))
IFF ŒF .n C 3/ is even IFF F .n C 2/ is even
(by strong ind. hyp. P .n/; P .n 1/)
IFF F .n C 3/ C F .n C 2/ is even (by (*))
IFF F .n C 4/ is even (by def of F .n C 4/):

This shows that
F .n C 1/ is even IFF F .n C 4/ is even;
which means that P .n C 1/ is true, as required.

There is a long standing community of Fibonacci number enthusiasts who have
been captivated by the many extraordinary properties of these number—a few fur-
ther illustrative properties appear in Problems 5.8, 5.25, and 5.30.

5.2.3 Products of Primes
We can use strong induction to re-prove Theorem 2.3.1 which we previously proved
using Well Ordering.

Theorem. Every integer greater than 1 is a product of primes.

Proof. We will prove the Theorem by strong induction, letting the induction hy-
pothesis P .n/ be
n is a product of primes:
So the Theorem will follow if we prove that P .n/ holds for all n 2.
Base Case: (n D 2): P .2/ is true because 2 is prime, so it is a length one product
of primes by convention.
Inductive step: Suppose that n 2 and that every number from 2 to n is a product
of primes. We must show that P .n C 1/ holds, namely, that n C 1 is also a product
of primes. We argue by cases:
If n C 1 is itself prime, then it is a length one product of primes by convention,
and so P .n C 1/ holds in this case.
Otherwise, n C 1 is not prime, which by definition means n C 1 D k m for some
integers k; m between 2 and n. Now by the strong induction hypothesis, we know
that both k and m are products of primes. By multiplying these products, it follows
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140 Chapter 5 Induction

Figure 5.5 One way to make 26 Sg using Strongian currency

immediately that k m D n C 1 is also a product of primes. Therefore, P .n C 1/
holds in this case as well.
So P .n C 1/ holds in any case, which completes the proof by strong induction
that P .n/ holds for all n 2.

5.2.4 Making Change
The country Inductia, whose unit of currency is the Strong, has coins worth 3Sg
(3 Strongs) and 5Sg. Although the Inductians have some trouble making small
change like 4Sg or 7Sg, it turns out that they can collect coins to make change for
any number that is at least 8 Strongs.
Strong induction makes this easy to prove for n C 1 11, because then .n C
1/ 3 8, so by strong induction the Inductians can make change for exactly
.n C 1/ 3 Strongs, and then they can add a 3Sg coin to get .n C 1/Sg. So the only
thing to do is check that they can make change for all the amounts from 8 to 10Sg,
which is not too hard to do.
Here’s a detailed writeup using the official format:

Proof. We prove by strong induction that the Inductians can make change for any
amount of at least 8Sg. The induction hypothesis P .n/ will be:

There is a collection of coins whose value is n C 8 Strongs.

We now proceed with the induction proof:
Base case: P .0/ is true because a 3Sg coin together with a 5Sg coin makes 8Sg.
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5.2. Strong Induction 141

Inductive step: We assume P .k/ holds for all k n, and prove that P .n C 1/
holds. We argue by cases:
Case (n C 1 = 1): We have to make .n C 1/ C 8 D 9Sg. We can do this using
three 3Sg coins.
Case (n C 1 = 2): We have to make .n C 1/ C 8 D 10Sg. Use two 5Sg coins.
Case (n C 1 3): Then 0 n 2 n, so by the strong induction hypothesis,
the Inductians can make change for .n 2/ C 8Sg. Now by adding a 3Sg coin, they
can make change for .n C 1/ C 8Sg, so P .n C 1/ holds in this case.
Since n 0, we know that n C 1 1 and thus that the three cases cover
every possibility. Since P .n C 1/ is true in every case, we can conclude by strong
induction that for all n 0, the Inductians can make change for n C 8 Strong. That
is, they can make change for any number of eight or more Strong.

5.2.5 The Stacking Game
Here is another exciting game that’s surely about to sweep the nation!
You begin with a stack of n boxes. Then you make a sequence of moves. In each
move, you divide one stack of boxes into two nonempty stacks. The game ends
when you have n stacks, each containing a single box. You earn points for each
move; in particular, if you divide one stack of height a C b into two stacks with
heights a and b, then you score ab points for that move. Your overall score is the
sum of the points that you earn for each move. What strategy should you use to
maximize your total score?
As an example, suppose that we begin with a stack of n D 10 boxes. Then the
game might proceed as shown in Figure 5.6. Can you find a better strategy?

Analyzing the Game
Let’s use strong induction to analyze the unstacking game. We’ll prove that your
score is determined entirely by the number of boxes—your strategy is irrelevant!

Theorem 5.2.1. Every way of unstacking n blocks gives a score of n.n 1/=2
points.

There are a couple technical points to notice in the proof:

The template for a strong induction proof mirrors the one for ordinary induc-
tion.

As with ordinary induction, we have some freedom to adjust indices. In this
case, we prove P .1/ in the base case and prove that P .1/; : : : ; P .n/ imply
P .n C 1/ for all n 1 in the inductive step.
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142 Chapter 5 Induction

Stack Heights Score
10
5 5 25 points
5 3 2 6
4 3 2 1 4
2 3 2 1 2 4
2 2 2 1 2 1 2
1 2 2 1 2 1 1 1
1 1 2 1 2 1 1 1 1
1 1 1 1 2 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
Total Score D 45 points

Figure 5.6 An example of the stacking game with n D 10 boxes. On each line,
the underlined stack is divided in the next step.

Proof. The proof is by strong induction. Let P .n/ be the proposition that every
way of unstacking n blocks gives a score of n.n 1/=2.
Base case: If n D 1, then there is only one block. No moves are possible, and so
the total score for the game is 1.1 1/=2 D 0. Therefore, P .1/ is true.
Inductive step: Now we must show that P .1/, . . . , P .n/ imply P .n C 1/ for all
n 1. So assume that P .1/, . . . , P .n/ are all true and that we have a stack of
n C 1 blocks. The first move must split this stack into substacks with positive sizes
a and b where a C b D n C 1 and 0 < a; b n. Now the total score for the game
is the sum of points for this first move plus points obtained by unstacking the two
resulting substacks:
total score D (score for 1st move)
C (score for unstacking a blocks)
C (score for unstacking b blocks)
a.a 1/ b.b 1/
D ab C C by P .a/ and P .b/
2 2
.a C b/2 .a C b/ .a C b/..a C b/ 1/
D D
2 2
.n C 1/n
D
2
This shows that P .1/, P .2/, . . . , P .n/ imply P .n C 1/.
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5.3. Strong Induction vs. Induction vs. Well Ordering 143

Therefore, the claim is true by strong induction.

5.3 Strong Induction vs. Induction vs. Well Ordering
Strong induction looks genuinely “stronger” than ordinary induction —after all,
you can assume a lot more when proving the induction step. Since ordinary in-
duction is a special case of strong induction, you might wonder why anyone would
bother with the ordinary induction.
But strong induction really isn’t any stronger, because a simple text manipula-
tion program can automatically reformat any proof using strong induction into a
proof using ordinary induction—just by decorating the induction hypothesis with
a universal quantifier in a standard way. Still, it’s worth distinguishing these two
kinds of induction, since which you use will signal whether the inductive step for
n C 1 follows directly from the case for n or requires cases smaller than n, and that
is generally good for your reader to know.
The template for the two kinds of induction rules looks nothing like the one for
the Well Ordering Principle, but this chapter included a couple of examples where
induction was used to prove something already proved using well ordering. In fact,
this can always be done. As the examples may suggest, any well ordering proof
can automatically be reformatted into an induction proof. So theoretically, no one
need bother with the Well Ordering Principle either.
But it’s equally easy to go the other way, and automatically reformat any strong
induction proof into a Well Ordering proof. The three proof methods—well order-
ing, induction, and strong induction—are simply different formats for presenting
the same mathematical reasoning!
So why three methods? Well, sometimes induction proofs are clearer because
they don’t require proof by contradiction. Also, induction proofs often provide
recursive procedures that reduce large inputs to smaller ones. On the other hand,
well ordering can come out slightly shorter and sometimes seem more natural and
less worrisome to beginners.
So which method should you use? There is no simple recipe. Sometimes the
only way to decide is to write up a proof using more than one method and compare
how they come out. But whichever method you choose, be sure to state the method
up front to help a reader follow your proof.
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144 Chapter 5 Induction

Figure 5.7 Gehry’s new tile.

Problems for Section 5.1
Practice Problems
Problem 5.1.
Prove by induction that every nonempty finite set of real numbers has a minimum
element.

Problem 5.2.
Frank Gehry has changed his mind. Instead of the L-shaped tiles shown in fig-
ure 5.3, he wants to use an odd offset pattern of tiles (or its mirror-image reflection),
as shown in 5.7. To prove this is possible, he uses reasoning similar to the proof
in 5.1.5. However, unlike the proof in the text, this proof is flawed. Which part of
the proof below contains a logical error?

False Claim. The proof is by induction. Let P .n/ be the proposition that for every
location of Bill in a 2n 2n courtyard, there exists a tiling of the remainder with
the offset tile pattern.

False proof. Base case: P .0/ is true because Bill fills the whole courtyard.
Inductive step: Assume that P .n/ is true for some n 0; that is, for every location
of Bill in a 2n 2n courtyard, there exists a tiling of the remainder. Divide the
2nC1 2nC1 courtyard into four quadrants, each 2n 2n . One quadrant contains
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5.3. Strong Induction vs. Induction vs. Well Ordering 145

Figure 5.8 The induction hypothesis for the false theorem.

Bill (B in the diagram below). Place a temporary Bill (X in the diagram) in each of
the three squares lying near this quadrant as shown in Figure 5.8.
We can tile each of the four quadrants by the induction assumption. Replacing
the three temporary Bills with a single offset tile completes the job. This proves
that P .n/ implies P .n C 1/ for all n 0. Thus P .m/ is true for all m 2 N, and the
ability to place Bill in the center of the courtyard follows as a special case where
we put Bill in a central square.

Class Problems
Problem 5.3.
Use induction to prove that
2
n.n C 1/
13 C 23 C C n3 D : (5.5)
2
for all n 1.
Remember to formally
1. Declare proof by induction.

2. Identify the induction hypothesis P .n/.

3. Establish the base case.
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146 Chapter 5 Induction

4. Prove that P .n/ ) P .n C 1/.
5. Conclude that P .n/ holds for all n 1.
as in the five part template.

Problem 5.4.
Prove by induction on n that
r nC1 1
1 C r C r2 C C rn D (5.6)
r 1
for all n 2 N and numbers r ¤ 1.

Problem 5.5.
Prove by induction:
1 1 1 1
1C C C C 2 < 2 ; (5.7)
4 9 n n
for all n > 1.

Problem 5.6. (a) Prove by induction that a 2n 2n courtyard with a 1 1 statue
of Bill in a corner can be covered with L-shaped tiles. (Do not assume or reprove
the (stronger) result of Theorem 5.1.2 that Bill can be placed anywhere. The point
of this problem is to show a different induction hypothesis that works.)

(b) Use the result of part (a) to prove the original claim that there is a tiling with
Bill in the middle.

Problem 5.7.
We’ve proved in two different ways that
n.n C 1/
1 C 2 C 3 C C n D
2
But now we’re going to prove a contradictory theorem!
False Theorem. For all n 0,
n.n C 1/
2 C 3 C 4 C C n D
2
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5.3. Strong Induction vs. Induction vs. Well Ordering 147

Proof. We use induction. Let P .n/ be the proposition that 2 C 3 C 4 C C n D
n.n C 1/=2.
Base case: P .0/ is true, since both sides of the equation are equal to zero. (Recall
that a sum with no terms is zero.)
Inductive step: Now we must show that P .n/ implies P .n C 1/ for all n 0. So
suppose that P .n/ is true; that is, 2 C 3 C 4 C C n D n.n C 1/=2. Then we can
reason as follows:

2 C 3 C 4 C C n C .n C 1/ D Œ2 C 3 C 4 C C n C .n C 1/
n.n C 1/
D C .n C 1/
2
.n C 1/.n C 2/
D
2
Above, we group some terms, use the assumption P .n/, and then simplify. This
shows that P .n/ implies P .n C 1/. By the principle of induction, P .n/ is true for
all n 2 N.

Where exactly is the error in this proof?

Homework Problems
Problem 5.8.
The Fibonacci numbers F .n/ are described in Section 5.2.2.
Prove by induction that for all n 1,

F .n 1/ F .n C 1/ F .n/2 D . 1/n : (5.8)

Problem 5.9.
For any binary string ˛ let num .˛/ be the nonnegative integer it represents in binary
notation. For example, num .10/ D 2, and num .0101/ D 5.
An n C 1-bit adder adds two n C 1-bit binary numbers. More precisely, an
n C 1-bit adder takes two length n C 1 binary strings

˛n WWD an : : : a1 a0 ;
ˇn WWD bn : : : b1 b0 ;

and a binary digit c0 as inputs, and produces a length-(n C 1) binary string

n WWD sn : : : s1 s0 ;
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148 Chapter 5 Induction

and a binary digit cnC1 as outputs, and satisfies the specification:

num .˛n / C num .ˇn / C c0 D 2nC1 cnC1 C num .n / : (5.9)

There is a straighforward way to implement an nC1-bit adder as a digital circuit:
an n C 1-bit ripple-carry circuit has 1 C 2.n C 1/ binary inputs

an ; : : : ; a1 ; a0 ; bn ; : : : ; b1 ; b0 ; c0 ;

and n C 2 binary outputs,
cnC1 ; sn ; : : : ; s1 ; s0 :
As in Problem 3.6, the ripple-carry circuit is specified by the following formulas:

si WWD ai XOR bi XOR ci (5.10)
ci C1 WWD .ai AND bi / OR .ai AND ci / OR .bi AND ci /; : (5.11)

for 0 i n.
(a) Verify that definitions (5.10) and (5.11) imply that

an C bn C cn D 2cnC1 C sn : (5.12)

for all n 2 N.

(b) Prove by induction on n that an n C 1-bit ripple-carry circuit really is an n C 1-
bit adder, that is, its outputs satisfy (5.9).
Hint: You may assume that, by definition of binary representation of integers,

num .˛nC1 / D anC1 2nC1 C num .˛n / : (5.13)

Problem 5.10.
Divided Equilateral Triangles5 (DETs) can be built up as follows:

A single equilateral triangle counts as a DET whose only subtriangle is itself.

If T WWD is a DET, then the equilateral triangle T 0 built out of four
copies of T as shown in in Figure 5.9 is also a DET, and the subtriangles of
T 0 are exactly the subtriangles of each of the copies of T .

5 Adapted from [46].
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5.3. Strong Induction vs. Induction vs. Well Ordering 149

[h]

Figure 5.9 DET T 0 from Four Copies of DET T

[h]

Figure 5.10 Trapezoid from Three Triangles

(a) Define the length of a DET to be the number of subtriangles with an edge on
its base. Prove by induction on length that the total number of subtriangles of a
DET is the square of its length.

(b) Show that a DET with one of its corner subtriangles removed can be tiled with
trapezoids built out of three subtriangles as in Figure 5.10.

Problem 5.11.
The Math for Computer Science mascot, Theory Hippotamus, made a startling
discovery while playing with his prized collection of unit squares over the weekend.
Here is what happened.
First, Theory Hippotamus put his favorite unit square down on the floor as in
Figure 5.11 (a). He noted that the length of the periphery of the resulting shape was
4, an even number. Next, he put a second unit square down next to the first so that
the two squares shared an edge as in Figure 5.11 (b). He noticed that the length
of the periphery of the resulting shape was now 6, which is also an even number.
(The periphery of each shape in the figure is indicated by a thicker line.) Theory
Hippotamus continued to place squares so that each new square shared an edge with
at least one previously-placed square and no squares overlapped. Eventually, he
arrived at the shape in Figure 5.11 (c). He realized that the length of the periphery
of this shape was 36, which is again an even number.
Our plucky porcine pal is perplexed by this peculiar pattern. Use induction on
the number of squares to prove that the length of the periphery is always even, no
matter how many squares Theory Hippotamus places or how he arranges them.
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150 Chapter 5 Induction

(a) (b) (c)

Figure 5.11 Some shapes that Theory Hippotamus created.

Problem 5.12.
Prove the Distributive Law of intersection over the union of n sets by induction:
n
[ n
[
A\ Bi D .A \ Bi /: (5.14)
i D1 i D1

Hint: Theorem 4.1.2 gives the n D 2 case.

Problem 5.13.
Here is an interesting construction of a geometric object known as the Koch snowflake.
Define a sequence of polygons S0 ; S1 recursively, starting with S0 equal to an equi-
lateral triangle with unit sides. We construct SnC1 by removing the middle third
of each edge of Sn and replacing it with two line segments of the same length, as
illustrated in Figure 5.12.
Let an be the area of Sn . Observe that pa0 is just the area of the unit equilateral
triangle which by elementary geometry is 3=4.
Prove by induction that for n 0, the area of the nth snowflake is given by:

8 3 4 n

an D a0 : (5.15)
5 5 9
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5.3. Strong Induction vs. Induction vs. Well Ordering 151

Figure 5.12 S0 ; S1 ; S2 and S3 .

Exam Problems
Problem 5.14.
Prove by induction that
n
X
k kŠ D .n C 1/Š 1: (5.16)
1

Problem 5.15.
Prove by induction:
2
n.n C 1/
03 C 13 C 23 C C n3 D ; 8n 0:
2

using the equation itself as the induction hypothesis P .n/.
(a) Prove the

base case .n D 0/.

(b) Now prove the

inductive step.
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152 Chapter 5 Induction

Problem 5.16.
Suppose P .n/ is a predicate on nonnegative numbers, and suppose

8k: P .k/ IMPLIES P .k C 2/: (5.17)

For P ’s that satisfy (5.17), some of the assertions below Can hold for some,
but not all, such P , other assertions Always hold no matter what the P may be,
and some Never hold for any such P . Indicate which case applies for each of the
assertions and briefly explain why.
(a) 8n 0: P .n/

(b) NOT.P .0// AND 8n 1: P .n/

(d) .8n 100: P .n// AND .8n > 100: NOT.P .n///

(e) .8n 100: NOT.P .n/// AND .8n > 100: P .n//

(f) P .0/ IMPLIES 8n: P .n C 2/

(g) Œ9n: P .2n/ IMPLIES 8n: P .2n C 2/

(h) P .1/ IMPLIES 8n: P .2n C 1/

(i) Œ9n: P .2n/ IMPLIES 8n: P .2n C 2/

(j) 9n: 9m > n: ŒP .2n/ AND NOT.P .2m//

(k) Œ9n: P .n/ IMPLIES 8n: 9m > n: P .m/

(l) NOT.P .0// IMPLIES 8n: NOT.P .2n//

Problem 5.17.
We examine a series of propositional formulas F1 ; F2 ; : : : ; Fn ; : : : containing propo-
sitional variables P1 ; P2 ; : : : ; Pn ; : : : constructed as follows
F1 .P1 / WWD P1
F2 .P1 ; P2 / WWD P1 IMPLIES P2
F3 .P1 ; P2 ; P3 / WWD .P1 IMPLIES P2 / IMPLIES P3
F4 .P1 ; P2 ; P3 ; P4 / WWD ..P1 IMPLIES P2 / IMPLIES P3 / IMPLIES P4
F5 .P1 ; P2 ; P3 ; P4 ; P5 / WWD ...P1 IMPLIES P2 / IMPLIES P3 / IMPLIES P4 / IMPLIES P5
::
:
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5.3. Strong Induction vs. Induction vs. Well Ordering 153

Let Tn be the number of different true/false settings of the variables P1 ; P2 ; : : : ; Pn
for which Fn .P1 ; P2 ; : : : ; Pn / is true. For example, T2 D 3 since F2 .P1 ; P2 / is
true for 3 different settings of the variables P1 and P2 :

P1 P2 F2 .P1 ; P2 /
T T T
T F F
F T T
F F T

(a) Explain why

TnC1 D 2nC1 Tn : (5.18)

(b) Use induction to prove that

2nC1 C . 1/n
Tn D (*)
3
for n 1.

Problem 5.18.
You are given n envelopes, numbered 0; 1; : : : ; n 1. Envelope 0 contains 20 D 1
dollar, Envelope 1 contains 21 D 2 dollars, . . . , and Envelope n 1 contains 2n 1
dollars. Let P .n/ be the assertion that:

For all nonnegative integers k < 2n , there is a subset of the n envelopes
whose contents total to exactly k dollars.

Prove by induction that P .n/ holds for all integers n 1.

Problem 5.19.
Prove by induction that

n.n C 1/.n C 2/
1 2 C 2 3 C 3 4 C C n.n C 1/ D (5.19)
3
for all integers n 1.
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154 Chapter 5 Induction

...

AND-circuit

NOT-gate

Figure 5.13 OR -circuit from AND-circuit.

Problem 5.20.
A k-bit AND-circuit is a digital circuit that has k 0-1 valued inputs6 d0 ; d1 ; : : : ; dk 1
and one 0-1-valued output variable whose value will be
d0 AND d1 AND AND dk 1:

OR -circuitsare defined in the same way, with “OR” replacing “AND.”
(a) Suppose we want an OR-circuit but only have a supply of AND-circuits and
some NOT-gates (“inverters”) that have one 0-1 valued input and one 0-1 valued
output. We can turn an AND-circuit into an OR-circuit by attaching a NOT-gate to
each input of the AND-circuit and also attaching a NOT-gate to the output of the
AND -circuit. This is illustrated in Figure 5.13. Briefly explain why this works.

Large digital circuits are built by connecting together smaller digital circuits as
components. One of the most basic components is a two-input/one-output AND-
gate that produces an output value equal to the AND of its two input values. So
according the definition in part (a), a single AND-gate is a 1-bit AND-circuit.
We can build up larger AND-circuits out of a collection of AND-gates in several
ways. For example, one way to build a 4-bit AND-circuit is to connect three AND-
gates as illustrated in Figure 5.14.
More generally, a depth-n tree-design AND-circuit—“depth-n circuit” for short—
has 2n inputs and is built from two depth-.n 1/ circuits by using the outputs of
the two depth-.n 1/ circuits as inputs to a single AND-gate. This is illustrated in
Figure 5.15. So the 4-bit AND-circuit in Figure 5.14 is a depth-2 circuit. A depth-1
circuit is defined simply to be a single AND-gate.
6 Following the usual conventions for digital circuits, we’re using 1 for the truth value T and 0 for
F.
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5.3. Strong Induction vs. Induction vs. Well Ordering 155

a2 a3

AND AND

AND

Figure 5.14 A 4-bit AND-circuit.

a2n-1-1 a2n-1 a2n-1+1 •.. a2n-1

depth- depth-
n-1 n-1
circuit circuit

AND

Figure 5.15 An n-bit tree-design AND-circuit.
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156 Chapter 5 Induction

(b) Let gate#.n/ be the number of AND-gates in a depth-n circuit. Prove by in-
duction that
gate#.n/ D 2n 1 (5.20)
for all n 1.

Problems for Section 5.2
Practice Problems
Problem 5.21.
Some fundamental principles for reasoning about nonnegative integers are:

1. The Induction Principle,

2. The Strong Induction Principle,

3. The Well Ordering Principle.

Identify which, if any, of the above principles is captured by each of the following
inference rules.
(a)
P .0/; 8m: .8k m: P .k// IMPLIES P .m C 1/
8n: P .n/

(b)
P .b/; 8k b: P .k/ IMPLIES P .k C 1/
8k b: P .k/

(d)
P .0/; 8k > 0: P .k/ IMPLIES P .k C 1/
8n: P .n/

(e)
8m: .8k < m: P .k// IMPLIES P .m/
8n: P .n/
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5.3. Strong Induction vs. Induction vs. Well Ordering 157

Problem 5.22.
The Fibonacci numbers F .n/ are described in Section 5.2.2.
Indicate exactly which sentence(s) in the following bogus proof contain logical
errors? Explain.

False Claim. Every Fibonacci number is even.

Bogus proof. Let all the variables n; m; k mentioned below be nonnegative integer
valued. Let Even.n/ mean that F .n/ is even. The proof is by strong induction with
induction hypothesis Even.n/.
base case: F .0/ D 0 is an even number, so Even.0/ is true.
inductive step: We assume may assume the strong induction hypothesis

Even.k/ for 0 k n;

and we must prove Even.n C 1/.
Then by strong induction hypothesis, Even.n/ and Even.n 1/ are true, that
is, F .n/ and F .n 1/ are both even. But by the definition, F .n C 1/ equals the
sum F .n/ C F .n 1/ of two even numbers, and so it is also even. This proves
Even.n C 1/ as required.
Hence, F .m/ is even for all m 2 N by the Strong Induction Principle.

Problem 5.23.
Alice wants to prove by induction that a predicate P holds for certain nonnegative
integers. She has proven that for all nonnegative integers n D 0; 1; : : :

P .n/ IMPLIES P .n C 3/:
(a) Suppose Alice also proves that P .5/ holds. Which of the following proposi-
tions can she infer?

1. P .n/ holds for all n 5
2. P .3n/ holds for all n 5
3. P .n/ holds for n D 8; 11; 14; : : :
4. P .n/ does not hold for n < 5
5. 8n: P .3n C 5/
6. 8n > 2: P .3n 1/
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158 Chapter 5 Induction

7. P .0/ IMPLIES 8n: P .3n C 2/
8. P .0/ IMPLIES 8n: P .3n/

(b) Which of the following could Alice prove in order to conclude that P .n/ holds
for all n 5?

1. P .0/
2. P .5/
3. P .5/ and P .6/
4. P .0/, P .1/ and P .2/
5. P .5/, P .6/ and P .7/
6. P .2/, P .4/ and P .5/
7. P .2/, P .4/ and P .6/
8. P .3/, P .5/ and P .7/

Problem 5.24.
Prove that every amount of postage of 12 cents or more can be formed using just
4-cent and 5-cent stamps.

Class Problems
Problem 5.25.
The Fibonacci numbers are described in Section 5.2.2.
Prove, using strong induction, the following closed-form formula for the Fi-
bonacci numbers.7
pn qn
F .n/ D p
5
p p
where p D 1C2 5 and q D 1 2 5 .
Hint: Note that p and q are the roots of x 2 x 1 D 0, and so p 2 D p C 1 and
2
q D q C 1.

Problem 5.26.
A sequence of numbers is weakly decreasing when each number in the sequence is
7 This mind-boggling formula is known as Binet’s formula. We’ll explain in Chapter 16, and again
in Chapter 22, how it comes about.
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5.3. Strong Induction vs. Induction vs. Well Ordering 159

the numbers after it. (This implies that a sequence of just one number is weakly
decreasing.)
Here’s a bogus proof of a very important true fact, every integer greater than 1 is
a product of a unique weakly decreasing sequence of primes—a pusp, for short.
Explain what’s bogus about the proof.

Lemma. Every integer greater than 1 is a pusp.

For example, 252 D 7 3 3 2 2, and no other weakly decreasing sequence of
primes will have a product equal to 252.

Bogus proof. We will prove the lemma by strong induction, letting the induction
hypothesis P .n/ be
n is a pusp:
So the lemma will follow if we prove that P .n/ holds for all n 2.
Base Case (n D 2): P .2/ is true because 2 is prime, and so it is a length one
product of primes, and this is obviously the only sequence of primes whose product
can equal 2.
Inductive step: Suppose that n 2 and that i is a pusp for every integer i where
2 i < n C 1. We must show that P .n C 1/ holds, namely, that n C 1 is also a
pusp. We argue by cases:
If n C 1 is itself prime, then it is the product of a length one sequence consisting
of itself. This sequence is unique, since by definition of prime, n C 1 has no other
prime factors. So n C 1 is a pusp, that is P .n C 1/ holds in this case.
Otherwise, n C 1 is not prime, which by definition means n C 1 D km for
some integers k; m such that 2 k; m < n C 1. Now by the strong induction
hypothesis, we know that k and m are pusps. It follows that by merging the unique
prime sequences for k and m, in sorted order, we get a unique weakly decreasing
sequence of primes whose product equals n C 1. So n C 1 is a pusp, in this case as
well.
So P .n C 1/ holds in any case, which completes the proof by strong induction
that P .n/ holds for all n 2.

Problem 5.27.
Define the potential p.S / of a stack of blocks S to be k.k 1/=2 where k is the
number of blocks in S. Define the potential p.A/ of a set of stacks A to be the sum
of the potentials of the stacks in A.
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160 Chapter 5 Induction

Generalize Theorem 5.2.1 about scores in the stacking game to show that for any
set of stacks A if a sequence of moves starting with A leads to another set of stacks
B then p.A/ p.B/, and the score for this sequence of moves is p.A/ p.B/.
Hint: Try induction on the number of moves to get from A to B.

Homework Problems
Problem 5.28.
A group of n 1 people can be divided into teams, each containing either 4 or
7 people. What are all the possible values of n? Use induction to prove that your
answer is correct.

Problem 5.29.
The following Lemma is true, but the proof given for it below is defective. Pin-
point exactly where the proof first makes an unjustified step and explain why it is
unjustified.

Lemma. For any prime p and positive integers n; x1 ; x2 ; : : : ; xn , if p j x1 x2 : : : xn ,
then p j xi for some 1 i n.

Bogus proof. Proof by strong induction on n. The induction hypothesis P .n/ is
that Lemma holds for n.
Base case n D 1: When n D 1, we have p j x1 , therefore we can let i D 1 and
conclude p j xi .
Induction step: Now assuming the claim holds for all k n, we must prove it
for n C 1.
So suppose p j x1 x2 xnC1 . Let yn D xn xnC1 , so x1 x2 xnC1 D x1 x2 xn 1 yn .
Since the right-hand side of this equality is a product of n terms, we have by induc-
tion that p divides one of them. If p j xi for some i < n, then we have the desired
i . Otherwise p j yn . But since yn is a product of the two terms xn ; xnC1 , we have
by strong induction that p divides one of them. So in this case p j xi for i D n or
i D n C 1.

Exam Problems
Problem 5.30.
The Fibonacci numbers F .n/ are described in Section 5.2.2.
These numbers satisfy many unexpected identities, such as

F .0/2 C F .1/2 C C F .n/2 D F .n/F .n C 1/: (5.21)
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5.3. Strong Induction vs. Induction vs. Well Ordering 161

Equation (5.21) can be proved to hold for all n 2 N by induction, using the equation
itself as the induction hypothesis P .n/.
(a) Prove the

base case .n D 0/.

(b) Now prove the

inductive step.

Problem 5.31.
Use strong induction to prove that n 3n=3 for every integer n 0.

Problem 5.32.
A class of any size of 18 or more can be assembled from student teams of sizes 4
and 7. Prove this by induction (of some kind), using the induction hypothesis:

S.n/ WWD a class of n C 18 students can be assembled from teams of sizes 4 and 7:

Problem 5.33.
Any amount of ten or more cents postage that is a multiple of five can be made
using only 10¢ and 15¢ stamps. Prove this by induction (ordinary or strong, but say
which) using the induction hypothesis

S.n/ WWD .5n C 10/¢ postage can be made using only 10¢ and 15¢ stamps:
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6 State Machines
State machines are a simple, abstract model of step-by-step processes. Since com-
puter programs can be understood as defining step-by-step computational processes,
it’s not surprising that state machines come up regularly in computer science. They
also come up in many other settings such as designing digital circuits and mod-
eling probabilistic processes. This section introduces Floyd’s Invariant Principle
which is a version of induction tailored specifically for proving properties of state
machines.
One of the most important uses of induction in computer science involves prov-
ing one or more desirable properties continues to hold at every step in a process.
A property that is preserved through a series of operations or steps is known as a
preserved invariant.
Examples of desirable invariants include properties such as a variable never ex-
ceeding a certain value, the altitude of a plane never dropping below 1,000 feet
without the wingflaps being deployed, and the temperature of a nuclear reactor
never exceeding the threshold for a meltdown.

6.1 States and Transitions
Formally, a state machine is nothing more than a binary relation on a set, except
that the elements of the set are called “states,” the relation is called the transition
relation, and an arrow in the graph of the transition relation is called a transition.
A transition from state q to state r will be written q ! r. The transition relation
is also called the state graph of the machine. A state machine also comes equipped
with a designated start state.
A simple example is a bounded counter, which counts from 0 to 99 and overflows
at 100. This state machine is pictured in Figure 6.1, with states pictured as circles,
transitions by arrows, and with start state 0 indicated by the double circle. To be

start
state
0 1 2 99 overflow

Figure 6.1 State transitions for the 99-bounded counter.
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164 Chapter 6 State Machines

precise, what the picture tells us is that this bounded counter machine has

states WWD f0; 1; : : : ; 99; overflowg;
start state WWD 0;
transitions WWD fn ! n C 1 j 0 n < 99g
[ f99 ! overflow; overflow ! overflowg:

This machine isn’t much use once it overflows, since it has no way to get out of its
overflow state.
State machines for digital circuits and string pattern matching algorithms, for in-
stance, usually have only a finite number of states. Machines that model continuing
computations typically have an infinite number of states. For example, instead of
the 99-bounded counter, we could easily define an “unbounded” counter that just
keeps counting up without overflowing. The unbounded counter has an infinite
state set, the nonnegative integers, which makes its state diagram harder to draw.
State machines are often defined with labels on states and/or transitions to indi-
cate such things as input or output values, costs, capacities, or probabilities. Our
state machines don’t include any such labels because they aren’t needed for our
purposes. We do name states, as in Figure 6.1, so we can talk about them, but the
names aren’t part of the state machine.

6.2 The Invariant Principle
6.2.1 A Diagonally-Moving Robot
Suppose we have a robot that starts at the origin and moves on an infinite 2-
dimensional integer grid. The state of the robot at any time can be specified by
the integer coordinates .x; y/ of the robot’s current position. So the start state
is .0; 0/. At each step, the robot may move to a diagonally adjacent grid point, as
illustrated in Figure 6.2.
To be precise, the robot’s transitions are:

f.m; n/ ! .m ˙ 1; n ˙ 1/ j m; n 2 Zg:

For example, after the first step, the robot could be in states .1; 1/, .1; 1/, . 1; 1/
or . 1; 1/. After two steps, there are 9 possible states for the robot, includ-
ing .0; 0/. The question is, can the robot ever reach position .1; 0/?
If you play around with the robot a bit, you’ll probably notice that the robot can
only reach positions .m; n/ for which m C n is even, which of course means that it
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6.2. The Invariant Principle 165

x
0 1 2 3

Figure 6.2 The Diagonally Moving Robot.

can’t reach .1; 0/. This follows because the evenness of the sum of the coordinates
is a property that is preserved by transitions. This is an example of a preserved
invariant.
This once, let’s go through this preserved invariant argument, carefully high-
lighting where induction comes in. Specifically, define the even-sum property of
states to be:
Even-sum..m; n// WWD Œm C n is even:

Lemma 6.2.1. For any transition q ! r of the diagonally-moving robot, if Even-
sum(q), then Even-sum(r).

This lemma follows immediately from the definition of the robot’s transitions:
.m; n/ ! .m ˙ 1; n ˙ 1/. After a transition, the sum of coordinates changes by
.˙1/ C .˙1/, that is, by 0, 2, or -2. Of course, adding 0, 2 or -2 to an even number
gives an even number. So by a trivial induction on the number of transitions, we
can prove:

Theorem 6.2.2. The sum of the coordinates of any state reachable by the diagonally-
moving robot is even.
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166 Chapter 6 State Machines

2
‹‹

0
goal

x
0 1 2 3

Figure 6.3 Can the Robot get to .1; 0/?
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6.2. The Invariant Principle 167

Proof. The proof is induction on the number of transitions the robot has made. The
induction hypothesis is

P .n/ WWD if q is a state reachable in n transitions, then Even-sum(q):

Base case: P .0/ is true since the only state reachable in 0 transitions is the start
state .0; 0/, and 0 C 0 is even.
Inductive step: Assume that P .n/ is true, and let r be any state reachable in n C 1
transitions. We need to prove that Even-sum(r) holds.
Since r is reachable in n C 1 transitions, there must be a state q reachable in n
transitions such that q ! r. Since P .n/ is assumed to be true, Even-sum(q) holds,
and so by Lemma 6.2.1, Even-sum(r) also holds. This proves that P .n/ IMPLIES
P .n C 1/ as required, completing the proof of the inductive step.
We conclude by induction that for all n 0, if q is reachable in n transitions, then
Even-sum(q). This implies that every reachable state has the Even-sum property.

Corollary 6.2.3. The robot can never reach position .1; 0/.
Proof. By Theorem 6.2.2, we know the robot can only reach positions with coor-
dinates that sum to an even number, and thus it cannot reach position .1; 0/.

6.2.2 Statement of the Invariant Principle
Using the Even-sum invariant to understand the diagonally-moving robot is a sim-
ple example of a basic proof method called The Invariant Principle. The Principle
summarizes how induction on the number of steps to reach a state applies to invari-
ants.
A state machine execution describes a possible sequence of steps a machine
might take.
Definition 6.2.4. An execution of the state machine is a (possibly infinite) sequence
of states with the property that
it begins with the start state, and

if q and r are consecutive states in the sequence, then q ! r.
A state is called reachable if it appears in some execution.
Definition 6.2.5. A preserved invariant of a state machine is a predicate P on
states, such that whenever P .q/ is true of a state q and q ! r for some state r
then P .r/ holds.
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168 Chapter 6 State Machines

The Invariant Principle
If a preserved invariant of a state machine is true for the start state,
then it is true for all reachable states.

The Invariant Principle is nothing more than the Induction Principle reformulated
in a convenient form for state machines. Showing that a predicate is true in the start
state is the base case of the induction, and showing that a predicate is a preserved
invariant corresponds to the inductive step.1
1 Preserved invariants are commonly just called “invariants” in the literature on program correct-
ness, but we decided to throw in the extra adjective to avoid confusion with other definitions. For
example, other texts (as well as another subject at MIT) use “invariant” to mean “predicate true of
all reachable states.” Let’s call this definition “invariant-2.” Now invariant-2 seems like a reason-
able definition, since unreachable states by definition don’t matter, and all we want to show is that
a desired property is invariant-2. But this confuses the objective of demonstrating that a property is
invariant-2 with the method of finding a preserved invariant—which is preserved even at unreachable
states—to show that it is invariant-2.
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6.2. The Invariant Principle 169

Robert W. Floyd

The Invariant Principle was formulated by Robert W. Floyd at Carnegie Tech
in 1967. (Carnegie Tech was renamed Carnegie-Mellon University the following
year.) Floyd was already famous for work on the formal grammars that trans-
formed the field of programming language parsing; that was how he got to be
a professor even though he never got a Ph.D. (He had been admitted to a PhD
program as a teenage prodigy, but flunked out and never went back.)
In that same year, Albert R. Meyer was appointed Assistant Professor in the
Carnegie Tech Computer Science Department, where he first met Floyd. Floyd
and Meyer were the only theoreticians in the department, and they were both de-
lighted to talk about their shared interests. After just a few conversations, Floyd’s
new junior colleague decided that Floyd was the smartest person he had ever met.
Naturally, one of the first things Floyd wanted to tell Meyer about was his new,
as yet unpublished, Invariant Principle. Floyd explained the result to Meyer, and
Meyer wondered (privately) how someone as brilliant as Floyd could be excited
by such a trivial observation. Floyd had to show Meyer a bunch of examples be-
fore Meyer understood Floyd’s excitement —not at the truth of the utterly obvious
Invariant Principle, but rather at the insight that such a simple method could be so
widely and easily applied in verifying programs.
Floyd left for Stanford the following year. He won the Turing award—the
“Nobel prize” of computer science—in the late 1970’s, in recognition of his work
on grammars and on the foundations of program verification. He remained at
Stanford from 1968 until his death in September, 2001. You can learn more about
Floyd’s life and work by reading the eulogy at

http://oldwww.acm.org/pubs/membernet/stories/floyd.pdf

written by his closest colleague, Don Knuth.
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170 Chapter 6 State Machines

6.2.3 The Die Hard Example
The movie Die Hard 3: With a Vengeance includes an amusing example of a state
machine. The lead characters played by Samuel L. Jackson and Bruce Willis have
to disarm a bomb planted by the diabolical Simon Gruber:

Simon: On the fountain, there should be 2 jugs, do you see them? A 5-
gallon and a 3-gallon. Fill one of the jugs with exactly 4 gallons of water
and place it on the scale and the timer will stop. You must be precise;
one ounce more or less will result in detonation. If you’re still alive in 5
minutes, we’ll speak.
Bruce: Wait, wait a second. I don’t get it. Do you get it?
Samuel: No.
Bruce: Get the jugs. Obviously, we can’t fill the 3-gallon jug with 4 gal-
lons of water.
Samuel: Obviously.
Bruce: All right. I know, here we go. We fill the 3-gallon jug exactly to
the top, right?
Samuel: Uh-huh.
Bruce: Okay, now we pour this 3 gallons into the 5-gallon jug, giving us
exactly 3 gallons in the 5-gallon jug, right?
Samuel: Right, then what?
Bruce: All right. We take the 3-gallon jug and fill it a third of the way...
Samuel: No! He said, “Be precise.” Exactly 4 gallons.
Bruce: Sh - -. Every cop within 50 miles is running his a - - off and I’m
out here playing kids games in the park.
Samuel: Hey, you want to focus on the problem at hand?

Fortunately, they find a solution in the nick of time. You can work out how.

The Die Hard 3 State Machine
The jug-filling scenario can be modeled with a state machine that keeps track of
the amount b of water in the big jug, and the amount l in the little jug. With the 3
and 5 gallon water jugs, the states formally will be pairs .b; l/ of real numbers such
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6.2. The Invariant Principle 171

that 0 b 5; 0 l 3. (We can prove that the reachable values of b and l will
be nonnegative integers, but we won’t assume this.) The start state is .0; 0/, since
both jugs start empty.
Since the amount of water in the jug must be known exactly, we will only con-
sider moves in which a jug gets completely filled or completely emptied. There are
several kinds of transitions:
1. Fill the little jug: .b; l/ ! .b; 3/ for l < 3.
2. Fill the big jug: .b; l/ ! .5; l/ for b < 5.
3. Empty the little jug: .b; l/ ! .b; 0/ for l > 0.
4. Empty the big jug: .b; l/ ! .0; l/ for b > 0.
5. Pour from the little jug into the big jug: for l > 0,
(
.b C l; 0/ if b C l 5,
.b; l/ !
.5; l .5 b// otherwise.

6. Pour from big jug into little jug: for b > 0,
(
.0; b C l/ if b C l 3,
.b; l/ !
.b .3 l/; 3/ otherwise.

Note that in contrast to the 99-counter state machine, there is more than one pos-
sible transition out of states in the Die Hard machine. Machines like the 99-counter
with at most one transition out of each state are called deterministic. The Die Hard
machine is nondeterministic because some states have transitions to several differ-
ent states.
The Die Hard 3 bomb gets disarmed successfully because the state (4,3) is reach-
able.

Die Hard Permanently
The Die Hard series is getting tired, so we propose a final Die Hard Permanently.
Here, Simon’s brother returns to avenge him, posing the same challenge, but with
the 5 gallon jug replaced by a 9 gallon one. The state machine has the same spec-
ification as the Die Hard 3 version, except all occurrences of “5” are replaced by
“9.”
Now, reaching any state of the form .4; l/ is impossible. We prove this using
the Invariant Principle. Specifically, we define the preserved invariant predicate
P ..b; l// to be that b and l are nonnegative integer multiples of 3.
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172 Chapter 6 State Machines

To prove that P is a preserved invariant of Die-Hard-Once-and-For-All machine,
we assume P .q/ holds for some state q WWD .b; l/ and that q ! r. We have to
show that P .r/ holds. The proof divides into cases, according to which transition
rule is used.
One case is a “fill the little jug” transition. This means r D .b; 3/. But P .q/
implies that b is an integer multiple of 3, and of course 3 is an integer multiple of
3, so P .r/ still holds.
Another case is a “pour from big jug into little jug” transition. For the subcase
when there isn’t enough room in the little jug to hold all the water, that is, when
b C l > 3, we have r D .b .3 l/; 3/. But P .q/ implies that b and l are integer
multiples of 3, which means b .3 l/ is too, so in this case too, P .r/ holds.
We won’t bother to crank out the remaining cases, which can all be checked
just as easily. Now by the Invariant Principle, we conclude that every reachable
state satisifies P . But since no state of the form .4; l/ satisifies P , we have proved
rigorously that Bruce dies once and for all!
By the way, notice that the state (1,0), which satisfies NOT.P /, has a transition
to (0,0), which satisfies P . So the negation of a preserved invariant may not be a
preserved invariant.

6.3 Partial Correctness & Termination
Floyd distinguished two required properties to verify a program. The first property
is called partial correctness; this is the property that the final results, if any, of the
process must satisfy system requirements.
You might suppose that if a result was only partially correct, then it might also
be partially incorrect, but that’s not what Floyd meant. The word “partial” comes
from viewing a process that might not terminate as computing a partial relation.
Partial correctness means that when there is a result, it is correct, but the process
might not always produce a result, perhaps because it gets stuck in a loop.
The second correctness property, called termination, is that the process does
always produce some final value.
Partial correctness can commonly be proved using the Invariant Principle. Termi-
nation can commonly be proved using the Well Ordering Principle. We’ll illustrate
this by verifying a Fast Exponentiation procedure.
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6.3. Partial Correctness & Termination 173

6.3.1 Fast Exponentiation
Exponentiating
The most straightforward way to compute the bth power of a number a is to multi-
ply a by itself b 1 times. But the solution can be found in considerably fewer mul-
tiplications by using a technique called Fast Exponentiation. The register machine
program below defines the fast exponentiation algorithm. The letters x; y; z; r de-
note registers that hold numbers. An assignment statement has the form “z WD a”
and has the effect of setting the number in register z to be the number a.

A Fast Exponentiation Program
Given inputs a 2 R; b 2 N, initialize registers x; y; z to a; 1; b respectively, and
repeat the following sequence of steps until termination:

if z D 0 return y and terminate
r WD remainder.z; 2/
z WD quotient.z; 2/
if r D 1, then y WD xy
x WD x 2

We claim this program always terminates and leaves y D ab .
To begin, we’ll model the behavior of the program with a state machine:
1. states WWD R R N,

2. start state WWD .a; 1; b/,

3. transitions are defined by the rule
(
.x 2 ; y; quotient.z; 2// if z is nonzero and even;
.x; y; z/ ! 2
.x ; xy; quotient.z; 2// if z is nonzero and odd:

The preserved invariant P ..x; y; z// will be

z 2 N AND yx z D ab : (6.1)

To prove that P is preserved, assume P ..x; y; z// holds and that .x; y; z/ !
.xt ; yt ; zt /. We must prove that P ..xt ; yt ; zt // holds, that is,

zt 2 N AND yt xtzt D ab : (6.2)
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174 Chapter 6 State Machines

Since there is a transition from .x; y; z/, we have z ¤ 0, and since z 2 N
by (6.1), we can consider just two cases:
If z is even, then we have that xt D x 2 ; yt D y; zt D z=2. Therefore, zt 2 N
and
yt xtzt D y.x 2 /z=2
D yx 2z=2
D yx z
D ab (by (6.1))
If z is odd, then we have that xt D x 2 ; yt D xy; zt D .z 1/=2. Therefore,
zt 2 N and
yt xtzt D xy.x 2 /.z 1/=2

D yx 1C2.z 1/=2

D yx 1C.z 1/

D yx z
D ab (by (6.1))
So in both cases, (6.2) holds, proving that P is a preserved invariant.
Now it’s easy to prove partial correctness: if the Fast Exponentiation program
terminates, it does so with ab in register y. This works because 1 ab D ab , which
means that the start state .a; 1; b/ satisifies P . By the Invariant Principle, P holds
for all reachable states. But the program only stops when z D 0. If a terminated
state .x; y; 0/ is reachable, then y D yx 0 D ab as required.
Ok, it’s partially correct, but what’s fast about it? The answer is that the number
of multiplications it performs to compute ab is roughly the length of the binary
representation of b. That is, the Fast Exponentiation program uses roughly log b 2
multiplications, compared to the naive approach of multiplying by a a total of b 1
times.
More precisely, it requires at most 2.dlog be C 1/ multiplications for the Fast
Exponentiation algorithm to compute ab for b > 1. The reason is that the number
in register z is initially b, and gets at least halved with each transition. So it can’t
be halved more than dlog be C 1 times before hitting zero and causing the program
to terminate. Since each of the transitions involves at most two multiplications, the
total number of multiplications until z D 0 is at most 2.dlog be C 1/ for b > 0 (see
Problem 6.6).
2 As usual in computer science, log b means the base two logarithm log2 b. We use, ln b for the
natural logarithm loge b, and otherwise write the logarithm base explicitly, as in log10 b.
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6.3. Partial Correctness & Termination 175

6.3.2 Derived Variables
The preceding termination proof involved finding a nonnegative integer-valued
measure to assign to states. We might call this measure the “size” of the state.
We then showed that the size of a state decreased with every state transition. By
the Well Ordering Principle, the size can’t decrease indefinitely, so when a mini-
mum size state is reached, there can’t be any transitions possible: the process has
terminated.
More generally, the technique of assigning values to states—not necessarily non-
negative integers and not necessarily decreasing under transitions—is often useful
in the analysis of algorithms. Potential functions play a similar role in physics. In
the context of computational processes, such value assignments for states are called
derived variables.
For example, for the Die Hard machines we could have introduced a derived
variable f W states ! R for the amount of water in both buckets, by setting
f ..a; b// WWD a C b. Similarly, in the robot problem, the position of the robot along
the x-axis would be given by the derived variable x-coord, where x-coord..i; j //WWD i .
There are a few standard properties of derived variables that are handy in ana-
lyzing state machines.
Definition 6.3.1. A derived variable f W states ! R is strictly decreasing iff

q ! q 0 IMPLIES f .q 0 / < f .q/:

It is weakly decreasing iff

q ! q 0 IMPLIES f .q 0 / f .q/:

Strictly increasingweakly increasing derived variables are defined similarly.3
We confirmed termination of the Fast Exponentiation procedure by noticing that
the derived variable z was nonnegative-integer-valued and strictly decreasing. We
can summarize this approach to proving termination as follows:
Theorem 6.3.2. If f is a strictly decreasing N-valued derived variable of a state
machine, then the length of any execution starting at state q is at most f .q/.
Of course, we could prove Theorem 6.3.2 by induction on the value of f .q/, but
think about what it says: “If you start counting down at some nonnegative integer
f .q/, then you can’t count down more than f .q/ times.” Put this way, it’s obvious.
3 Weakly increasing variables are often also called nondecreasing. We will avoid this terminology

to prevent confusion between nondecreasing variables and variables with the much weaker property
of not being a decreasing variable.
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176 Chapter 6 State Machines

6.3.3 Termination with Well ordered Sets (Optional)
Theorem 6.3.2 generalizes straightforwardly to derived variables taking values in a
well ordered set (Section 2.4.
Theorem 6.3.3. If there exists a strictly decreasing derived variable whose range
is a well ordered set, then every execution terminates.
Theorem 6.3.3 follows immediately from the observation that a set of numbers
is well ordered iff it has no infinite decreasing sequences (Problem 2.23).
Note that the existence of a weakly decreasing derived variable does not guaran-
tee that every execution terminates. An infinite execution could proceed through
states in which a weakly decreasing variable remained constant.

6.3.4 A Southeast Jumping Robot (Optional)
Here’s a simple, contrived example of a termination proof based on a variable that
is strictly decreasing over a well ordered set. Let’s think about a robot that travels
around the nonnegative integer quadrant N2 .
If the robot is at some position .x; y/ different from the origin .0; 0/, the robot
must make a move, which may be
a unit distance West—that is, .x; y/ ! .x 1; y/ for x > 0, or

a unit distance South combined with an arbitrary jump East—that is, .x; y/ !
.z; y 1/ for z x,
providing the move does not leave the quadrant.
Claim 6.3.4. The robot will always get stuck at the origin.
If we think of the robot as a nondeterministic state machine, then Claim 6.3.4 is
a termination assertion. The Claim may seem obvious, but it really has a different
character than termination based on nonnegative integer-valued variables. That’s
because, even knowing that the robot is at position .0; 1/, for example, there is no
way to bound the time it takes for the robot to get stuck. It can delay getting stuck
for as many seconds as it wants by making its next move to a distant point in the
Far East. This rules out proving termination using Theorem 6.3.2.
So does Claim 6.3.4 still seem obvious?
Well it is if you see the trick. Define a derived variable v mapping robot states
to the numbers in the well ordered set N C F of Lemma 2.4.5. In particular, define
v W N2 ! N C F as follows
x
v.x; y/ WWD y C :
xC1
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6.4. The Stable Marriage Problem 177

Brad 2 1 Jennifer
1 2

2 1
Billy Bob 1 2 Angelina

Figure 6.4 Preferences for four people. Both men like Angelina best and both
women like Brad best.

Now it’s easy to check that if .x; y/ ! .x 0 ; y 0 / is a legitimate robot move, then
v..x 0 ; y 0 // < v..x; y//. In particular, v is a strictly decreasing derived variable, so
Theorem 6.3.3 implies that the robot always get stuck—even though we can’t say
how many moves it will take until it does.

6.4 The Stable Marriage Problem
Suppose we have a population of men and women in which each person has pref-
erences of the opposite-gender person they would like to marry: each man has his
preference list of all the women, and each woman has her preference list of all of
the men.
The preferences don’t have to be symmetric. That is, Jennifer might like Brad
best, but Brad doesn’t necessarily like Jennifer best. The goal is to marry every-
one: every man must marry exactly one woman and vice versa—no polygamy and
heterosexual marriages only.4 Moreover, we would like to find a matching between
men and women that is stable in the sense that there is no pair of people who prefer
one another to their spouses.
For example, suppose Brad likes Angelina best, and Angelina likes Brad best, but
Brad and Angelina are married to other people, say Jennifer and Billy Bob. Now
Brad and Angelina prefer each other to their spouses, which puts their marriages
at risk. Pretty soon, they’re likely to start spending late nights together working on
problem sets!
This unfortunate situation is illustrated in Figure 6.4, where the digits “1” and “2”
near a man shows which of the two women he ranks first and second, respectively,
and similarly for the women.
4 Same-sex marriage is an interesting but separate case.
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178 Chapter 6 State Machines

More generally, in any matching, a man and woman who are not married to each
other and who like each other better than their spouses is called a rogue couple. In
the situation shown in Figure 6.4, Brad and Angelina would be a rogue couple.
Having a rogue couple is not a good thing, since it threatens the stability of the
marriages. On the other hand, if there are no rogue couples, then for any man and
woman who are not married to each other, at least one likes their spouse better than
the other, and so there won’t be any mutual temptation to start an affair.

Definition 6.4.1. A stable matching is a matching with no rogue couples.

The question is, given everybody’s preferences, can you find a stable set of mar-
riages? In the example consisting solely of the four people in Figure 6.4, we could
let Brad and Angelina both have their first choices by marrying each other. Now
neither Brad nor Angelina prefers anybody else to their spouse, so neither will be
in a rogue couple. This leaves Jen not-so-happily married to Billy Bob, but neither
Jen nor Billy Bob can entice somebody else to marry them, and so this is a stable
matching.
It turns out there always is a stable matching among a group of men and women.
We don’t know of any immediate way to recognize this, and it seems surprising. In
fact, in the apparently similar same-sex or “buddy” matching problem where people
are supposed to be paired off as buddies, regardless of gender, a stable matching
may not be possible. An example of preferences among four people where there is
no stable buddy match is given in Problem 6.22. But when men are only allowed
to marry women, and vice versa, then there is a simple procedure to produce a
stable matching and the concept of preserved invariants provides an elegant way to
understand and verify the procedure.

6.4.1 The Mating Ritual
The procedure for finding a stable matching can be described in a memorable way
as a Mating Ritual that takes place over several days. On the starting day, each man
has his full preference list of all the women, and likewise each woman has her full
preference list of all the men. Then following events happen each day:
Morning: Each man stands under the balcony of the woman on the top of his
list, that is the woman he prefers above all the other remaining women. The he
serenades her. He is said to be her suitor. If a man has no women left on his list, he
stays home and does his math homework.
Afternoon: Each woman who has one or more suitors says to her favorite among
them, “We might get engaged. Please stay around.” To the other suitors, she says,
“No. I will never marry you! Take a hike!”
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6.4. The Stable Marriage Problem 179

Evening: Any man who is told by a woman to take a hike crosses that woman
off his preference list.
Termination condition: When a day arrives in which every woman has at most
one suitor, the ritual ends with each woman marrying her suitor, if she has one.
There are a number of facts about this Mating Ritual that we would like to prove:

The Ritual eventually reaches the termination condition.

Everybody ends up married.

The resulting marriages are stable.

To prove these facts, it will be helpful to recognize the Ritual as the description
of a state machine. The state at the start of any day is determined by knowing for
each man, which woman, if any, he will serenade that day—that is, the woman
at the top of his preference list after he has crossed out all the women who have
rejected him on earlier days.

Mating Ritual at Akamai

The Internet infrastructure company Akamai, cofounded by Tom Leighton, also
uses a variation of the Mating Ritual to assign web traffic to its servers.
In the early days, Akamai used other combinatorial optimization algorithms
that got to be too slow as the number of servers (over 65,000 in 2010) and requests
(over 800 billion per day) increased. Akamai switched to a Ritual-like approach,
since a Ritual is fast and can be run in a distributed manner. In this case, web
requests correspond to women and web servers correspond to men. The web
requests have preferences based on latency and packet loss, and the web servers
have preferences based on cost of bandwidth and co-location.
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180 Chapter 6 State Machines

6.4.2 There is a Marriage Day
It’s easy to see why the Mating Ritual has a terminal day when people finally get
married. Every day on which the ritual hasn’t terminated, at least one man crosses
a woman off his list. (If the ritual hasn’t terminated, there must be some woman
serenaded by at least two men, and at least one of them will have to cross her off his
list). If we start with n men and n women, then each of the n men’s lists initially
has n women on it, for a total of n2 list entries. Since no women ever gets added
to a list, the total number of entries on the lists decreases every day that the Ritual
continues, and so the Ritual can continue for at most n2 days.

6.4.3 They All Live Happily Ever After. . .
We will prove that the Mating Ritual leaves everyone in a stable marriage. To do
this, we note one very useful fact about the Ritual: if on some morning a woman has
any suitor, then her favorite suitor will still be serenading her the next morning—his
list won’t have changed. So she is sure to have today’s favorite suitor among her
suitors tomorrow. That means she will be able to choose a favorite suitor tomorrow
who is at least as desirable to her as today’s favorite. So day by day, her favorite
suitor can stay the same or get better, never worse. This sounds like an invariant,
and it is. Namely, let P be the predicate

For every woman w and man m, if w is crossed off m’s list, then w has
a suitor whom she prefers over m.

Lemma 6.4.2. P is a preserved invariant for The Mating Ritual.

Proof. Woman w gets crossed off m’s list only when w has a suitor she prefers to
m. Thereafter, her favorite suitor doesn’t change until one she likes better comes
along. So if her favorite suitor was preferable to m, then any new favorite suitor
will be as well.

Notice that the invariant P holds vacuously at the beginning since no women are
crossed off to start. So by the Invariant Principle, P holds throughout the Ritual.
Now we can prove:

Theorem 6.4.3. Everyone is married at the end of the Mating Ritual.

Proof. Assume to the contrary that on the last day of the Mating Ritual, some
man—call him Bob—is not married. This means Bob can’t be serenading anybody,
that is, his list must be empty. So every woman must have been crossed off his
list and, since P is true, every woman has a suitor whom she prefers to Bob. In
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6.4. The Stable Marriage Problem 181

particular, every woman has some suitor, and since it is the last day, they have only
one suitor, and this is who they marry. But there are an equal number of men and
women, so if all women are married, so are all men, contradicting the assumption
that Bob is not married.

Theorem 6.4.4. The Mating Ritual produces a stable matching.

Proof. Let Brad and Jen be any man and woman, respectively, that are not married
to each other on the last day of the Mating Ritual. We will prove that Brad and Jen
are not a rogue couple, and thus that all marriages on the last day are stable. There
are two cases to consider.

Case 1: Jen is not on Brad’s list by the end. Then by invariant P , we know that
Jen has a suitor (and hence a husband) whom she prefers to Brad. So she’s
not going to run off with Brad—Brad and Jen cannot be a rogue couple.

Case 2: Jen is on Brad’s list. Since Brad picks women to serenade by working
down his list, his wife must be higher on his preference list than Jen. So
he’s not going to run off with Jen—once again, Brad and Jen are not a rogue
couple.

6.4.4 . . . Especially the Men
Who is favored by the Mating Ritual, the men or the women? The women seem
to have all the power: each day they choose their favorite suitor and reject the rest.
What’s more, we know their suitors can only change for the better as the Ritual
progresses. Similarly, a man keeps serenading the woman he most prefers among
those on his list until he must cross her off, at which point he serenades the next
most preferred woman on his list. So from the man’s perspective, the woman he is
serenading can only change for the worse. Sounds like a good deal for the women.
But it’s not! We will show that the men are by far the favored gender under the
Mating Ritual.
While the Mating Ritual produces one stable matching, stable matchings need
not be unique. For example, reversing the roles of men and women will often yield
a different stable matching among them. So a man may have different wives in
different sets of stable marriages. In some cases, a man can stably marry every one
of the women, but in most cases, there are some women who cannot be a man’s wife
in any stable matching. For example, given the preferences shown in Figure 6.4,
Jennifer cannot be Brad’s wife in any stable matching because if he was married to
her, then he and Angelina would be a rogue couple. It is not feasible for Jennifer to
be stably married to Brad.
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182 Chapter 6 State Machines

Definition 6.4.5. Given a set of preferences for the men and women, one person is
a feasible spouse for another person when there is a stable matching in which these
two people are married.

Definition 6.4.6. Let Q be the predicate: for every woman w and man m, if w is
crossed off m’s list, then w is not a feasible spouse for m.

Lemma 6.4.7. Q is a preserved invariant5 for The Mating Ritual.

Proof. Suppose Q holds at some point in the Ritual and some woman Alice is
about to be crossed off some man’s, Bob’s, list. We claim that Alice must not be
feasible for Bob. Therefore Q will still hold after Alice is crossed off, proving that
Q is invariant.
To verify the claim, notice that when Alice gets crossed of Bob’s list, it’s because
Alice has a suitor, Ted, she prefers to Bob. What’s more, since Q holds, all Ted’s
feasible wives are still on his list, and Alice is at the top. So Ted likes Alice better
than all his other feasible spouses. Now if Alice could be married to Bob in some
set of stable marriages, then Ted must be married to a wife he likes less than Alice,
making Alice and Ted a rogue couple and contradicting stability. So Alice can’t be
married to Bob, that is, Alice is not a feasible wife for Bob, as claimed.

Definition 6.4.8. A person’s optimal spouse is their most preferred feasible spouse.
A person’s pessimal spouse is their least preferred feasible spouse.

Everybody has an optimal and a pessimal spouse, since we know there is at least
one stable matching, namely, the one produced by the Mating Ritual. Lemma 6.4.7
implies a key property the Mating Ritual:

Theorem 6.4.9. The Mating Ritual marries every man to his optimal spouse and
every woman to her pessimal spouse.

Proof. If Bob is married to Alice on the final day of the Ritual, then everyone above
Alice on Bob’s preference list was crossed off, and by property Q, all these crossed
off women were infeasible for Bob. So Alice is Bob’s highest ranked feasible
spouse, that is, his optimal spouse.
Further, since Bob likes Alice better than any other feasible wife, Alice and Bob
would be a rogue couple if Alice was married to a husband she liked less than Bob.
So Bob must be Alice’s least preferred feasible husband.
5 We appeal to P in justifying Q, so technically it is P AND Q which is actually the preserved
invariant. But let’s not be picky.
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6.4. The Stable Marriage Problem 183

6.4.5 Applications
The Mating Ritual was first announced in a paper by D. Gale and L.S. Shapley in
1962, but ten years before the Gale-Shapley paper was published, and unknown to
them, a similar algorithm was being used to assign residents to hospitals by the Na-
tional Resident Matching Program (NRMP). The NRMP has, since the turn of the
twentieth century, assigned each year’s pool of medical school graduates to hospi-
tal residencies (formerly called “internships”), with hospitals and graduates playing
the roles of men and women.6 Before the Ritual-like algorithm was adopted, there
were chronic disruptions and awkward countermeasures taken to preserve unsta-
ble assignments of graduates to residencies. The Ritual resolved these problems so
successfully, that it was used essentially without change at least through 1989.7 For
this and related work, Shapley was awarded the 2012 Nobel prize in Economics.
Not surprisingly, the Mating Ritual is also used by at least one large online dat-
ing agency. Of course there is no serenading going on—everything is handled by
computer.

Problems for Section 6.3
Practice Problems
Problem 6.1.
Which states of the Die Hard 3 machine below have transitions to exactly two
states?

Die Hard Transitions

1. Fill the little jug: .b; l/ ! .b; 3/ for l < 3.

2. Fill the big jug: .b; l/ ! .5; l/ for b < 5.

3. Empty the little jug: .b; l/ ! .b; 0/ for l > 0.

4. Empty the big jug: .b; l/ ! .0; l/ for b > 0.
6 Inthis case there may be multiple women married to one man, but this is a minor complication,
see Problem 6.23.
7 Much more about the Stable Marriage Problem can be found in the very readable mathematical

monograph by Dan Gusfield and Robert W. Irving, [25].
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184 Chapter 6 State Machines

5. Pour from the little jug into the big jug: for l > 0,
(
.b C l; 0/ if b C l 5,
.b; l/ !
.5; l .5 b// otherwise.

6. Pour from big jug into little jug: for b > 0,
(
.0; b C l/ if b C l 3,
.b; l/ !
.b .3 l/; 3/ otherwise.

Homework Problems
Problem 6.2.
In the late 1960s, the military junta that ousted the government of the small re-
public of Nerdia completely outlawed built-in multiplication operations, and also
forbade division by any number other than 3. Fortunately, a young dissident found
a way to help the population multiply any two nonnegative integers without risking
persecution by the junta. The procedure he taught people is:

procedure multiply.x; y: nonnegative integers/
r WD x;
s WD y;
a WD 0;
while s ¤ 0 do
if 3 j s then
r WD r C r C r;
s WD s=3;
else if 3 j .s 1/ then
a WD a C r;
r WD r C r C r;
s WD .s 1/=3;
else
a WD a C r C r;
r WD r C r C r;
s WD .s 2/=3;
return a;

We can model the algorithm as a state machine whose states are triples of non-
negative integers .r; s; a/. The initial state is .x; y; 0/. The transitions are given by
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6.4. The Stable Marriage Problem 185

the rule that for s > 0:
8
<.3r; s=3; a/
ˆ if 3 j s
.r; s; a/ ! .3r; .s 1/=3; a C r/ if 3 j .s 1/
ˆ
.3r; .s 2/=3; a C 2r/ otherwise:
:

(a) List the sequence of steps that appears in the execution of the algorithm for
inputs x D 5 and y D 10.

(b) Use the Invariant Method to prove that the algorithm is partially correct—that
is, if s D 0, then a D xy.

Problem 6.3.
A robot named Wall-E wanders around a two-dimensional grid. He starts out at
.0; 0/ and is allowed to take four different types of steps:

1. .C2; 1/

2. .C1; 2/

3. .C1; C1/

4. . 3; 0/

Thus, for example, Wall-E might walk as follows. The types of his steps are
listed above the arrows.
1 3 2 4
.0; 0/ ! .2; 1/ ! .3; 0/ ! .4; 2/ ! .1; 2/ ! : : :
Wall-E’s true love, the fashionable and high-powered robot, Eve, awaits at .0; 2/.
(a) Describe a state machine model of this problem.

(b) Will Wall-E ever find his true love? Either find a path from Wall-E to Eve, or
use the Invariant Principle to prove that no such path exists.

Problem 6.4.
A hungry ant is placed on an unbounded grid. Each square of the grid either con-
tains a crumb or is empty. The squares containing crumbs form a path in which,
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186 Chapter 6 State Machines

except at the ends, every crumb is adjacent to exactly two other crumbs. The ant is
placed at one end of the path and on a square containing a crumb. For example, the
figure below shows a situation in which the ant faces North, and there is a trail of
food leading approximately Southeast. The ant has already eaten the crumb upon
which it was initially placed.

The ant can only smell food directly in front of it. The ant can only remember
a small number of things, and what it remembers after any move only depends on
what it remembered and smelled immediately before the move. Based on smell and
memory, the ant may choose to move forward one square, or it may turn right or
left. It eats a crumb when it lands on it.
The above scenario can be nicely modelled as a state machine in which each state
is a pair consisting of the “ant’s memory” and “everything else”—for example,
information about where things are on the grid. Work out the details of such a
model state machine; design the ant-memory part of the state machine so the ant
will eat all the crumbs on any finite path at which it starts and then signal when it
is done. Be sure to clearly describe the possible states, transitions, and inputs and
outputs (if any) in your model. Briefly explain why your ant will eat all the crumbs.
Note that the last transition is a self-loop; the ant signals done for eternity. One
could also add another end state so that the ant signals done only once.

Problem 6.5.
Suppose that you have a regular deck of cards arranged as follows, from top to
bottom:
A~ 2~ : : : K~ A 2 : : : K A| 2| : : : K| A} 2} : : : K}
Only two operations on the deck are allowed: inshuffling and outshuffling. In
both, you begin by cutting the deck exactly in half, taking the top half into your
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6.4. The Stable Marriage Problem 187

right-hand and the bottom into your left. Then you shuffle the two halves together
so that the cards are perfectly interlaced; that is, the shuffled deck consists of one
card from the left, one from the right, one from the left, one from the right, etc. The
top card in the shuffled deck comes from the right-hand in an outshuffle and from
the left-hand in an inshuffle.
(a) Model this problem as a state machine.

(b) Use the Invariant Principle to prove that you cannot make the entire first half
of the deck black through a sequence of inshuffles and outshuffles.
Note: Discovering a suitable invariant can be difficult! This is the part of a
correctness proof that generally requires some insight, and there is no simple recipe
for finding invariants. A standard initial approach is to identify a bunch of reachable
states and then look for a pattern—some feature that they all share.

Problem 6.6.
Prove that the fast exponentiation state machine of Section 6.3.1 will halt after

dlog2 ne C 1 (6.3)

transitions starting from any state where the value of z is n 2 ZC .
Hint: Strong induction.

Class Problems
Problem 6.7.
In this problem you will establish a basic property of a puzzle toy called the Fifteen
Puzzle using the method of invariants. The Fifteen Puzzle consists of sliding square
tiles numbered 1; : : : ; 15 held in a 4 4 frame with one empty square. Any tile
adjacent to the empty square can slide into it.
The standard initial position is

1 2 3 4
5 6 7 8
9 10 11 12
13 14 15

We would like to reach the target position (known in the oldest author’s youth as
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188 Chapter 6 State Machines

“the impossible”):
15 14 13 12
11 10 9 8
7 6 5 4
3 2 1
A state machine model of the puzzle has states consisting of a 4 4 matrix with
16 entries consisting of the integers 1; : : : ; 15 as well as one “empty” entry—like
each of the two arrays above.
The state transitions correspond to exchanging the empty square and an adjacent
numbered tile. For example, an empty at position .2; 2/ can exchange position with
tile above it, namely, at position .1; 2/:

n1 n2 n3 n4 n1 n3 n4
n5 n6 n7 n5 n2 n6 n7
!
n8 n9 n10 n11 n8 n9 n10 n11
n12 n13 n14 n15 n12 n13 n14 n15

We will use the invariant method to prove that there is no way to reach the target
state starting from the initial state.
We begin by noting that a state can also be represented as a pair consisting of
two things:

1. a list of the numbers 1; : : : ; 15 in the order in which they appear—reading
rows left-to-right from the top row down, ignoring the empty square, and

2. the coordinates of the empty square—where the upper left square has coor-
dinates .1; 1/, the lower right .4; 4/.

(a) Write out the “list” representation of the start state and the “impossible” state.
Let L be a list of the numbers 1; : : : ; 15 in some order. A pair of integers is
an out-of-order pair in L when the first element of the pair both comes earlier in
the list and is larger, than the second element of the pair. For example, the list
1; 2; 4; 5; 3 has two out-of-order pairs: (4,3) and (5,3). The increasing list 1; 2 : : : n
has no out-of-order pairs.
Let a state S be a pair .L; .i; j // described above. We define the parity of S to
be 0 or 1 depending on whether the sum of the number of out-of-order pairs in L
and the row-number of the empty square is even or odd. that is
(
0 if p.L/ C i is even;
parity.S / WWD
1 otherwise:
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6.4. The Stable Marriage Problem 189

(b) Verify that the parity of the start state and the target state are different.

(c) Show that the parity of a state is preserved under transitions. Conclude that
“the impossible” is impossible to reach.
By the way, if two states have the same parity, then in fact there is a way to get
from one to the other. If you like puzzles, you’ll enjoy working this out on your
own.

Problem 6.8.
The Massachusetts Turnpike Authority is concerned about the integrity of the new
Zakim bridge. Their consulting architect has warned that the bridge may collapse
if more than 1000 cars are on it at the same time. The Authority has also been
warned by their traffic consultants that the rate of accidents from cars speeding
across bridges has been increasing.
Both to lighten traffic and to discourage speeding, the Authority has decided to
make the bridge one-way and to put tolls at both ends of the bridge (don’t laugh, this
is Massachusetts). So cars will pay tolls both on entering and exiting the bridge, but
the tolls will be different. In particular, a car will pay $3 to enter onto the bridge and
will pay $2 to exit. To be sure that there are never too many cars on the bridge, the
Authority will let a car onto the bridge only if the difference between the amount
of money currently at the entry toll booth and the amount at the exit toll booth is
strictly less than a certain threshold amount of $T0 .
The consultants have decided to model this scenario with a state machine whose
states are triples .A; B; C / of nonnegative integers, where
A is an amount of money at the entry booth,
B is an amount of money at the exit booth, and
C is a number of cars on the bridge.
Any state with C > 1000 is called a collapsed state, which the Authority dearly
hopes to avoid. There will be no transition out of a collapsed state.
Since the toll booth collectors may need to start off with some amount of money
in order to make change, and there may also be some number of “official” cars
already on the bridge when it is opened to the public, the consultants must be ready
to analyze the system started at any uncollapsed state. So let A0 be the initial
number of dollars at the entrance toll booth, B0 the initial number of dollars at the
exit toll booth, and C0 1000 the number of official cars on the bridge when it is
opened. You should assume that even official cars pay tolls on exiting or entering
the bridge after the bridge is opened.
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190 Chapter 6 State Machines

(a) Give a mathematical model of the Authority’s system for letting cars on and off
the bridge by specifying a transition relation between states of the form .A; B; C /
above.

(b) Characterize each of the following derived variables

A; B; A C B; A B; 3C A; 2A 3B; B C 3C; 2A 3B 6C; 2A 2B 3C

as one of the following
constant C
strictly increasing SI
strictly decreasing SD
weakly increasing but not constant WI
weakly decreasing but not constant WD
none of the above N
and briefly explain your reasoning.
The Authority has asked their engineering consultants to determine T and to
verify that this policy will keep the number of cars from exceeding 1000.
The consultants reason that if C0 is the number of official cars on the bridge
when it is opened, then an additional 1000 C0 cars can be allowed on the bridge.
So as long as A B has not increased by 3.1000 C0 /, there shouldn’t more than
1000 cars on the bridge. So they recommend defining

T0 WWD 3.1000 C0 / C .A0 B0 /; (6.4)

where A0 is the initial number of dollars at the entrance toll booth, B0 is the initial
number of dollars at the exit toll booth.
(c) Use the results of part (b) to define a simple predicate P on states of the tran-
sition system which is satisfied by the start state—that is P .A0 ; B0 ; C0 / holds—is
not satisfied by any collapsed state, and is a preserved invariant of the system. Ex-
plain why your P has these properties. Conclude that the traffic won’t cause the
bridge to collapse.

(d) A clever MIT intern working for the Turnpike Authority agrees that the Turn-
pike’s bridge management policy will be safe: the bridge will not collapse. But she
warns her boss that the policy will lead to deadlock—a situation where traffic can’t
move on the bridge even though the bridge has not collapsed.
Explain more precisely in terms of system transitions what the intern means, and
briefly, but clearly, justify her claim.
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6.4. The Stable Marriage Problem 191

Problem 6.9.
Start with 102 coins on a table, 98 showing heads and 4 showing tails. There are
two ways to change the coins:

(i) flip over any ten coins, or

(ii) let n be the number of heads showing. Place n C 1 additional coins, all
showing tails, on the table.

For example, you might begin by flipping nine heads and one tail, yielding 90
heads and 12 tails, then add 91 tails, yielding 90 heads and 103 tails.
(a) Model this situation as a state machine, carefully defining the set of states, the
start state, and the possible state transitions.

(b) Explain how to reach a state with exactly one tail showing.

C WWD the number of coins on the table; H WWD the number of heads;
T WWD the number of tails; C2 WWD remainder.C =2/;
H2 WWD remainder.H=2/; T2 WWD remainder.T =2/:

Which of these variables is

1. strictly increasing
2. weakly increasing
3. strictly decreasing
4. weakly decreasing
5. constant

(d) Prove that it is not possible to reach a state in which there is exactly one head
showing.

Problem 6.10.
A classroom is designed so students sit in a square arrangement. An outbreak of
beaver flu sometimes infects students in the class; beaver flu is a rare variant of bird
flu that lasts forever, with symptoms including a yearning for more quizzes and the
thrill of late night problem set sessions.
Here is an illustration of a 66-seat classroom with seats represented by squares.
The locations of infected students are marked with an asterisk.
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192 Chapter 6 State Machines

Outbreaks of infection spread rapidly step by step. A student is infected after a
step if either

the student was infected at the previous step (since beaver flu lasts forever),
or

the student was adjacent to at least two already-infected students at the pre-
vious step.

Here adjacent means the students’ individual squares share an edge (front, back,
left or right); they are not adjacent if they only share a corner point. So each student
is adjacent to 2, 3 or 4 others.
In the example, the infection spreads as shown below.

) )

In this example, over the next few time-steps, all the students in class become
infected.

Theorem. If fewer than n students among those in an nn arrangment are initially
infected in a flu outbreak, then there will be at least one student who never gets
infected in this outbreak, even if students attend all the lectures.

Prove this theorem.
Hint: Think of the state of an outbreak as an n n square above, with asterisks
indicating infection. The rules for the spread of infection then define the transitions
of a state machine. Find a weakly decreasing derived variable that leads to a proof
of this theorem.
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6.4. The Stable Marriage Problem 193

Exam Problems
Problem 6.11.
Token replacing-1-2 is a single player game using a set of tokens, each colored
black or white. Except for color, the tokens are indistinguishable. In each move,
a player can replace one black token with two white tokens, or replace one white
token with two black tokens.
We can model this game as a state machine whose states are pairs .nb ; nw / where
nb 0 equals the number of black tokens, and nw 0 equals the number of white
tokens.
(a) List the numbers of the following predicates that are preserved invariants.

nb C nw rem.nb C nw ; 3/ ¤ 2 (6.5)
nw nb rem.nw nb ; 3/ D 2 (6.6)
nb nw rem.nb n2 ; 3/ D 2 (6.7)
nb C nw > 5 (6.8)
nb C nw < 5 (6.9)

Now assume the game starts with a single black token, that is, the start state is
.1; 0/.

(b) List the numbers of the predicates above are true for all reachable states:

T .nb ; nw / WWD rem.nw nb ; 3/ D 2:

We will now prove the following:
Claim. If T .nb ; nw /, then state .nb ; nw / is reachable.
Note that this claim is different from the claim that T is a preserved invariant.
The proof of the Claim will be by induction in n using induction hypothesis P .n/WWD

8.nb ; nw /: Œ.nb C nw D n/ AND T .nb ; nw / IMPLIES .nb ; nw / is reachable:

The base cases will be when n 2.
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194 Chapter 6 State Machines

Assuming that the base cases have been verified, complete the Inductive
Step.
Now verify the Base Cases: P .n/ for n 2.

Problem 6.12.
Token Switching is a process for updating a set of black and white tokens. The
process starts with a single black token. At each step,

(i) one black token can be replaced with two white tokens, or

(ii) if the numbers of white and black tokens are not the same, the colors of all
the tokens can be switched: all the black tokens become white, and the white
tokens become black.

We can model Token Switching as a state machine whose states are pairs .b; w/
of nonnegative integers, where b equals the number of black tokens, and w equals
the number of white tokens. So the start state is .1; 0/.
(a) Indicate which of the following states can be reached from the start state in
exactly two steps:

.0; 0/; .1; 0/; .0; 1/; .1; 1/; .0; 2/; .2; 0/; .2; 1/; .1; 2/; .0; 3/; .3; 0/

(b) Define the predicate F .b; w/ by the rule:

F .b; w/ WWD .b w/is not a multiple of 3:

Prove the following
Claim. If F .b; w/, then state .b; w/ is reachable from the start state.
7777 88
(c) Explain why state .116 ; 510 / is not a reachable state.
Hint: Do not assume F is a preserved invariant without proving it.

Problem 6.13.
Token replacing-1-3 is a single player game using a set of tokens, each colored
black or white. In each move, a player can replace a black token with three white
tokens, or replace a white token with three black tokens. We can model this game
as a state machine whose states are pairs .b; w/ of nonnegative integers, where b is
the number of black tokens and w the number of white ones.
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6.4. The Stable Marriage Problem 195

The game has two possible start states: .5; 4/ or .4; 3/.
We call a state .b; w/ eligible when

rem.b w; 4/ D 1; AND (6.10)
minfb; wg 3: (6.11)

This problem examines the connection between eligible states and states that are
reachable from either of the possible start states.
(a) Give an example of a reachable state that is not eligible.

(b) Show that the derived variable b C w is strictly increasing. Conclude that state
.3; 2/ is not reachable.

For the rest of the problem, you may—and should—assume the following Fact:

Fact. If maxfb; wg 5 and .b; w/ is eligible, then .b; w/ is reachable.

(This is easy to verify since there are only nine states with b; w 2 f3; 4; 5g, but
don’t waste time doing this.)
(d) Define the predicate P .n/ to be:

8.b; w/:Œb C w D n AND .b; w/ is eligible IMPLIES .b; w/ is reachable:

Prove that P .n 1/ IMPLIES P .n C 1/ for all n 1.

(e) Conclude that all eligible states are reachable.

(f) Prove that .47 C 1; 45 C 2/ is not reachable.

(g) Verify that rem.3b w; 8/ is a derived variable that is constant. Conclude that
no state is reachable from both start states.

Problem 6.14.
There is a bucket containing more blue balls than red balls. As long as there are
more blues than reds, any one of the following rules may be applied to add and/or
remove balls from the bucket:

(i) Add a red ball.
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196 Chapter 6 State Machines

(ii) Remove a blue ball.

(iii) Add two reds and one blue.

(iv) Remove two blues and one red.

(a) Starting with 10 reds and 16 blues, what is the largest number of balls the
bucket will contain by applying these rules?
Let b be the number of blue balls and r be the number of red balls in the bucket
at any given time.
(b) Prove that b r 0 is a preserved invariant of the process of adding and
removing balls according to rules (i)–(iv).

(c) Prove that no matter how many balls the bucket contains, repeatedly applying
rules (i)–(iv) will eventually lead to a state where no further rule can be applied.

Problem 6.15.
The following problem is a twist on the Fifteen-Puzzle analyzed in Problem 6.7.
Let A be a sequence consisting of the numbers 1; : : : ; n in some order. A pair
of integers in A is called an out-of-order pair when the first element of the pair
both comes earlier in the sequence, and is larger, than the second element of the
pair. For example, the sequence .1; 2; 4; 5; 3/ has two out-of-order pairs: .4; 3/
and .5; 3/. We let t .A/ equal the number of out-of-order pairs in A. For example,
t ..1; 2; 4; 5; 3// D 2.
The elements in A can be rearranged using the Rotate-Triple operation, in which
three consecutive elements of A are rotated to move the smallest of them to be first.
For example, in the sequence .2; 4; 1; 5; 3/, the Rotate-Triple operation could
rotate the consecutive numbers 4; 1; 5, into 1; 5; 4 so that

.2; 4; 1; 5; 3/ ! .2; 1; 5; 4; 3/:

The Rotate-Triple could also rotate the consecutive numbers 2; 4; 1 into 1; 2; 4
so that
.2; 4; 1; 5; 3/ ! .1; 2; 4; 5; 3/:
We can think of a sequence A as a state of a state machine whose transitions
correspond to possible applications of the Rotate-Triple operation.
(a) Argue that the derived variable t is weakly decreasing.

(b) Prove that having an even number of out-of-order pairs is a preserved invariant
of this machine.
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6.4. The Stable Marriage Problem 197

T WWD .1; 2; : : : ; 2012; 2013; 2014/:

Problems for Section 6.4
Practice Problems
Problem 6.16.
Four Students want separate assignments to four VI-A Companies. Here are their
preference rankings:
Student Companies
Albert: HP, Bellcore, AT&T, Draper
Sarah: AT&T, Bellcore, Draper, HP
Tasha: HP, Draper, AT&T, Bellcore
Elizabeth: Draper, AT&T, Bellcore, HP

Company Students
AT&T: Elizabeth, Albert, Tasha, Sarah
Bellcore: Tasha, Sarah, Albert, Elizabeth
HP: Elizabeth, Tasha, Albert, Sarah
Draper: Sarah, Elizabeth, Tasha, Albert

(a) Use the Mating Ritual to find two stable assignments of Students to Compa-
nies.

(b) Describe a simple procedure to determine whether any given stable marriage
problem has a unique solution, that is, only one possible stable matching. Briefly
explain why it works.

Problem 6.17.
Suppose that Harry is one of the boys and Alice is one of the girls in the Mating
Ritual. Which of the properties below are preserved invariants? Why?

a. Alice is the only girl on Harry’s list.
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198 Chapter 6 State Machines

b. There is a girl who does not have any boys serenading her.

c. If Alice is not on Harry’s list, then Alice has a suitor that she prefers to Harry.

d. Alice is crossed off Harry’s list, and Harry prefers Alice to anyone he is
serenading.

e. If Alice is on Harry’s list, then she prefers Harry to any suitor she has.

Problem 6.18.
Prove that in a stable set of marriages, every man is the pessimal husband of his
optimal wife.
Hint: Follows directly from the definition of “rogue couple.”

Problem 6.19.
In the Mating Ritual for stable marriages between an equal number of boys and
girls, explain why there must be a girl to whom no boy proposes (serenades) until
the last day.

Class Problems
Problem 6.20.
The preferences among 4 boys and 4 girls are partially specified in the following
table:
B1: G1 G2 – –
B2: G2 G1 – –
B3: – – G4 G3
B4: – – G3 G4
G1: B2 B1 – –
G2: B1 B2 – –
G3: – – B3 B4
G4: – – B4 B3

(a) Verify that
.B1; G1/; .B2; G2/; .B3; G3/; .B4; G4/
will be a stable matching whatever the unspecified preferences may be.

(b) Explain why the stable matching above is neither boy-optimal nor boy-pessimal
and so will not be an outcome of the Mating Ritual.
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6.4. The Stable Marriage Problem 199

(c) Describe how to define a set of marriage preferences among n boys and n girls
which have at least 2n=2 stable assignments.
Hint: Arrange the boys into a list of n=2 pairs, and likewise arrange the girls into
a list of n=2 pairs of girls. Choose preferences so that the kth pair of boys ranks
the kth pair of girls just below the previous pairs of girls, and likewise for the kth
pair of girls. Within the kth pairs, make sure each boy’s first choice girl in the pair
prefers the other boy in the pair.

Problem 6.21.
The Mating Ritual of Section 6.4.1 for finding stable marriages works even when
the numbers of men and women are not equal. As before, a set of (monogamous)
marriages between men and women is called stable when it has no “rogue couples.”
(a) Extend the definition of rogue couple so it covers the case of unmarried men
and women. Verify that in a stable set of marriages, either all the men are married
or all the women are married.

(b) Explain why even in the case of unequal numbers of men and women, applying
the Mating Ritual will yield a stable matching.

Homework Problems
Problem 6.22.
Suppose we want to assign pairs of “buddies,” who may be of the sex, where each
person has a preference rank for who they would like to be buddies with. For the
preference ranking given in Figure 6.5, show that there is no stable buddy assign-
ment. In this figure Mergatroid’s preferences aren’t shown because they don’t even
matter.

Problem 6.23.
The most famous application of stable matching was in assigning graduating med-
ical students to hospital residencies. Each hospital has a preference ranking of
students, and each student has a preference ranking of hospitals, but unlike finding
stable marriages between an equal number of boys and girls, hospitals generally
have differing numbers of available residencies, and the total number of residen-
cies may not equal the number of graduating students.
Explain how to adapt the Stable Matching problem with an equal number of boys
and girls to this more general situation. In particular, modify the definition of stable
matching so it applies in this situation, and explain how to adapt the Mating Ritual
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200 Chapter 6 State Machines

Alex
2 1
3

1 2
Robin Bobby Joe
3 2 1 3

Mergatroid

Figure 6.5 Some preferences with no stable buddy matching.

to handle it.

Problem 6.24.
Give an example of a stable matching between 3 boys and 3 girls where no person
gets their first choice. Briefly explain why your matching is stable. Can your
matching be obtained from the Mating Ritual or the Ritual with boys and girls
reversed?

Problem 6.25.
In a stable matching between an equal number of boys and girls produced by the
Mating Ritual, call a person lucky if they are matched up with someone in the top
half of their preference list. Prove that there must be at least one lucky person.
Hint: The average number of times a boy gets rejected by girls.

Problem 6.26.
Suppose there are two stable sets of marriages. So each man has a first wife and a
second wife , and likewise each woman has a first husband and a second husband.
Someone in a given marriage is a winner when they prefer their current spouse
to their other spouse, and they are a loser when they prefer their other spouse to
their current spouse. (If someone has the same spouse in both of their marriages,
then they will be neither a winner nor a loser.)
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6.4. The Stable Marriage Problem 201

We will show that

In every marriage, someone is a winner iff their spouse is a loser. (WL)

This will lead to an alternative proof of Theorem 6.4.9 that when men are married
to their optimal spouses, women must be married to their pessimal spouses. This
alternative proof does not depend on the Mating Ritual of Section 6.4.1.
(a) The left to right direction of (WL) is equivalent to the assertion that married
partners cannot both be winners. Explain why this follows directly from the defini-
tion of rogue couple.

The right to left direction of (WL) is equivalent to the assertion that a married
couple cannot both be losers. This will follow by comparing the number of winners
and losers among the marriages.
(b) Explain why the number of winners must equal the number of losers among
the two sets of marriages.

(c) Complete the proof of (WL) by showing that if some married couple were both
losers, then there must be another couple who were both winners.

(d) Conclude that in a stable set of marriages, someone’s spouse is optimal iff they
are pessimal for their spouse.

Problem 6.27.
Suppose there are two stable sets of marriages, a first set and a second set. So
each man has a first wife and a second wife (they may be the same), and likewise
each woman has a first husband and a second husband. We can form a third set
of marriages by matching each man with the wife he prefers among his first and
second wives.
(a) Prove that this third set of marriages is an exact matching: no woman is mar-
ried to two men.

(b) Prove that this third marriage set is stable.
Hint: You may assume the following fact from Problem 6.26.

In every marriage, someone is a winner iff their spouse is a loser. (SL)
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202 Chapter 6 State Machines

Problem 6.28.
A state machine has commuting transitions if for any states p; q; r

.p ! q AND p ! r/ IMPLIES 9t: q ! t AND r ! t:

The state machine is confluent if

.p ! q AND p ! r/ IMPLIES 9t: q ! t AND r ! t:

(a) Prove that if a state machine has commuting transitions, then it is confluent.
Hint: By induction on the number of moves from p to q plus the number from p
to r.

(b) A final state of a state machine is one from which no transition is possible.
Explain why, if a state machine is confluent, then at most one final state is reachable
from the start state.

Problem 6.29.
According to the day-by-day description of the Mating Ritual of Section 6.4.1, at
the end of each day, every the man’s list is updated to remove the name of the
woman he who rejected him. But it’s easier, and more flexible, simply to let one
women reject one suitor at a time.
In particular, the states of this Flexible Mating Ritual state machine will be the
same as for the day-by-day Ritual: a state will be a list, for each man, of the women
who have not rejected him. But now a transition will be to choose two men who
are serenading the same woman—that is, who have the same woman at the top
of their current lists—and then have the woman reject whichever of the two she
likes less. So the only change in state is that the name of the serenaded woman
gets deleted from the top of the list of the man she liked less among two of her
serenaders—everything else stays the same.
It’s a worthwhile review to verify that the same preserved invariants used to es-
tablish the properties of the Mating Ritual will apply to the Flexible Mating Ritual.
This ensures that the Flexible Ritual will also terminate with a stable set of mar-
riages.
But now a new issue arises: we know that there can be many sets of possible
sets of stable marriages for the same set of men/women preferences. So it seems
possible that the Flexible Ritual might terminate with different stable marriage sets,
depending on which choice of transition was made at each state. But this does
not happen: the Flexible Ritual will always terminate with the same set of stable
marriages as the day-by-day Ritual.
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6.4. The Stable Marriage Problem 203

To prove this, we begin with a definition: a state machine has commuting transi-
tions if for any states p; q; r,

.p ! q AND p ! r/ IMPLIES 9t: q ! t AND r ! t:

(a) Verify that the Flexible Mating Ritual has commuting transitions.

(b) Now conclude from Problem 6.28 that the Flexible Mating Ritual always ter-
minate with the same set of stable marriages as the day-by-day Ritual.

Exam Problems
Problem 6.30.
Four unfortunate children want to be adopted by four foster families of ill repute.
A child can only be adopted by one family, and a family can only adopt one child.
Here are their preference rankings (most-favored to least-favored):
Child Families
Bottlecap: Hatfields, McCoys, Grinches, Scrooges
Lucy: Grinches, Scrooges, McCoys, Hatfields
Dingdong: Hatfields, Scrooges, Grinches, McCoys
Zippy: McCoys, Grinches, Scrooges, Hatfields

Family Children
Grinches: Zippy, Dingdong, Bottlecap, Lucy
Hatfields: Zippy, Bottlecap, Dingdong, Lucy
Scrooges: Bottlecap, Lucy, Dingdong, Zippy
McCoys: Lucy, Zippy, Bottlecap, Dingdong

(a) Exhibit two different stable matching of Children and Families.
Family Child in 1st match Child in 2nd match
Grinches:
Hatfields:
Scrooges:
McCoys:

(b) Examine the matchings from part a, and explain why these matchings are the
only two possible stable matchings between Children and Families.
Hint: In general, there may be many more than two stable matchings for the same
set of preferences.
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204 Chapter 6 State Machines

Problem 6.31.
The Mating Ritual 6.4.1 for finding stable marriages works without change when
there are at least as many, and possibly more, men than women. You may assume
this. So the Ritual ends with all the women married and no rogue couples for these
marriages, where an unmarried man and a married woman who prefers him to her
spouse is also considered to be a “rogue couple.”
Let Alice be one of the women, and Bob be one of the men. Indicate which of
the properties below that are preserved invariants of the Mating Ritual 6.4 when
there are at least as many men as women. Briefly explain your answers.
(a) Alice has a suitor (man who is serenading her) whom she prefers to Bob.

(b) Alice is the only woman on Bob’s list.

(d) Bob prefers Alice to the women he is serenading.

(e) Bob is serenading Alice.

(f) Bob is not serenading Alice.

(g) Bob’s list of women to serenade is empty.

Problem 6.32.
We want a stable matching between n boys and n girls for a positive integer n.
(a) Explain how to define preference rankings for the boys and the girls that allow
only one possible stable matching. Briefly justify your answer.

(b) Mark each of the following predicates about the Stable Marriage Ritual P if it
is a Preserved Invariant, N if it is not, and “U” if you are very unsure. “Bob’s list”
refers to the list of the women he has not crossed off.

(i) Alice is not on Bob’s list.
(ii) No girl is on Bob’s list.
(iii) Bob is the only boy serenading Alice.
(iv) Bob has fewer than 5 girls on his list.
(v) Bob prefers Alice to his favorite remaining girl.
(vi) Alice prefers her favorite current suitor to Bob.
(vii) Bob is serenading his optimal spouse.
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6.4. The Stable Marriage Problem 205

(viii) Bob is serenading his pessimal spouse.
(ix) Alice’s optimal spouse is serenading her.
(x) Alice’s pessimal spouse is serenading her.
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7 Recursive Data Types
Recursive data types play a central role in programming, and induction is really all
about them.
Recursive data types are specified by recursive definitions, which say how to
construct new data elements from previous ones. Along with each recursive data
type there are recursive definitions of properties or functions on the data type. Most
importantly, based on a recursive definition, there is a structural induction method
for proving that all data of the given type have some property.
This chapter examines a few examples of recursive data types and recursively
defined functions on them:
strings of characters,

“balanced” strings of brackets,

the nonnegative integers, and

arithmetic expressions.

two-player games with perfect information.

7.1 Recursive Definitions and Structural Induction
We’ll start off illustrating recursive definitions and proofs using the example of
character strings. Normally we’d take strings of characters for granted, but it’s
informative to treat them as a recursive data type. In particular, strings are a nice
first example because you will see recursive definitions of things that are easy to
understand, or that you already know, so you can focus on how the definitions work
without having to figure out what they are supposed to mean.
Definitions of recursive data types have two parts:

Base case(s) specifying that some known mathematical elements are in the
data type, and

Constructor case(s) that specify how to construct new data elements from
previously constructed elements or from base elements.
The definition of strings over a given character set A follows this pattern:
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208 Chapter 7 Recursive Data Types

Definition 7.1.1. Let A be a nonempty set called an alphabet, whose elements are
referred to as characters (also called letters, symbols, or digits). The recursive data
type A of strings over alphabet A is defined as follows:

Base case: the empty string is in A .

Constructor case: If a 2 A and s 2 A , then the pair ha; si 2 A .

So f0; 1g are the binary strings.
The usual way to treat binary strings is as sequences of 0’s and 1’s. For example,
we have identified the length-4 binary string 1011 as a sequence of bits, the 4-tuple
.1; 0; 1; 1/. But according to the recursive Definition 7.1.1, this string would be
represented by nested pairs, namely

h1; h0; h1; h1; iiii :

These nested pairs are definitely cumbersome and may also seem bizarre, but they
actually reflect the way that such lists of characters would be represented in pro-
gramming languages like Scheme or Python, where ha; si would correspond to
cons.a; s/.
Notice that we haven’t said exactly how the empty string is represented. It really
doesn’t matter, as long as we can recognize the empty string and not confuse it with
any nonempty string.
Continuing the recursive approach, let’s define the length of a string.

Definition 7.1.2. The length jsj of a string s is defined recursively based on Defi-
nition 7.1.1.

Base case: jj WWD 0.

Constructor case: j ha; si j WWD 1 C jsj.

This definition of length follows a standard pattern: functions on recursive data
types can be defined recursively using the same cases as the data type definition.
Specifically, to define a function f on a recursive data type, define the value of f
for the base cases of the data type definition, then define the value of f in each
constructor case in terms of the values of f on the component data items.
Let’s do another example: the concatenation s t of the strings s and t is the
string consisting of the letters of s followed by the letters of t. This is a per-
fectly clear mathematical definition of concatenation (except maybe for what to do
with the empty string), and in terms of Scheme/Python lists, s t would be the list
append.s; t /. Here’s a recursive definition of concatenation.
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7.1. Recursive Definitions and Structural Induction 209

Definition 7.1.3. The concatenation s t of the strings s; t 2 A is defined recur-
sively based on Definition 7.1.1:

Base case:
t WWD t:

Constructor case:
ha; si t WWD ha; s ti :

7.1.1 Structural Induction
Structural induction is a method for proving that all the elements of a recursively
defined data type have some property. A structural induction proof has two parts
corresponding to the recursive definition:

Prove that each base case element has the property.

Prove that each constructor case element has the property, when the construc-
tor is applied to elements that have the property.

For example, in the base case of the definition of concatenation 7.1.3, we defined
concatenation so the empty string was a “left identity,” namely, s WWD s. We want
the empty string also to be “right identity,” namely, s D s. Being a right identity
is not part of Definition 7.1.3, but we can prove it easily by structural induction:

Lemma 7.1.4.
sDs
for all s 2 A .

Proof. The proof is by structural induction on the recursive definition 7.1.3 of con-
catenation. The induction hypothesis will be

P .s/ WWD Œs D s:

Base case: (s D ).

sD
D ( is a left identity by Def 7.1.3)
D s:
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210 Chapter 7 Recursive Data Types

Constructor case: (s D a t ).
s D .a t /
D a .t / (Constructor case of Def 7.1.3)
Dat by induction hypothesis P .t /
D s:
So P .s/ holds. This completes the proof of the constructor case, and we conclude
by structural induction that equation (7.1.4) holds for all s 2 A .

We can also verify properties of recursive functions by structural induction on
their definitions. For example, let’s verify the familiar fact that the length of the
concatenation of two strings is the sum of their lengths:
Lemma.
js t j D jsj C jtj
for all s; t 2 A .
Proof. By structural induction on the definition of s 2 A . The induction hypoth-
esis is
P .s/ WWD 8t 2 A : js t j D jsj C jtj:

Base case (s D ):
js tj D j tj
D jt j (base case of Def 7.1.3 of concatenation)
D 0 C jt j
D jsj C jt j (Def of jj):

Constructor case: (s WWD ha; ri).
js tj D j ha; ri tj
D j ha; r t i j (constructor case of Def of concat)
D 1 C jr t j (constructor case of def length)
D 1 C .jrj C jtj/ (ind. hyp. P .r/)
D .1 C jrj/ C jtj
D j ha; ri j C jtj (constructor case, def of length)
D jsj C jtj:
This proves that P .s/ holds, completing the constructor case. By structural induc-
tion, we conclude that P .s/ holds for all strings s 2 A .
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7.2. Strings of Matched Brackets 211

These proofs illustrate the general principle:

The Principle of Structural Induction.
Let P be a predicate on a recursively defined data type R. If

P .b/ is true for each base case element b 2 R, and

for all two-argument constructors c,

ŒP .r/ AND P .s/ IMPLIES P .c.r; s//

for all r; s 2 R,
and likewise for all constructors taking other numbers of arguments,

then
P .r/ is true for all r 2 R:

7.2 Strings of Matched Brackets
Let f] ; [ g be the set of all strings of square brackets. For example, the following
two strings are in f] ; [ g :

[]][[[[[]] and [ [ [ ] ] [ ] ] [ ] (7.1)

A string s 2 f] ; [ g is called a matched string if its brackets “match up” in
the usual way. For example, the left-hand string above is not matched because its
second right bracket does not have a matching left bracket. The string on the right
is matched.
We’re going to examine several different ways to define and prove properties
of matched strings using recursively defined sets and functions. These properties
are pretty straightforward, and you might wonder whether they have any particular
relevance in computer science. The honest answer is “not much relevance any
more.” The reason for this is one of the great successes of computer science, as
explained in the text box below.
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212 Chapter 7 Recursive Data Types

Expression Parsing
During the early development of computer science in the 1950’s and 60’s, creation
of effective programming language compilers was a central concern. A key aspect
in processing a program for compilation was expression parsing. One significant
problem was to take an expression like

x C y z2 y C 7

and put in the brackets that determined how it should be evaluated—should it be

ŒŒx C y z 2 y C 7; or;
x C Œy z 2 Œy C 7; or;
Œx C Œy z 2 Œy C 7; or : : :‹

The Turing award (the “Nobel Prize” of computer science) was ultimately be-
stowed on Robert W. Floyd, for, among other things, discovering simple proce-
dures that would insert the brackets properly.
In the 70’s and 80’s, this parsing technology was packaged into high-level
compiler-compilers that automatically generated parsers from expression gram-
mars. This automation of parsing was so effective that the subject no longer
demanded attention. It had largely disappeared from the computer science cur-
riculum by the 1990’s.

The matched strings can be nicely characterized as a recursive data type:
Definition 7.2.1. Recursively define the set RecMatch of strings as follows:

Base case: 2 RecMatch.

Constructor case: If s; t 2 RecMatch, then
[ s ] t 2 RecMatch:

Here [ s ] t refers to the concatenation of strings which would be written in full
as
[ .s .] t //:
From now on, we’ll usually omit the “’s.”
Using this definition, 2 RecMatch by the base case, so letting s D t D in
the constructor case implies
[ ] D [ ] 2 RecMatch:
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7.2. Strings of Matched Brackets 213

Now,

[ ] [ ] D [ ] [ ] 2 RecMatch (letting s D ; t D [ ] )
[ [ ] ] D [ [ ] ] 2 RecMatch (letting s D [ ] ; t D )
[ [ ] ] [ ] 2 RecMatch (letting s D [ ] ; t D [ ] )

are also strings in RecMatch by repeated applications of the constructor case; and
so on.
It’s pretty obvious that in order for brackets to match, there had better be an equal
number of left and right ones. For further practice, let’s carefully prove this from
the recursive definitions, beginning with a recursive definition of the number #c .s/
of occurrences of the character c 2 A in a string s:

Definition 7.2.2.
Base case: #c ./ WWD 0.
Constructor case:
(
#c .s/ if a ¤ c;
#c .ha; si/ WWD
1 C #c .s/ if a D c:

The following Lemma follows directly by structural induction on Definition 7.2.2.
We’ll leave the proof for practice (Problem 7.9).

Lemma 7.2.3.
#c .s t / D #c .s/ C #c .t /:

Lemma. Every string in RecMatch has an equal number of left and right brackets.

Proof. The proof is by structural induction with induction hypothesis
h i
P .s/ WWD #[ .s/ D #] .s/ :

Base case: P ./ holds because

#[ ./ D 0 D #] ./

by the base case of Definition 7.2.2 of #c ./.
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214 Chapter 7 Recursive Data Types

Constructor case: By structural induction hypothesis, we assume P .s/ and P .t /
and must show P .[ s ] t /:

#[ .[ s ] t / D #[ .[ / C #[ .s/ C #[ .] / C #[ .t / (Lemma 7.2.3)
D 1 C #[ .s/ C 0 C #[ .t / (def #[ ./)
D 1 C #] .s/ C 0 C #] .t / (by P .s/ and P .t /)
D 0 C #] .s/ C 1 C #] .t /
D #] .[ / C #] .s/ C #] .] / C #] .t / (def #] ./)
D #] .[ s ] t / (Lemma 7.2.3)

This completes the proof of the constructor case. We conclude by structural induc-
tion that P .s/ holds for all s 2 RecMatch.

Warning: When a recursive definition of a data type allows the same element
to be constructed in more than one way, the definition is said to be ambiguous.
We were careful to choose an unambiguous definition of RecMatch to ensure that
functions defined recursively on its definition would always be well-defined. Re-
cursively defining a function on an ambiguous data type definition usually will not
work. To illustrate the problem, here’s another definition of the matched strings.

Definition 7.2.4. Define the set, AmbRecMatch f] ; [ g recursively as follows:

Base case: 2 AmbRecMatch,

Constructor cases: if s; t 2 AmbRecMatch, then the strings [ s ] and st are
also in AmbRecMatch.

It’s pretty easy to see that the definition of AmbRecMatch is just another way
to define RecMatch, that is AmbRecMatch D RecMatch (see Problem 7.19). The
definition of AmbRecMatch is arguably easier to understand, but we didn’t use it
because it’s ambiguous, while the trickier definition of RecMatch is unambiguous.
Here’s why this matters. Let’s define the number of operations f .s/ to construct a
matched string s recursively on the definition of s 2 AmbRecMatch:

f ./ WWD 0; (f base case)
f .[ s ] / WWD 1 C f .s/;
f .st / WWD 1 C f .s/ C f .t /: (f concat case)
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7.3. Recursive Functions on Nonnegative Integers 215

This definition may seem ok, but it isn’t: f ./ winds up with two values, and
consequently:

0 D f ./ (f base case))
D f . / (concat def, base case)
D 1 C f ./ C f ./ (f concat case);
D1C0C0D1 (f base case):

This is definitely not a situation we want to be in!

7.3 Recursive Functions on Nonnegative Integers
The nonnegative integers can be understood as a recursive data type.

Definition 7.3.1. The set N is a data type defined recursively as:

0 2 N.

If n 2 N, then the successor n C 1 of n is in N.

The point here is to make it clear that ordinary induction is simply the special
case of structural induction on the recursive Definition 7.3.1. This also justifies the
familiar recursive definitions of functions on the nonnegative integers.

7.3.1 Some Standard Recursive Functions on N
Example 7.3.2. The factorial function. This function is often written “nŠ.” You will
see a lot of it in later chapters. Here, we’ll use the notation fac.n/:

fac.0/ WWD 1.

fac.n C 1/ WWD .n C 1/ fac.n/ for n 0.

Example 7.3.3. Summation notation. Let “S.n/” abbreviate the expression “ niD1 f .i /.”
P
We can recursively define S.n/ with the rules

S.0/ WWD 0.

S.n C 1/ WWD f .n C 1/ C S.n/ for n 0.
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216 Chapter 7 Recursive Data Types

7.3.2 Ill-formed Function Definitions
There are some other blunders to watch out for when defining functions recursively.
The main problems come when recursive definitions don’t follow the recursive def-
inition of the underlying data type. Below are some function specifications that re-
semble good definitions of functions on the nonnegative integers, but really aren’t.

f1 .n/ WWD 2 C f1 .n 1/: (7.2)

This “definition” has no base case. If some function f1 satisfied (7.2), so would a
function obtained by adding a constant to the value of f1 . So equation (7.2) does
not uniquely define an f1 .

(
0; if n D 0;
f2 .n/ WWD (7.3)
f2 .n C 1/ otherwise:

This “definition” has a base case, but still doesn’t uniquely determine f2 . Any
function that is 0 at 0 and constant everywhere else would satisfy the specification,
so (7.3) also does not uniquely define anything.
In a typical programming language, evaluation of f2 .1/ would begin with a re-
cursive call of f2 .2/, which would lead to a recursive call of f2 .3/, . . . with recur-
sive calls continuing without end. This “operational” approach interprets (7.3) as
defining a partial function f2 that is undefined everywhere but 0.

8
<0; if n is divisible by 2,
ˆ
f3 .n/ WWD 1; if n is divisible by 3, (7.4)
ˆ
2; otherwise.
:

This “definition” is inconsistent: it requires f3 .6/ D 0 and f3 .6/ D 1, so (7.4)
doesn’t define anything.
Mathematicians have been wondering about this function specification, known
as the Collatz conjecture for a while:
8
<1;
ˆ if n 1;
f4 .n/ WWD f4 .n=2/ if n > 1 is even; (7.5)
ˆ
f4 .3n C 1/ if n > 1 is odd:
:

For example, f4 .3/ D 1 because

f4 .3/ WWD f4 .10/ WWD f4 .5/ WWD f4 .16/ WWD f4 .8/ WWD f4 .4/ WWD f4 .2/ WWD f4 .1/ WWD 1:
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7.4. Arithmetic Expressions 217

The constant function equal to 1 will satisfy (7.5), but it’s not known if another
function does as well. The problem is that the third case specifies f4 .n/ in terms
of f4 at arguments larger than n, and so cannot be justified by induction on N. It’s
known that any f4 satisfying (7.5) equals 1 for all n up to over 1018 .
A final example is the Ackermann function, which is an extremely fast-growing
function of two nonnegative arguments. Its inverse is correspondingly slow-growing—
it grows slower than log n, log log n, log log log n, . . . , but it does grow unboundly.
This inverse actually comes up analyzing a useful, highly efficient procedure known
as the Union-Find algorithm. This algorithm was conjectured to run in a number
of steps that grew linearly in the size of its input, but turned out to be “linear”
but with a slow growing coefficient nearly equal to the inverse Ackermann func-
tion. This means that pragmatically, Union-Find is linear, since the theoretically
growing coefficient is less than 5 for any input that could conceivably come up.
The Ackermann function can be defined recursively as the function A given by
the following rules:

A.m; n/ D 2n if m D 0 or n 1; (7.6)
A.m; n/ D A.m 1; A.m; n 1// otherwise: (7.7)

Now these rules are unusual because the definition of A.m; n/ involves an eval-
uation of A at arguments that may be a lot bigger than m and n. The definitions
of f2 above showed how definitions of function values at small argument values in
terms of larger one can easily lead to nonterminating evaluations. The definition of
the Ackermann function is actually ok, but proving this takes some ingenuity (see
Problem 7.25).

7.4 Arithmetic Expressions
Expression evaluation is a key feature of programming languages, and recognition
of expressions as a recursive data type is a key to understanding how they can be
processed.
To illustrate this approach we’ll work with a toy example: arithmetic expressions
like 3x 2 C 2x C 1 involving only one variable, “x.” We’ll refer to the data type of
such expressions as Aexp. Here is its definition:

Definition 7.4.1.

Base cases:
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218 Chapter 7 Recursive Data Types

– The variable x is in Aexp.
– The arabic numeral k for any nonnegative integer k is in Aexp.

Constructor cases: If e; f 2 Aexp, then

– [ e + f ] 2 Aexp. The expression [ e + f ] is called a sum. The Aexp’s
e and f are called the components of the sum; they’re also called the
summands.
– [ e * f ] 2 Aexp. The expression [ e * f ] is called a product. The
Aexp’s e and f are called the components of the product; they’re also
called the multiplier and multiplicand.
– - [ e ] 2 Aexp. The expression - [ e ] is called a negative.

Notice that Aexp’s are fully bracketed, and exponents aren’t allowed. So the
Aexp version of the polynomial expression 3x 2 C2x C1 would officially be written
as
[ [ 3 * [ x * x ] ] + [ [ 2 * x ] + 1] ] : (7.8)
These brackets and ’s clutter up examples, so we’ll often use simpler expressions
like “3x 2 C2xC1” instead of (7.8). But it’s important to recognize that 3x 2 C2xC1
is not an Aexp; it’s an abbreviation for an Aexp.

7.4.1 Evaluation and Substitution with Aexp’s
Evaluating Aexp’s
Since the only variable in an Aexp is x, the value of an Aexp is determined by the
value of x. For example, if the value of x is 3, then the value of 3x 2 C 2x C 1
is 34. In general, given any Aexp e and an integer value n for the variable x we
can evaluate e to finds its value eval.e; n/. It’s easy, and useful, to specify this
evaluation process with a recursive definition.

Definition 7.4.2. The evaluation function, eval W Aexp Z ! Z, is defined recur-
sively on expressions e 2 Aexp as follows. Let n be any integer.

Base cases:

eval.x; n/ WWD n (value of variable x is n), (7.9)
eval.k; n/ WWD k (value of numeral k is k, regardless of x.) (7.10)
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7.4. Arithmetic Expressions 219

Constructor cases:

eval.[ e1 + e2 ] ; n/ WWD eval.e1 ; n/ C eval.e2 ; n/; (7.11)
eval.[ e1 * e2 ] ; n/ WWD eval.e1 ; n/ eval.e2 ; n/; (7.12)
eval.- [ e1 ] ; n/ WWD eval.e1 ; n/: (7.13)

For example, here’s how the recursive definition of eval would arrive at the value
of 3 C x 2 when x is 2:

eval.[ 3 + [ x * x ] ] ; 2/ D eval.3; 2/ C eval.[ x * x ] ; 2/ (by Def 7.4.2.7.11)
D 3 C eval.[ x * x ] ; 2/ (by Def 7.4.2.7.10)
D 3 C .eval.x; 2/ eval.x; 2// (by Def 7.4.2.7.12)
D 3 C .2 2/ (by Def 7.4.2.7.9)
D 3 C 4 D 7:

Substituting into Aexp’s
Substituting expressions for variables is a standard operation used by compilers
and algebra systems. For example, the result of substituting the expression 3x for
x in the expression x.x 1/ would be 3x.3x 1/. We’ll use the general notation
subst.f; e/ for the result of substituting an Aexp f for each of the x’s in an Aexp
e. So as we just explained,

subst.3x; x.x 1// D 3x.3x 1/:

This substitution function has a simple recursive definition:

Definition 7.4.3. The substitution function from Aexp Aexp to Aexp is defined
recursively on expressions e 2 Aexp as follows. Let f be any Aexp.

Base cases:

subst.f; x/ WWD f (subbing f for variable x just gives f ,) (7.14)
subst.f; k/ WWD k (subbing into a numeral does nothing.) (7.15)

Constructor cases:
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220 Chapter 7 Recursive Data Types

subst.f; [ e1 + e2 ] / WWD [ subst.f; e1 / + subst.f; e2 /] (7.16)
subst.f; [ e1 * e2 ] / WWD [ subst.f; e1 / * subst.f; e2 /] (7.17)
subst.f; - [ e1 ] / WWD - [ subst.f; e1 /] : (7.18)

Here’s how the recursive definition of the substitution function would find the
result of substituting 3x for x in the expression x.x 1/:

subst.3x; x.x 1//
D subst.[ 3 * x ] ; [ x * [ x + - [ 1] ] ] / (unabbreviating)
D [ subst.[ 3 * x ] ; x/ *
subst.[ 3 * x ] ; [ x + - [ 1] ] /] (by Def 7.4.3 7.17)
D [ [ 3 * x ] * subst.[ 3 * x ] ; [ x + - [ 1] ] /] (by Def 7.4.3 7.14)
D [ [ 3 * x ] * [ subst.[ 3 * x ] ; x/
+ subst.[ 3 * x ] ; - [ 1] /] ] (by Def 7.4.3 7.16)
D [ [ 3 * x ] * [ [ 3 * x ] + - [ subst.[ 3 * x ] ; 1/] ] ] (by Def 7.4.3 7.14 & 7.18)
D [ [ 3 * x ] * [ [ 3 * x ] + - [ 1] ] ] (by Def 7.4.3 7.15)
D 3x.3x 1/ (abbreviation)

Now suppose we have to find the value of subst.3x; x.x 1// when x D 2.
There are two approaches. First, we could actually do the substitution above to get
3x.3x 1/, and then we could evaluate 3x.3x 1/ when x D 2, that is, we could
recursively calculate eval.3x.3x 1/; 2/ to get the final value 30. This approach is
described by the expression

eval.subst.3x; x.x 1//; 2/: (7.19)

In programming jargon, this would be called evaluation using the Substitution
Model. With this approach, the formula 3x appears twice after substitution, so
the multiplication 3 2 that computes its value gets performed twice.
The second approach is called evaluation using the Environment Model. Here, to
compute the value of (7.19), we evaluate 3x when x D 2 using just 1 multiplication
to get the value 6. Then we evaluate x.x 1/ when x has this value 6 to arrive at
the value 6 5 D 30. This approach is described by the expression

eval.x.x 1/; eval.3x; 2//: (7.20)

The Environment Model only computes the value of 3x once, and so it requires one
fewer multiplication than the Substitution model to compute (7.20).
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7.4. Arithmetic Expressions 221

This is a good place to stop and work this example out yourself (Problem 7.26).
The fact that the final integer values of (7.19) and (7.20) agree is no surprise. The
substitution model and environment models will always produce the same final. We
can prove this by structural induction directly following the definitions of the two
approaches. More precisely, what we want to prove is

Theorem 7.4.4. For all expressions e; f 2 Aexp and n 2 Z,

eval.subst.f; e/; n/ D eval.e; eval.f; n//: (7.21)

Proof. The proof is by structural induction on e.1
Base cases:

Case[x]
The left-hand side of equation (7.21) equals eval.f; n/ by this base case in
Definition 7.4.3 of the substitution function; the right-hand side also equals
eval.f; n/ by this base case in Definition 7.4.2 of eval.

Case[k].
The left-hand side of equation (7.21) equals k by this base case in Defini-
tions 7.4.3 and 7.4.2 of the substitution and evaluation functions. Likewise,
the right-hand side equals k by two applications of this base case in the Def-
inition 7.4.2 of eval.

Constructor cases:

Case[[ e1 + e2 ] ]
By the structural induction hypothesis (7.21), we may assume that for all
f 2 Aexp and n 2 Z,

eval.subst.f; ei /; n/ D eval.ei ; eval.f; n// (7.22)
for i D 1; 2. We wish to prove that

eval.subst.f; [ e1 + e2 ] /; n/ D eval.[ e1 + e2 ] ; eval.f; n//: (7.23)

The left-hand side of (7.23) equals

eval.[ subst.f; e1 / + subst.f; e2 /] ; n/
1 This is an example of why it’s useful to notify the reader what the induction variable is—in this
case it isn’t n.
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222 Chapter 7 Recursive Data Types

by Definition 7.4.3.7.16 of substitution into a sum expression. But this equals

eval.subst.f; e1 /; n/ C eval.subst.f; e2 /; n/

by Definition 7.4.2.(7.11) of eval for a sum expression. By induction hypoth-
esis (7.22), this in turn equals

eval.e1 ; eval.f; n// C eval.e2 ; eval.f; n//:

Finally, this last expression equals the right-hand side of (7.23) by Defini-
tion 7.4.2.(7.11) of eval for a sum expression. This proves (7.23) in this case.

Case[[ e1 * e2 ] ] Similar.

Case[ [ e1 ] ] Even easier.

This covers all the constructor cases, and so completes the proof by structural
induction.

7.5 Games as a Recursive Data Type
Chess, Checkers, Go, and Nim are examples of two-person games of perfect in-
formation. These are games where two players, Player-1 and Player-2, alternate
moves, and “perfect information” means that the situation at any point in the game
is completely visible to both players. In Chess, for example, the visible positions
of the pieces on the chess board completely determine how the rest of the game can
be played by each player. By contrast, most card games are not games of perfect
information because neither player can see the other’s hand.
In the section we’ll examine the win-lose two-person games of perfect informa-
tion, WL-2PerGm. We will define WL-2PerGm as a recursive data type, and then
we will prove, by structural induction, a fundamental theorem about winning strate-
gies for these games. The idea behind the recursive definition is to recognize that
the situation at any point during game play can itself be treated as the start of a new
game. This is clearest for the game of Nim.
A Nim game starts with several piles of stones. A move in the game consists of
removing some positive number of stones from a single pile. Player-1 and player-2
alternate making moves, and whoever takes the last stone wins. So if there is only
one pile, then the first player to move wins by taking the whole pile. On the hand,
if the game starts with just two piles, each with the same number of stones, then the
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7.5. Games as a Recursive Data Type 223

player who moves second can guarantee a win simply by mimicking the first player.
For example, this means that if the first player removes three stones from one pile,
then the second player removes three stones from the other pile. At this point, it’s
worth thinking for a moment about why the mimicking strategy guarantees a
win for the second player.
We can think of the first move in a Nim game as simply picking another Nim
game with different piles of stone to play next. For the Nim game Nimh3;4;5i that
starts with piles of 3, 4 and 5 stones, the first player can remove between one and
three stones from the first pile leading to three possible piles of stones

h2; 4; 5i ; h1; 4; 5i ; h4; 5i :

Similarly, the first player has five possible ways to remove stones from the last pile,
leading to five possible piles of stones

h3; 4; 4i ; h3; 4; 3i ; h3; 4; 2i ; h3; 4; 1i ; h3; 4i :

So all the properties of Nimh3;4;5i are captured by the set of 3 C 4 C 5 D 12 Nim
games that can result from the first move.
With this idea in mind, we now give the formal definition.
Definition 7.5.1. The class WL-2PerGm of two-person win-lose games of perfect
information is defined recursively as follows:
Base case: win and lose are WL-2PerGm’s.
Constructor case: If G is a nonempty set of WL-2PerGm’s, then G is a WL-2PerGm
game. Each game M 2 G is called a possible first move of G.
A play of a WL-2PerGm game is a sequence of moves that ends with a win or
loss for the first player, or goes on forever without arriving at an outcome.2 More
formally:
Definition. A play of a WL-2PerGm game G and its outcome is defined recursively
on the definition of WL-2PerGm:
Base case: (G D win). The sequence hwini of length one is a play of G. Its
outcome is a win.
Base case: (G D lose). The sequence hlosei of length one is a play of G. Its
outcome is a loss.
2 In English, “Nim game” might refer to the rules that define the game, but it might also refer
to a particular play of the game—as in the once famous third game in the 1961 movie Last Year at
Marienbad. It’s usually easy to figure out which way the phrase in being used, and we won’t worry
about it.
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224 Chapter 7 Recursive Data Types

Constructor case: (G is a nonempty set of WL-2PerGm’s). A play of G is a
sequence that starts with G followed by a play PM of some game M 2 G. The
outcome of the play, if any, is the outcome of PM .

The basic rules of some games do allow plays that go on forever. In Chess for
example, a player might just keep moving the same piece back and forth, and if his
opponent did the same, the play could go on forever.3 But the recursive definition
of WL-2PerGm games actually rules out the possibility of infinite play.

Lemma 7.5.2. Every play of a game G 2 WL-2PerGm has an outcome.

Proof. We prove Lemma 7.5.2 by structural induction, using the statement of the
Lemma as the induction hypothesis.
Base case: (G D win). There is only one play of G, namely the length one play
hwini, whose outcome is a win.
Base case: (G D lose). Likewise with the outcome being a loss.
Constructor case: (G is a nonempty set of WL-2PerGm’s). A play of G by defini-
tion consists G followed by a play PM for some M 2 G. By structural induction,
PM must be a sequence of some finite length n that ends with an outcome. So this
play of G is a length n C 1 sequence that finishes with the same outcome.

Among the games of Checker, Chess, Go and Nim, only Nim is genuinely a win-
lose game, The other games might end in a tie (draw, stalemate, jigo) rather than a
win or loss. However, by treating a tie in these games as a loss for the first player,
the results about win-lose games will apply to games with ties.

7.5.1 Game Strategies
A strategy for a player is a rule that tells the player which move to make whenever
it is their turn. More precisely, a strategy s is a function from games to games with
the property that s.G/ 2 G for all games G. A pair of strategies for the two players
determines exactly which moves the players choose, and so it determines a unique
play of the game, depending on who moves first.
A key question about a game is what strategy will ensure that a player will win.
The Player-1 wants a strategy whose outcome is guaranteed to be a win, and Player-
2 wants a strategy whose outcome is guaranteed to be a loss for Player-1.
3 Real chess tournaments rule this out by setting an advance limit on the number of moves, or by

forbidding repetitions of the same position more than twice.
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7.5. Games as a Recursive Data Type 225

7.5.2 Fundamental Theorem for Win-Lose Games
The Fundamental Theorem for WL-2PerGm games says that one of the players
always has a fixed “winning” strategy that guarantees a win against every possible
opponent strategy.
Thinking about Chess for instance, this seems surprising. Serious chess players
are typically secretive about their intended play strategies, believing that an oppo-
nent could take advantage of knowing their strategy. It seems to them that for any
strategy they choose, their opponent can tailor a strategy to beat it.
But the Fundamental Theorem says otherwise. In theory, in any win-lose-tie
game like Chess or Checkers, each of the players will have a strategy that guar-
antees a win or a stalemate, even if the strategy is known to their opponent. That
is,

there is winning strategy for one of the players, or

both players have strategies that guarantee them at worst a draw.

Even though the Fundamental Theorem reveals a profound fact about games, it
has a very simple proof by structural induction.

Theorem 7.5.3. [Fundamental Theorem for Win-Lose Games] For any WL-2PerGm
game G, one of the players has a winning strategy.

Proof. The proof is by structural induction on the definition of a G 2 WL-2PerGm.
The induction hypothesis is that one of the players has a winning strategy for G.
Base case: (G D win or lose). Then there is only one possible strategy for each
player, namely, do nothing and finish with outcome G.
Constructor case: (G is a nonempty set of WL-2PerGm’s). By structural induction
we may assume that for each M 2 G one of the players has a winning strategy.
Notice that since players alternate moves, the first player in G becomes the second
player in M .
Now if there is a move M0 2 G where the second player in M0 has a winning
strategy, then the first player in G has a simple winning strategy: pick M0 as the
first move, and then follow the second player’s winning strategy for M0 .
On the other hand, if no M 2 G has a winning strategy for the second player in
M , then we can conclude by induction that every M 2 G has a winning strategy for
the first player in M . Now the second player in G has a simple winning strategy,
namely if the first player in G makes the move M , then the second player in G
should follow the follow the winning strategy for the first player in M .
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226 Chapter 7 Recursive Data Types

Infinite Games
So where do we come upon games with an infinite number of first moves? Well,
suppose we play a tournament of n chess games for some positive integer n. This
tournament will be a WL-2PerGm if we agree on a rule for combining the payoffs
of the n individual chess games into a final payoff for the whole tournament.
There still are only a finite number of possible moves at any stage of the n-game
chess tournament, but we can define a meta-chess-tournament, whose first move is
a choice of any positive integer n, after which we play an n-game tournament. Now
the meta-chess-tournament has an infinite number of first moves.
Of course only the first move in the meta-chess-tournament is infinite, but then
we could set up a tournament consisting of n meta-chess-tournaments. This would
be a game with n possible infinite moves. And then we could have a meta-meta-
chess-tournament whose first move was to choose how many meta-chess-tournaments
to play. This meta-meta-chess-tournament will have an infinite number of infinite
moves. Then we could move on to meta-meta-meta-chess-tournaments . . . .
As silly or weird as these meta games may seem, their weirdness doesn’t dis-
qualify the Fundamental Theorem: in each of these games, one of the players will
have winning strategy.
Notice that although Theorem 7.5.3 guarantees a winning strategy, its proof gives
no clue which player has it. For the Subset Takeaway Game of Problem 4.7 and
most familiar 2PerGm’s like Chess, Go, . . . , no one knows which player has a
winning strategy.4

7.6 Induction in Computer Science
Induction is a powerful and widely applicable proof technique, which is why we’ve
devoted two entire chapters to it. Strong induction and its special case of ordinary
induction are applicable to any kind of thing with nonnegative integer sizes—which
is an awful lot of things, including all step-by-step computational processes.
Structural induction then goes beyond number counting, and offers a simple,
natural approach to proving things about recursive data types and recursive compu-
tation.
In many cases, a nonnegative integer size can be defined for a recursively defined
datum, such as the length of a string, or the number of operations in an Aexp. It is
then possible to prove properties of data by ordinary induction on their size. But
4 Checkers used to be in this list, but there has been a recent announcement that each player has a
strategy that forces a tie. (reference TBA)
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7.6. Induction in Computer Science 227

this approach often produces more cumbersome proofs than structural induction.
In fact, structural induction is theoretically more powerful than ordinary induc-
tion. However, it’s only more powerful when it comes to reasoning about infinite
data types—like infinite trees, for example—so this greater power doesn’t matter in
practice. What does matter is that for recursively defined data types, structural in-
duction is a simple and natural approach. This makes it a technique every computer
scientist should embrace.

Problems for Section 7.1
Practice Problems
Problem 7.1.
The set OBT of Ordered Binary Trees is defined recursively as follows:
Base case: hleafi is an OBT, and
Constructor case: if R and S are OBT’s, then hnode; R; S i is an OBT.
If T is an OBT, let nT be the number of node labels in T and lT be the number
of leaf labels in T .
Prove by structural induction that for all T 2 OBT,

lT D nT C 1: (7.24)

Class Problems
Problem 7.2.
Prove by structural induction on the recursive definition(7.1.1) of A that concate-
nation is associative:
.r s/ t D r .s t / (7.25)
for all strings r; s; t 2 A .

Problem 7.3.
The reversal of a string is the string written backwards, for example, rev.abcde/ D
edcba.
(a) Give a simple recursive definition of rev.s/ based on the recursive defini-
tions 7.1.1 of s 2 A and of the concatenation operation 7.1.3.
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228 Chapter 7 Recursive Data Types

(b) Prove that
rev.s t / D rev.t / rev.s/; (7.26)
for all strings s; t 2 A . You may assume that concatenation is associative:

.r s/ t D r .s t /

for all strings r; s; t 2 A (Problem 7.2).

Problem 7.4.
The Elementary 18.01 Functions (F18’s) are the set of functions of one real variable
defined recursively as follows:
Base cases:

The identity function id.x/ WWD x is an F18,

any constant function is an F18,

the sine function is an F18,

Constructor cases:
If f; g are F18’s, then so are

1. f C g fg 2g ,

2. the inverse function f 1,

3. the composition f ı g.

(a) Prove that the function 1=x is an F18.
1
Warning: Don’t confuse 1=x D x 1 with the inverse id of the identity function
id.x/. The inverse id 1 is equal to id.

(b) Prove by Structural Induction on this definition that the Elementary 18.01
Functions are closed under taking derivatives. That is, show that if f .x/ is an F18,
then so is f 0 WWD df =dx. (Just work out 2 or 3 of the most interesting constructor
cases; you may skip the less interesting ones.)

Problem 7.5.
Here is a simple recursive definition of the set E of even integers:
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7.6. Induction in Computer Science 229

Definition. Base case: 0 2 E.
Constructor cases: If n 2 E, then so are n C 2 and n.
Provide similar simple recursive definitions of the following sets:
(a) The set S WWD f2k 3m 5n 2 N j k; m; n 2 Ng.

(b) The set T WWD f2k 32kCm 5mCn 2 N j k; m; n 2 Ng.

(c) The set L WWD f.a; b/ 2 Z2 j .a b/ is a multiple of 3g.
Let L0 be the set defined by the recursive definition you gave for L in the previous
part. Now if you did it right, then L0 D L, but maybe you made a mistake. So let’s
check that you got the definition right.
(d) Prove by structural induction on your definition of L0 that

L0 L:

(e) Confirm that you got the definition right by proving that

L L0 :

(f) See if you can give an unambiguous recursive definition of L.

Problem 7.6.

Definition. The recursive data type binary-2PG of binary trees with leaf labels L
is defined recursively as follows:

Base case: hleaf; li 2 binary-2PG, for all labels l 2 L.

Constructor case: If G1 ; G2 2 binary-2PG, then

hbintree; G1 ; G2 i 2 binary-2PG:

The size jGj of G 2 binary-2PG is defined recursively on this definition by:

Base case:
j hleaf; li j WWD 1; for all l 2 L:

Constructor case:

j hbintree; G1 ; G2 i j WWD jG1 j C jG2 j C 1:
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230 Chapter 7 Recursive Data Types

G1
win

G1,2
win

lose win

Figure 7.1 A picture of a binary tree G.

For example, the size of the binary-2PG G pictured in Figure 7.1, is 7.
(a) Write out (using angle brackets and labels bintree, leaf, etc.) the binary-2PG
G pictured in Figure 7.1.
The value of flatten.G/ for G 2 binary-2PG is the sequence of labels in L of the
leaves of G. For example, for the binary-2PG G pictured in Figure 7.1,

flatten.G/ D .win; lose; win; win/:

(b) Give a recursive definition of flatten. (You may use the operation of concate-
nation (append) of two sequences.)

2 length.flatten.G// D jGj C 1: (7.27)

Homework Problems
Problem 7.7.
The string reversal function, rev W A ! A has a simple recursive definition.
Base case: rev./ WWD .
Constructor case: rev.as/ WWD rev.s/a for s 2 A and a 2 A.
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7.6. Induction in Computer Science 231

A string s is a palindrome when rev.s/ D s. The palindromes also have a simple
recursive definition as the set RecPal.
Base cases: 2 RecPal and a 2 RecPal for a 2 A.
Constructor case: If s 2 RecPal, then asa 2 RecPal for a 2 A.
Verifying that the two definitions agree offers a nice exercise in structural induc-
tion and also induction on length of strings. The verification rests on three basic
properties of concatenation and reversal proved in separate problems 7.2 and 7.3.
Fact.

.rs D uv AND jrj D juj/ IFF .r D u AND s D v/ (7.28)
r .s t / D .r s/ t (7.29)
rev.st / D rev.t / rev.s/ (7.30)

(a) Prove that s D rev.s/ for all s 2 RecPal.

(b) Prove conversely that if s D rev.s/, then s 2 RecPal.
Hint: By induction on n D jsj.

Problem 7.8.
Let m; n be integers, not both zero. Define a set of integers, Lm;n , recursively as
follows:
Base cases: m; n 2 Lm;n .

Constructor cases: If j; k 2 Lm;n , then

1. j 2 Lm;n ,
2. j C k 2 Lm;n .

Let L be an abbreviation for Lm;n in the rest of this problem.
(a) Prove by structural induction that every common divisor of m and n also di-
vides every member of L.

(b) Prove that any integer multiple of an element of L is also in L.

(d) Show that there is a positive integer g 2 L that divides every member of L.
Hint: The least positive integer in L.
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232 Chapter 7 Recursive Data Types

Figure 7.2 Constructing the Koch Snowflake.

(e) Conclude that g from part (d) is gcd.m; n/, the greatest common divisor, of m
and n.

Problem 7.9.

Definition. Define the number #c .s/ of occurrences of the character c 2 A in the
string s recursively on the definition of s 2 A :
base case: #c ./ WWD 0.
constructor case:
(
#c .s/ if a ¤ c;
#c .ha; si/ WWD
1 C #c .s/ if a D c:

Prove by structural induction that for all s; t 2 A and c 2 A

#c .s t / D #c .s/ C #c .t /:

Problem 7.10.
Fractals are an example of mathematical objects that can be defined recursively.
In this problem, we consider the Koch snowflake. Any Koch snowflake can be
constructed by the following recursive definition.

Base case: An equilateral triangle with a positive integer side length is a
Koch snowflake.

Constructor case: Let K be a Koch snowflake, and let l be a line segment
on the snowflake. Remove the middle third of l, and replace it with two line
segments of the same length jlj, as is done in Figure 7.2
The resulting figure is also a Koch snowflake.

Prove
pby structural induction that the area inside any Koch snowflake is of the
form q 3, where q is a rational number.
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7.6. Induction in Computer Science 233

Problem 7.11.
The set RBT of Red-Black Trees is defined recursively as follows:
Base cases:

hredi 2 RBT, and

hblacki 2 RBT.

Constructor cases: A; B are RBT’s, then

if A; B start with black, then hred; A; Bi is an RBT.

if A; B start with red, then hblack; A; Bi is an RBT.

For any RBT T , let

rT be the number of red labels in T ,

bT be the number of black labels in T , and

nT WWD rT C bT be the total number of labels in T .

Prove that
nT 2nT C 1
If T starts with a red label; then rT ; (7.31)
3 3

Hint:
n=3 r IFF .2=3/n n r

Exam Problems

Problem 7.12.
The Arithmetic Trig Functions (Atrig’s) are the set of functions of one real variable
defined recursively as follows:
Base cases:

The identity function id.x/ WWD x is an Atrig,

any constant function is an Atrig,

the sine function is an Atrig,
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234 Chapter 7 Recursive Data Types

Constructor cases:
If f; g are Atrig’s, then so are

1. f C g

2. f g

3. the composition f ı g.

Prove by structural induction on this definition that if f .x/ is an Atrig, then so is
f 0 WWD df =dx.

Problem 7.13.

Definition. The set RAF of rational functions of one real variable is the set of
functions defined recursively as follows:
Base cases:

The identity function, id.r/ WWD r for r 2 R (the real numbers), is an RAF,

any constant function on R is an RAF.

Constructor cases: If f; g are RAF’s, then so is f ~ g, where ~ is one of the
operations

1. addition C,

2. multiplication or

3. division =.

(a) Describe how to construct functions e; f; g 2 RAF such that

e ı .f C g/ ¤ .e ı f / C .e ı g/: (7.32)

(b) Prove that for all real-valued functions e; f; g (not just those in RAF):

.e ~ f / ı g D .e ı g/ ~ .f ı g/; (7.33)

Hint: .e ~ f /.x/ WWD e.x/ ~ f .x/.
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7.6. Induction in Computer Science 235

P .h/ WWD 8g 2 RAF: h ı g 2 RAF:

Prove by structural induction on the definition of RAF that P .h/ holds for all h 2
RAF.
Make sure to indicate explicitly

each of the base cases, and
each of the constructor cases.

Problem 7.14.
The 2-3-averaged numbers are a subset, N23, of the real interval Œ0; 1 defined
recursively as follows:
Base cases: 0; 1 2 N23.
Constructor case: If a; b are in N23, then so is L.a; b/ where

2a C 3b
L.a; b/ WWD :
5
(a) Use ordinary induction or the Well-Ordering Principle to prove that
n
3
2 N23
5

for all nonnegative integers n.

(b) Prove by Structural Induction that the product of two 2-3-averaged numbers is
also a 2-3-averaged number.
Hint: Prove by structural induction on c that, if d 2 N23, then cd 2 N23.

Problem 7.15.
This problem is about binary strings s 2 f0; 1g .
Let’s call a recursive definition of a set of strings cat-OK when all its constructors
are defined as concatenations of strings.5
5 The concatenation of two strings x and y, written xy, is the string obtained by appending x to
the left end of y. For example, the concatenation of 01 and 101 is 01101.
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236 Chapter 7 Recursive Data Types

For example, the set, One1, of strings with exactly one 1 has the cat-OK defini-
tion:
Base case: The length-one string 1 is in One1.
Constructor case: If s is in One1, then so is 0s and s0.
(a) Give a cat-OK definition of the set E of even length strings consisting solely
of 0’s.

(b) Let rev.s/ be the reversal of the string s. For example, rev.001/ D 100. A
palindrome is a string s such that s D rev.s/. For example, 11011 and 010010
are palindromes.
Give a cat-OK definition of the palindromes.

Problems for Section 7.2
Practice Problems
Problem 7.16.
Define the sets F1 and F2 recursively:

F1 :

– 5 2 F1 ,
– if n 2 F1 , then 5n 2 F1 .

F2 :

– 5 2 F2 ,
– if n; m 2 F1 , then nm 2 F2 .

(a) Show that one of these definitions is technically ambiguous. (Remember that
“ambiguous recursive definition” has a technical mathematical meaning which does
not imply that the ambiguous definition is unclear.)

(b) Briefly explain what advantage unambiguous recursive definitions have over
ambiguous ones.
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7.6. Induction in Computer Science 237

(c) A way to prove that F1 D F2 , is to show first that F1 F2 and second that
F2 F1 . One of these containments follows easily by structural induction. Which
one? What would be the induction hypothesis? (You do not need to complete a
proof.)

Problem 7.17. (a) To prove that the set RecMatch, of matched strings of Defini-
tion 7.2.1 equals the set AmbRecMatch of ambiguous matched strings of Defini-
tion 7.2.4, you could first prove that

8r 2 RecMatch: r 2 AmbRecMatch;

and then prove that

8u 2 AmbRecMatch: u 2 RecMatch:

Of these two statements, indicate the one that would be simpler to prove by struc-
tural induction directly from the definitions.

(b) Suppose structural induction was being used to prove that AmbRecMatch
RecMatch. Indicate the one predicate below that would fit the format for a structural
induction hypothesis in such a proof.

P0 .n/ WWD jsj n IMPLIES s 2 RecMatch.
P1 .n/ WWD jsj n IMPLIES s 2 AmbRecMatch.
P2 .s/ WWD s 2 RecMatch.
P3 .s/ WWD s 2 AmbRecMatch.
P4 .s/ WWD .s 2 RecMatch IMPLIES s 2 AmbRecMatch/.

(c) The recursive definition AmbRecMatch is ambiguous because it allows the
s t constructor to apply when s or t is the empty string. But even fixing that,
ambiguity remains. Demonstrate this by giving two different derivations for the
string ”[ ] [ ] [ ] according to AmbRecMatch but only using the s t constructor
when s ¤ and t ¤ .

Class Problems
Problem 7.18.
Let p be the string [ ] . A string of brackets is said to be erasable iff it can be
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238 Chapter 7 Recursive Data Types

reduced to the empty string by repeatedly erasing occurrences of p. For example,
to erase the string
[[[]][]][];
start by erasing the three occurrences of p to obtain

[[]]:

Then erase the single occurrence of p to obtain,

[];

which can now be erased to obtain the empty string .
On the other hand the string

[]][[[[[]] (7.34)

is not erasable, because when we try to erase, we get stuck. Namely, start by erasing
the two occurrences of p in (7.34) to obtain

][[[[]:

The erase the one remaining occurrence of p to obtain.

][[[:

At this point we are stuck with no remaining occurrences of p. 6
Let Erasable be the set of erasable strings of brackets. Let RecMatch be the
recursive data type of strings of matched brackets given in Definition 7.2.1
(a) Use structural induction to prove that

RecMatch Erasable:

(b) Supply the missing parts (labeled by “(*)”) of the following proof that

Erasable RecMatch:
6 Notice that there are many ways to erase a string, depending on when and which occurrences
of p are chosen to be erased. It turns out that given any initial string, the final string reached after
performing all possible erasures will be the same, no matter how erasures are performed. We take
this for granted here, although it is not altogether obvious. (See Problem 6.28 for a proof).
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7.6. Induction in Computer Science 239

Proof. We prove by strong induction that every length n string in Erasable is also
in RecMatch. The induction hypothesis is

P .n/ WWD 8x 2 Erasable: jxj D n IMPLIES x 2 RecMatch:

Base case:
(*) What is the base case? Prove that P is true in this case.
Inductive step: To prove P .n C 1/, suppose jxj D n C 1 and x 2 Erasable. We
need to show that x 2 RecMatch.
Let’s say that a string y is an erase of a string z iff y is the result of erasing a single
occurrence of p in z.
Since x 2 Erasable and has positive length, there must be an erase, y 2 Erasable,
of x. So jyj D n 1 0, and since y 2 Erasable, we may assume by induction
hypothesis that y 2 RecMatch.
Now we argue by cases:
Case (y is the empty string):
(*) Prove that x 2 RecMatch in this case.
Case (y D [ s ] t for some strings s; t 2 RecMatch): Now we argue by subcases.

Subcase(x D py):
(*) Prove that x 2 RecMatch in this subcase.
Subcase (x is of the form [ s 0 ] t where s is an erase of s 0 ):
Since s 2 RecMatch, it is erasable by part (b), which implies that s 0 2
Erasable. But js 0 j < jxj, so by induction hypothesis, we may assume that
s 0 2 RecMatch. This shows that x is the result of the constructor step of
RecMatch, and therefore x 2 RecMatch.
Subcase (x is of the form [ s ] t 0 where t is an erase of t 0 ):
(*) Prove that x 2 RecMatch in this subcase.

(*) Explain why the above cases are sufficient.
This completes the proof by strong induction on n, so we conclude that P .n/ holds
for all n 2 N. Therefore x 2 RecMatch for every string x 2 Erasable. That is,
Erasable RecMatch. Combined with part (a), we conclude that

Erasable D RecMatch:

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240 Chapter 7 Recursive Data Types

Problem 7.19. (a) Prove that the set RecMatch of matched strings of Definition 7.2.1
is closed under string concatenation. Namely, if s; t 2 RecMatch, then s t 2
RecMatch.

(b) Prove AmbRecMatch RecMatch, where AmbRecMatch is the set of am-
biguous matched strings of Definition 7.2.4.

Homework Problems
Problem 7.20.
One way to determine if a string has matching brackets, that is, if it is in the set,
RecMatch, of Definition 7.2.1 is to start with 0 and read the string from left to right,
adding 1 to the count for each left bracket and subtracting 1 from the count for each
right bracket. For example, here are the counts for two sample strings:
[ ] ] [ [ [ [ [ ] ] ] ]
0 1 0 1 0 1 2 3 4 3 2 1 0

[ [ [ ] ] [ ] ] [ ]
0 1 2 3 2 1 2 1 0 1 0
A string has a good count if its running count never goes negative and ends with 0.
So the second string above has a good count, but the first one does not because its
count went negative at the third step. Let
GoodCount WWD fs 2 f] ; [ g j s has a good countg:
The empty string has a length 0 running count we’ll take as a good count by
convention, that is, 2 GoodCount. The matched strings can now be characterized
precisely as this set of strings with good counts.
(a) Prove that GoodCount contains RecMatch by structural induction on the defi-
nition of RecMatch.

(b) Conversely, prove that RecMatch contains GoodCount.
Hint: By induction on the length of strings in GoodCount. Consider when the
running count equals 0 for the second time.

Problem 7.21.
Divided Equilateral Triangles (DETs) were defined in Problem 5.10 as follows:
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7.6. Induction in Computer Science 241

[h]

Figure 7.3 DET T 0 from Four Copies of DET T

[h]

Figure 7.4 Trapezoid from Three Triangles

Base case: A single equilateral triangle is a DET whose only subtriangle is
itself.

If T WWD is a DET, then the equilateral triangle T 0 built out of four
copies of T as shown in in Figure 7.3 is also a DET, and the subtriangles of
T 0 are exactly the subtriangles of each of the copies of T .

Properties of DETs were proved earlier by induction on the length of a side of the
triangle. Recognizing that the definition of DETs is recursive, we can instead prove
properties of DETs by structural induction.

(a) Prove by structural induction that a DET with one of its corner subtriangles
removed can be tiled with trapezoids built out of three subtriangles as in Figure 7.4.

(b) Explain why a DET with a triangle removed from the middle of one side can
also be tiled by trapezoids.

(c) In tiling a large square using L-shaped blocks as described in Section 5.1.5,
there was a tiling with any single subsquare removed. Part (b) indicates that trapezoid-
tilings are possible for DETs with a non-corner subtriangle removed, so it’s natural
to make the mistaken guess that DETs have a corresponding property:
False Claim. A DET with any single subtriangle removed can be trapezoid-tiled.
We can try to prove the claim by structural induction as in part (a).
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242 Chapter 7 Recursive Data Types

Bogus proof. The claim holds vacuously in the base case of a DET with a single
subtriangle.
Now let T 0 be a DET made of four copies of a DET T , and suppose we remove an
arbitrary subtriangle from T 0 .
The removed subtriangle must be a subtriangle of one of the copies of T . The
copies are the same, so for definiteness we assume the subtriangle was removed
from copy 1. Then by structural induction hypothesis, copy 1 can be trapezoid-
tiled, and then the other three copies of T can be trapezoid-tiled exactly as in the
solution to part(a). This yields a complete trapezoid-tiling of T 0 with the arbitrary
subtriangle removed.
We conclude by structural induction that any DET with any subtriangle removed
can be trapezoid-tiled.

What’s wrong with the proof?
Hint: Find a counter-example and show where the proof breaks down.
We don’t know if there is a simple characterization of exactly which subtriangles
can be removed to allow a trapezoid tiling.

Problem 7.22.
A binary word is a finite sequence of 0’s and 1’s. In this problem, we’ll simply call
them “words.” For example, .1; 1; 0/ and .1/ are words of length three and one,
respectively. We usually omit the parentheses and commas in the descriptions of
words, so the preceding binary words would just be written as 110 and 1.
The basic operation of placing one word immediately after another is called con-
catentation. For example, the concatentation of 110 and 1 is 1101, and the con-
catentation of 110 with itself is 110110.
We can extend this basic operation on words to an operation on sets of words. To
emphasize the distinction between a word and a set of words, from now on we’ll
refer to a set of words as a language. Now if R and S are languages, then R S is
the language consisting of all the words you can get by concatenating a word from
R with a word from S . That is,

R S WWD frs j r 2 R AND s 2 S g:

For example,
f0; 00g f00; 000g D f000; 0000; 00000g
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7.6. Induction in Computer Science 243

Another example is D D, abbreviated as D 2 , where D WWD f1; 0g.

D 2 D f00; 01; 10; 11g:

In other words, D 2 is the language consisting of all the length-two words. More
generally, D n will be the language of length-n words.
If S is a language, the language you can get by concatenating any number of
copies of words in S is called S —pronounced “S star.” (By convention, the empty
word always included in S .) For example, f0; 11g is the language consisting
of all the words you can make by stringing together 0’s and 11’s. This language
could also be described as consisting of the words whose blocks of 1’s are always
of even length. Another example is .D 2 / , which consists of all the even length
words. Finally, the language B of all binary words is just D .
The Concatenation-Definable (C-D) languages are defined recursively:

Base case: Every finite language is a C-D.

Constructor cases: If L and M are C-D’s, then

L M; L [ M; and L

are C-D’s.

Note that the -operation is not allowed. For this reason, the C-D languages are
also called the “star-free languages,” [33].
Lots of interesting languages turn out to be concatenation-definable, but some
very simple languages are not. This problem ends with the conclusion that the
language f00g of even length words whose bits are all 0’s is not a C-D language.
(a) Show that the set B of all binary words is C-D. Hint: The empty set is finite.
Now a more interesting example of a C-D set is the language of all binary words
that include three consecutive 1’s:

B111B:

Notice that the proper expression here is “B f111gB.” But it causes no confusion
and helps readability to omit the dots in concatenations and the curly braces for sets
with only one element.
(b) Show that the language consisting of the binary words that start with 0 and
end with 1 is C-D.

244 Chapter 7 Recursive Data Types

(d) Show that if R and S are C-D, then so is R \ S .

(e) Show that f01g is C-D.
Let’s say a language S is 0-finite when it includes only a finite number of words
whose bits are all 0’s, that is, when S \ 0 is a finite set of words. A langauge S is
0-boring—boring, for short—when either S or S is 0-finite.
(f) Explain why f00g is not boring.

(g) Verify that if R and S are boring, then so is R [ S .

(h) Verify that if R and S are boring, then so is R S .
Hint: By cases: whether R and S are both 0-finite, whether R or S contains no
all-0 words at all (including the empty word ), and whether neither of these cases
hold.

(i) Conclude by structural induction that all C-D languages are boring.
So we have proved that the set .00/ of even length all-0 words is not a C-D
language.

Problem 7.23.
We can explain in a simple and precise way how digital circuits work, and gain the
powerful proof method of structural induction to verify their properties, by defining
digital circuits as a recursive data type DigCirc. The definition is a little easier to
state if all the gates in the circuit take two inputs, so we will use the two-input NOR
gate rather than a one-input NOT, and let the set of gates be

Gates WWD fNOR; AND; OR; XORg:

A digital circuit will be a recursively defined list of gate connections of the form
.x; y; G; I / where G is a gate, x and y are the input wires, and I is the set of wires
that the gate output feeds into as illustrated in Figure 7.5.

Formally, we let W be a set w0 ; w1 ; : : : whose elements are called wires, and
O … W be an object called the output.

Definition. The set of digital circuit DigCirc, and their inputs and internal wires,
are defined recursively as follows:
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7.6. Induction in Computer Science 245

Figure 7.5 Digital Circuit Constructor Step

Base case: If x; y 2 W , then C 2 DigCirc, where

C D list..x; y; G; fOg// for some G 2 Gates;
inputs.C/ WWD fx; yg;
internal.C/ WWD ;:

Constructor cases: If

C 2 DigCirc;
I inputs.C/; I ¤ ;;
x; y 2 W .I [ internal.C//

then D 2 DigCirc, where

D D cons..x; y; G; I/; C/ for some G 2 Gates;
inputs.D/ WWD fx; yg [ .inputs.C/ I /;
internal.D/ WWD internal.C/ [ I:

For any circuit C define

wires.C/ WWD inputs.C/ [ internal.C/ [ fOg:
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246 Chapter 7 Recursive Data Types

A wire assignment for C is a function

˛ W wires.C/ ! fT; Fg

such that for each gate connection .x; y; G; I / 2 C ,

˛.i / D .˛.x/ G ˛.y// for all i 2 I:

(a) Define an environment for C to be a function e W inputs.C/ ! fT; Fg. Prove
that if two wire assignments for C are equal for each wire in inputs.C/, then the
wire assignments are equal for all wires.

Part (a) implies that for any environment e for C , there is a unique wire assign-
ment ˛e such that

˛e .w/ D e.w/ for all w 2 inputs.C/:

So for any input environment e, the circuit computes a unique output

eval.C; e/ WWD ˛e .O/:

Now suppose F is a propositional formula whose propositional variables are the
input wires of some circuit C . Then C and F are defined to be equivalent iff

eval.C; e/ D eval.F; e/

for all environments e for C .
(b) Define a function E.C / recursively on the definition of circuit C , such that
E.C / is a propositional formula equivalent to C . Then verify the recursive defini-
tion by proving the equivalence using structural induction.

Exam Problems
Problem 7.24.
Let P be a propositional variable.
(a) Show how to express NOT.P / using P and a selection from among the con-
stant True, and the connectives XOR and AND.

The use of the constant True above is essential. To prove this, we begin with
a recursive definition of XOR-AND formulas that do not use True, called the PXA
formulas.
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7.6. Induction in Computer Science 247

Definition. Base case: The propositional variable P is a PXA formula.
Constructor cases If R; S 2 PXA, then
R XOR S ,
R AND S
are PXA’s.
For example,
...P XOR P / AND P / XOR .P AND P // XOR .P XOR P /
is a PXA.
(b) Prove by structural induction on the definition of PXA that every PXA formula
A is equivalent to P or to False.

Problems for Section 7.3
Homework Problems
Problem 7.25.
One version of the the Ackermann function A W N2 ! N is defined recursively by
the following rules:
A.m; n/ WWD 2n if m D 0 or n 1, (A-base)
A.m; n/ WWD A.m 1; A.m; n 1// otherwise: (AA)
Prove that if B W N2 ! N is a partial function that satisfies this same definition,
then B is total and B D A.

Problems for Section 7.4
Practice Problems
Problem 7.26. (a) Write out the evaluation of
eval.subst.3x; x.x 1//; 2/
according to the Environment Model and the Substitution Model, indicating where
the rule for each case of the recursive definitions of eval.; / and ŒWD] or substitution
is first used. Compare the number of arithmetic operations and variable lookups.
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248 Chapter 7 Recursive Data Types

(b) Describe an example along the lines of part (a) where the Environment Model
would perform 6 fewer multiplications than the Substitution model. You need not
carry out the evaluations.

(c) Describe an example along the lines of part (a) where the Substitution Model
would perform 6 fewer multiplications than the Environment model. You need not
carry out the evaluations.

Class Problems
Problem 7.27.
In this problem we’ll need to be careful about the propositional operations that
apply to truth values and the corresponding symbols that appear in formulas. We’ll
restrict ourselves to formulas with symbols And and Not that correspond to the
operations AND, NOT. We will also allow the constant symbols True and False.
(a) Give a simple recursive definition of propositional formula F and the set
pvar.F / of propositional variables that appear in it.
Let V be a set of propositional variables. A truth environment e over V assigns
truth values to all these variables. In other words, e is a total function,

e W V ! fT; Fg:

(b) Give a recursive definition of the truth value, eval.F; e/, of propositional for-
mula F in an environment e over a set of variables V pvar.F /.
Clearly the truth value of a propositional formula only depends on the truth val-
ues of the variables in it. How could it be otherwise? But it’s good practice to work
out a rigorous definition and proof of this assumption.
(c) Give an example of a propositional formula containing the variable P but
whose truth value does not depend on P . Now give a rigorous definition of the as-
sertion that “the truth value of propositional formula F does not depend on propo-
sitional variable P .”
Hint: Let e1 ; e2 be two environments whose values agree on all variables other than
P.

(d) Give a rigorous definition of the assertion that “the truth value of a proposi-
tional formula only depends on the truth values of the variables that appear in it,”
and then prove it by structural induction on the definition of propositional formula.

(e) Now we can formally define F being valid. Namely, F is valid iff

8e: eval.F; e/ D T:
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7.6. Induction in Computer Science 249

Give a similar formal definition of formula G being unsatisfiable. Then use the
definition of eval to prove that a formula F is valid iff Not.F / is unsatisfiable.

Homework Problems
Problem 7.28. (a) Give a recursive definition of a function erase.e/ that erases all
the symbols in e 2 Aexp but the brackets. For example

erase.[ [ 3 * [ x * x ] ] + [ [ 2 * x ] + 1] ] / D [ [ [ ] ] [ [ 2 * x ] + 1] ] :

(b) Prove that erase.e/ 2 RecMatch for all e 2 Aexp.

Problems for Section 7.5
Practice Problems
Problem 7.29.
In the game tree for the game Tic-Tac-Toe, the root has nine children corresponding
to the nine boxes that the first player could mark with an “X”.
Each of these nine nodes will have eight children in the second level of the tree,
indicating where the second player can mark his “O”, giving a total of 72 nodes.
Answer the following questions about the game tree for Tic-Tac-Toe.
(a) How many nodes will be in the third level of the tree?

(b) What is the first level where this simple pattern of calculating nodes stops
working?

Homework Problems
Problem 7.30.
We’re going to characterize a large category of games as a recursive data type and
then prove, by structural induction, a fundamental theorem about game strategies.
We are interested in two person games of perfect information that end with a nu-
merical score. Chess and Checkers would count as value games using the values
1; 1; 0 for a win, loss or draw for the first player. The game of Go really does end
with a score based on the number of white and black stones that remain at the end.
Here’s the formal definition:
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250 Chapter 7 Recursive Data Types

Definition. Let V be a nonempty set of real numbers. The class VG of V -valued
two-person deterministic games of perfect information is defined recursively as fol-
lows:
Base case: A value v 2 V is a VG known as a payoff.
Constructor case: If G is a nonempty set of VG’s, then G is a VG. Each game
M 2 G is called a possible first move of G.

A strategy for a player is a rule that tells the player which move to make when-
ever it is their turn. That is, a strategy is a function s from games to games with the
property that s.G/ 2 G for all games G. Given which player has the first move, a
pair of strategies for the two players determines exactly which moves the players
will choose. So the strategies determine a unique play of the game and a unique
payoff.7
The max-player wants a strategy that guarantees as high a payoff as possible,
and the min-player wants a strategy that guarantees as low a payoff as possible.
The Fundamental Theorem for deterministic games of perfect information says
that in any game, each player has an optimal strategy, and these strategies lead to
the same payoff. More precisely,

Theorem (Fundamental Theorem for VG’s). Let V be a finite set of real numbers
and G be a V -valued VG. Then there is a value v 2 V , called a max-value maxG
for G, such that if the max-player moves first,

the max-player has a strategy that will finish with a payoff of at least maxG ,
no matter what strategy the min-player uses, and

the min-player has a strategy that will finish with a payoff of at most maxG ,
no matter what strategy the max-player uses.

It’s worth a moment for the reader to observe that the definition of maxG implies
that if there is one for G, it is unique. So if the max-player has the first move,
the Fundamental Theorem means that there’s no point in playing the game: the
min-player may just as well pay the max-value to the max-player.
(a) Prove the Fundamental Theorem for VG’s.
Hint: VG’s are a recursively defined data type, so the basic method for proving that
all VG’s have some property is structural induction on the definition of VG. Since
the min-player moves first in whichever game the max-player picks for their first
move, the induction hypothesis will need to cover that case as well.
7 We take for granted the fact that no VG has an infinite play. The proof of this by structural
induction is essentially the same as that for win-lose games given in Lemma 7.5.2.
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7.6. Induction in Computer Science 251

(b) (OPTIONAL). State some reasonable generalization of the Fundamental The-
orem to games with an infinite set V of possible payoffs.

Problem 7.31.
Nim is a two-person game that starts with some piles of stones. A player’s move
consists of removing one or more stones from a single pile. The players alternate
making moves, and whoever takes the last stone wins.
It turns out there is a winning strategy for one of the players that is easy to carry
out but is not so obvious.
To explain the winning strategy, we need to think of a number in two ways: as
a nonnegative integer and as the bit string equal to the binary representation of the
number—possibly with leading zeroes.
For example, the XOR of numbers r; s; ::: is defined in terms of their binary repre-
sentations: combine the corresponding bits of the binary representations of r; s; :::
using XOR, and then interpret the resulting bit-string as a number. For example,

2 XOR 7 XOR 9 D 12

because, taking XOR’s down the columns, we have

0 0 1 0 (binary rep of 2)
0 1 1 1 (binary rep of 7)
1 0 0 1 (binary rep of 9)
1 1 0 0 (binary rep of 12)

This is the same as doing binary addition of the numbers, but throwing away the
carries (see Problem 3.6).
The XOR of the numbers of stones in the piles is called their Nim sum. In this
problem we will verify that if the Nim sum is not zero on a player’s turn, then the
player has a winning strategy. For example, if the game starts with five piles of
equal size, then the first player has a winning strategy, but if the game starts with
four equal-size piles, then the second player can force a win.
(a) Prove that if the Nim sum of the piles is zero, then any one move will leave a
nonzero Nim sum.

(b) Prove that if there is a pile with more stones than the Nim sum of all the other
piles, then there is a move that makes the Nim sum equal to zero.

(c) Prove that if the Nim sum is not zero, then one of the piles is bigger than the
Nim sum of the all the other piles.
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252 Chapter 7 Recursive Data Types

Hint: Notice that the largest pile may not be the one that is bigger than the Nim
sum of the others; three piles of sizes 2,2,1 is an example.

(d) Conclude that if the game begins with a nonzero Nim sum, then the first player
has a winning strategy.
Hint: Describe a preserved invariant that the first player can maintain.

(e) (Extra credit) Nim is sometimes played with winners and losers reversed, that
is, the person who takes the last stone loses. This is called the misère version of the
game. Use ideas from the winning strategy above for regular play to find one for
misère play.
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8 Infinite Sets
This chapter is about infinite sets and some challenges in proving things about
them.
Wait a minute! Why bring up infinity in a Mathematics for Computer Science
text? After all, any data set in a computer is limited by the size of the computer’s
memory, and there is a bound on the possible size of computer memory, for the
simple reason that the universe is (or at least appears to be) bounded. So why not
stick with finite sets of some large, but bounded, size? This is a good question, but
let’s see if we can persuade you that dealing with infinite sets is inevitable.
You may not have noticed, but up to now you’ve already accepted the routine use
of the integers, the rationals and irrationals, and sequences of them—infinite sets
all. Further, do you really want Physics or the other sciences to give up the real
numbers on the grounds that only a bounded number of bounded measurements
can be made in a bounded universe? It’s pretty convincing—and a lot simpler—to
ignore such big and uncertain bounds (the universe seems to be getting bigger all
the time) and accept theories using real numbers.
Likewise in computer science, it’s implausible to think that writing a program to
add nonnegative integers with up to as many digits as, say, the stars in the sky—
billions of galaxies each with billions of stars—would be different from writing a
program that would add any two integers, no matter how many digits they had. The
same is true in designing a compiler: it’s neither useful nor sensible to make use of
the fact that in a bounded universe, only a bounded number of programs will ever
be compiled.
Infinite sets also provide a nice setting to practice proof methods, because it’s
harder to sneak in unjustified steps under the guise of intuition. And there has
been a truly astonishing outcome of studying infinite sets. Their study led to the
discovery of fundamental, logical limits on what computers can possibly do. For
example, in Section 8.2, we’ll use reasoning developed for infinite sets to prove
that it’s impossible to have a perfect type-checker for a programming language.
So in this chapter, we ask you to bite the bullet and start learning to cope with
infinity.
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254 Chapter 8 Infinite Sets

8.1 Infinite Cardinality
In the late nineteenth century, the mathematician Georg Cantor was studying the
convergence of Fourier series and found some series that he wanted to say con-
verged “most of the time,” even though there were an infinite number of points
where they didn’t converge. As a result, Cantor needed a way to compare the size
of infinite sets. To get a grip on this, he got the idea of extending the Mapping Rule
Theorem 4.5.4 to infinite sets: he regarded two infinite sets as having the “same
size” when there was a bijection between them. Likewise, an infinite set A should
be considered “as big as” a set B when A surj B. So we could consider A to be
“strictly smaller” than B, which we abbreviate as A strict B, when A is not “as big
as” B:

Definition 8.1.1. A strict B iff NOT .A surj B/.

On finite sets, this strict relation really does mean “strictly smaller.” This follows
immediately from the Mapping Rule Theorem 4.5.4.

Corollary 8.1.2. For finite sets A; B,

A strict B iff jAj < jBj:

Proof.

A strict B iff NOT .A surj B/ (Def 8.1.1)
iff NOT .jAj jBj/ (Theorem 4.5.4.(4.5))
iff jAj < jBj:

Cantor got diverted from his study of Fourier series by his effort to develop a
theory of infinite sizes based on these ideas. His theory ultimately had profound
consequences for the foundations of mathematics and computer science. But Can-
tor made a lot of enemies in his own time because of his work: the general mathe-
matical community doubted the relevance of what they called “Cantor’s paradise”
of unheard-of infinite sizes.
A nice technical feature of Cantor’s idea is that it avoids the need for a definition
of what the “size” of an infinite set might be—all it does is compare “sizes.”
Warning: We haven’t, and won’t, define what the “size” of an infinite set is.
The definition of infinite “sizes” requires the definition of some infinite sets called
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8.1. Infinite Cardinality 255

ordinals with special well-ordering properties. The theory of ordinals requires get-
ting deeper into technical set theory than we want to go, and we can get by just
fine without defining infinite sizes. All we need are the “as big as” and “same size”
relations, surj and bij, between sets.
But there’s something else to watch out for: we’ve referred to surj as an “as big
as” relation and bij as a “same size” relation on sets. Of course, most of the “as big
as” and “same size” properties of surj and bij on finite sets do carry over to infinite
sets, but some important ones don’t—as we’re about to show. So you have to be
careful: don’t assume that surj has any particular “as big as” property on infinite
sets until it’s been proved.
Let’s begin with some familiar properties of the “as big as” and “same size”
relations on finite sets that do carry over exactly to infinite sets:
Lemma 8.1.3. For any sets A; B; C ,
1. A surj B iff B inj A.

2. If A surj B and B surj C , then A surj C .

3. If A bij B and B bij C , then A bij C .

4. A bij B iff B bij A.
Part 1. follows from the fact that R has the Œ 1 out; 1 in surjective function
property iff R 1 has the Œ 1 out; 1 in total, injective property. Part 2. follows
from the fact that compositions of surjections are surjections. Parts 3. and 4. fol-
low from the first two parts because R is a bijection iff R and R 1 are surjective
functions. We’ll leave verification of these facts to Problem 4.22.
Another familiar property of finite sets carries over to infinite sets, but this time
some real ingenuity is needed to prove it:
Theorem 8.1.4. [Schröder-Bernstein] For any sets A; B, if A surj B and B surj A,
then A bij B.
That is, the Schröder-Bernstein Theorem says that if A is at least as big as B
and conversely, B is at least as big as A, then A is the same size as B. Phrased
this way, you might be tempted to take this theorem for granted, but that would be
a mistake. For infinite sets A and B, the Schröder-Bernstein Theorem is actually
pretty technical. Just because there is a surjective function f W A ! B—which
need not be a bijection—and a surjective function g W B ! A—which also need
not be a bijection—it’s not at all clear that there must be a bijection e W A ! B. The
idea is to construct e from parts of both f and g. We’ll leave the actual construction
to Problem 8.10.
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256 Chapter 8 Infinite Sets

Another familiar set property is that for any two sets, either the first is at least
as big as the second, or vice-versa. For finite sets this follows trivially from the
Mapping Rule. It’s actually still true for infinite sets, but assuming it was obvious
would be mistaken again.

Theorem 8.1.5. For all sets A; B,

A surj B OR B surj A:

Theorem 8.1.5 lets us prove that another basic property of finite sets carries over
to infinite ones:

Lemma 8.1.6.
A strict B AND B strict C (8.1)
implies
A strict C
for all sets A; B; C .

Proof. (of Lemma 8.1.6)
Suppose 8.1 holds, and assume for the sake of contradiction that NOT.A strict
C /, which means that A surj C . Now since B strict C , Theorem 8.1.5 lets us
conclude that C surj B. So we have

A surj C AND C surj B;

and Lemma 8.1.3.2 lets us conclude that A surj B, contradicting the fact that
A strict B.

We’re omitting a proof of Theorem 8.1.5 because proving it involves technical
set theory—typically the theory of ordinals again—that we’re not going to get into.
But since proving Lemma 8.1.6 is the only use we’ll make of Theorem 8.1.5, we
hope you won’t feel cheated not to see a proof.

8.1.1 Infinity is different
A basic property of finite sets that does not carry over to infinite sets is that adding
something new makes a set bigger. That is, if A is a finite set and b … A, then
jA [ fbgj D jAj C 1, and so A and A [ fbg are not the same size. But if A is
infinite, then these two sets are the same size!

Lemma 8.1.7. Let A be a set and b … A. Then A is infinite iff A bij A [ fbg.
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8.1. Infinite Cardinality 257

Proof. Since A is not the same size as A [ fbg when A is finite, we only have to
show that A [ fbg is the same size as A when A is infinite.
That is, we have to find a bijection between A [ fbg and A when A is infinite.
Here’s how: since A is infinite, it certainly has at least one element; call it a0 . But
since A is infinite, it has at least two elements, and one of them must not equal to
a0 ; call this new element a1 . But since A is infinite, it has at least three elements,
one of which must not equal both a0 and a1 ; call this new element a2 . Continuing
in this way, we conclude that there is an infinite sequence a0 ; a1 ; a2 ; : : : ; an ; : : : of
different elements of A. Now it’s easy to define a bijection e W A [ fbg ! A:

e.b/ WWD a0 ;
e.an / WWD anC1 for n 2 N;
e.a/ WWD a for a 2 A fb; a0 ; a1 ; : : :g:

8.1.2 Countable Sets
A set C is countable iff its elements can be listed in order, that is, the elements in
C are precisely the elements in the sequence

c0 ; c1 ; : : : ; cn ; : : : :

Assuming no repeats in the list, saying that C can be listed in this way is formally
the same as saying that the function, f W N ! C defined by the rule that f .i /WWDci ,
is a bijection.
Definition 8.1.8. A set C is countably infinite iff N bij C . A set is countable iff it
is finite or countably infinite. A set is uncountable iff it is not countable.
We can also make an infinite list using just a finite set of elements if we allow
repeats. For example, we can list the elements in the three-element set f2; 4; 6g as

2; 4; 6; 6; 6; : : : :

This simple observation leads to an alternative characterization of countable sets
that does not make separate cases of finite and infinite sets. Namely, a set C is
countable iff there is a list
c0 ; c1 ; : : : ; cn ; : : :
of the elements of C , possibly with repeats.
Lemma 8.1.9. A set C is countable iff N surj C . In fact, a nonempty set C is
countable iff there is a total surjective function g W N ! C .
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258 Chapter 8 Infinite Sets

The proof is left to Problem 8.11.
The most fundamental countably infinite set is the set N itself. But the set Z of
all integers is also countably infinite, because the integers can be listed in the order:
0; 1; 1; 2; 2; 3; 3; : : : : (8.2)
In this case, there is a simple formula for the nth element of the list (8.2). That is,
the bijection f W N ! Z such that f .n/ is the nth element of the list can be defined
as: (
n=2 if n is even;
f .n/ WWD
.n C 1/=2 if n is odd:
There is also a simple way to list all pairs of nonnegative integers, which shows
that .N N/ is also countably infinite (Problem 8.17). From this, it’s a small
step to reach the conclusion that the set Q0 of nonnegative rational numbers is
countable. This may be a surprise—after all, the rationals densely fill up the space
between integers, and for any two, there’s another in between. So it might seem as
though you couldn’t write out all the rationals in a list, but Problem 8.9 illustrates
how to do it. More generally, it is easy to show that countable sets are closed under
unions and products (Problems 8.16 and 8.17) which implies the countability of a
bunch of familiar sets:
Corollary 8.1.10. The following sets are countably infinite:
ZC ; Z; N N; QC ; Z Z; Q:
A small modification of the proof of Lemma 8.1.7 shows that countably infinite
sets are the “smallest” infinite sets. Namely,
Lemma 8.1.11. If A is an infinite set, and B is countable, then A surj B.
We leave the proof to Problem 8.8.
Also, since adding one new element to an infinite set doesn’t change its size,
you can add any finite number of elements without changing the size by simply
adding one element after another. Something even stronger is true: you can add a
countably infinite number of new elements to an infinite set and still wind up with
just a set of the same size (Problem 8.13).
By the way, it’s a common mistake to think that, because you can add any finite
number of elements to an infinite set and have a bijection with the original set, that
you can also throw in infinitely many new elements. In general it isn’t true that just
because it’s OK to do something any finite number of times, it’s also OK to do it an
infinite number of times. For example, starting from 3, you can increment by 1 any
finite number of times, and the result will be some integer greater than or equal to
3. But if you increment an infinite number of times, you don’t get an integer at all.
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8.1. Infinite Cardinality 259

8.1.3 Power sets are strictly bigger
Cantor’s astonishing discovery was that not all infinite sets are the same size. In
particular, he proved that for any set A the power set pow.A/ is “strictly bigger”
than A. That is,
Theorem 8.1.12. [Cantor] For any set A,
A strict pow.A/:
Proof. To show that A is strictly smaller than pow.A/, we have to show that if g is
a function from A to pow.A/, then g is not a surjection. To do this, we’ll simply
find a subset Ag A that is not in the range of g. The idea is, for any element
a 2 A, to look at the set g.a/ A and ask whether or not a happens to be in g.a/.
First, define
Ag WWD fa 2 A j a … g.a/g:
Ag is now a well-defined subset of A, which means it is a member of pow.A/. But
Ag can’t be in the range of g, because if it were, we would have
Ag D g.a0 /
for some a0 2 A, so by definition of Ag ,
a 2 g.a0 / iff a 2 Ag iff a … g.a/
for all a 2 A. Now letting a D a0 yields the contradiction
a0 2 g.a0 / iff a0 … g.a0 /:
So g is not a surjection, because there is an element in the power set of A, specifi-
cally the set Ag , that is not in the range of g.

Cantor’s Theorem immediately implies:
Corollary 8.1.13. pow.N/ is uncountable.
Proof. By Lemma 8.1.9, U is uncountable iff N strict U .

The bijection between subsets of an n-element set and the length n bit-strings
f0; 1gn used to prove Theorem 4.5.5, carries over to a bijection between subsets of
a countably infinite set and the infinite bit-strings, f0; 1g! . That is,
pow.N/ bij f0; 1g! :
This immediately implies
Corollary 8.1.14. f0; 1g! is uncountable.
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260 Chapter 8 Infinite Sets

More Countable and Uncountable Sets
Once we have a few sets we know are countable or uncountable, we can get lots
more examples using Lemma 8.1.3. In particular, we can appeal to the following
immediate corollary of the Lemma:

Corollary 8.1.15.

(a) If U is an uncountable set and A surj U , then A is uncountable.

(b) If C is a countable set and C surj A, then A is countable.

For example, now that we know that the set f0; 1g! of infinite bit strings is un-
countable, it’s a small step to conclude that

Corollary 8.1.16. The set R of real numbers is uncountable.

To prove this, think about the infinite decimal expansion of a real number:
p
2 D 1:4142 : : : ;
5 D 5:000 : : : ;
1=10 D 0:1000 : : : ;
1=3 D 0:333 : : : ;
1=9 D 0:111 : : : ;
1
4 D 4:010101 : : : :
99
Let’s map any real number r to the infinite bit string b.r/ equal to the sequence
of bits in the decimal expansion of r, starting at the decimal point. If the decimal
expansion of r happens to contain a digit other than 0 or 1, leave b.r/ undefined.
For example,

b.5/ D 000 : : : ;
b.1=10/ D 1000 : : : ;
b.1=9/ D 111 : : : ;
1
b.4 / D 010101 : : :
p 99
b. 2/; b.1=3/ are undefined:
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8.1. Infinite Cardinality 261

Now b is a function from real numbers to infinite bit strings.1 It is not a total
function, but it clearly is a surjection. This shows that

R surj f0; 1g! ;

and the uncountability of the reals now follows by Corollary 8.1.15.(a).
For another example, let’s prove

Corollary 8.1.17. The set .ZC / of all finite sequences of positive integers is count-
able.

To prove this, think about the prime factorization of a nonnegative integer:

20 D 22 30 51 70 110 130 ;
6615 D 20 33 51 72 110 130 :

Let’s map any nonnegative integer n to the finite sequence e.n/ of nonzero expo-
nents in its prime factorization. For example,

e.20/ D .2; 1/;
e.6615/ D .3; 1; 2/;
e.513 119 47817 10344 / D .13; 9; 817; 44/;
e.1/ D ; (the empty string)
e.0/ is undefined:

Now e is a function from N to .ZC / . It is defined on all positive integers, and it
clearly is a surjection. This shows that

N surj .ZC / ;

and the countability of the finite strings of positive integers now follows by Corol-
lary 8.1.15.(b).
1 Some rational numbers can be expanded in two ways—as an infinite sequence ending in all 0’s
or as an infinite sequence ending in all 9’s. For example,

5 D 5:000 D 4:999 : : : ;
1
D 0:1000 D 0:0999 : : : :
10
In such cases, define b.r/ to be the sequence that ends with all 0’s.
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262 Chapter 8 Infinite Sets

Larger Infinities
There are lots of different sizes of infinite sets. For example, starting with the
infinite set N of nonnegative integers, we can build the infinite sequence of sets

N strict pow.N/ strict pow.pow.N// strict pow.pow.pow.N/// strict : : : :

By Cantor’s Theorem 8.1.12, each of these sets is strictly bigger than all the pre-
ceding ones. But that’s not all: the union of all the sets in the sequence is strictly
bigger than each set in the sequence (see Problem 8.24). In this way you can keep
going indefinitely, building “bigger” infinities all the way.

8.1.4 Diagonal Argument
Theorem 8.1.12 and similar proofs are collectively known as “diagonal arguments”
because of a more intuitive version of the proof described in terms of on an infinite
square array. Namely, suppose there was a bijection between N and f0; 1g! . If such
a relation existed, we would be able to display it as a list of the infinite bit strings
in some countable order or another. Once we’d found a viable way to organize
this list, any given string in f0; 1g! would appear in a finite number of steps, just
as any integer you can name will show up a finite number of steps from 0. This
hypothetical list would look something like the one below, extending to infinity
both vertically and horizontally:

A0 D 1 0 0 0 1 1
A1 D 0 1 1 1 0 1
A2 D 1 1 1 1 1 1
A3 D 0 1 0 0 1 0
A4 D 0 0 1 0 0 0
A5 D 1 0 0 1 1 1
:: :: :: :: :: :: :: ::
: : : : : : : :

But now we can exhibit a sequence that’s missing from our allegedly complete list
of all the sequences. Look at the diagonal in our sample list:

A0 D 1 0 0 0 1 1
A1 D 0 1 1 1 0 1
A2 D 1 1 1 1 1 1
A3 D 0 1 0 0 1 0
A4 D 0 0 1 0 0 0
A5 D 1 0 0 1 1 1
:: :: :: :: :: :: :: ::
: : : : : : : :
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8.2. The Halting Problem 263

Here is why the diagonal argument has its name: we can form a sequence D con-
sisting of the bits on the diagonal.

D D 1 1 1 0 0 1 ;

Then, we can form another sequence by switching the 1’s and 0’s along the diago-
nal. Call this sequence C :

C D 0 0 0 1 1 0 :

Now if nth term of An is 1 then the nth term of C is 0, and vice versa, which
guarantees that C differs from An . In other words, C has at least one bit different
from every sequence on our list. So C is an element of f0; 1g! that does not appear
in our list—our list can’t be complete!
This diagonal sequence C corresponds to the set fa 2 A j a … g.a/g in the
proof of Theorem 8.1.12. Both are defined in terms of a countable subset of the
uncountable infinity in a way that excludes them from that subset, thereby proving
that no countable subset can be as big as the uncountable set.

8.2 The Halting Problem
Although towers of larger and larger infinite sets are at best a romantic concern for
a computer scientist, the reasoning that leads to these conclusions plays a critical
role in the theory of computation. Diagonal arguments are used to show that lots of
problems can’t be solved by computation, and there is no getting around it.
This story begins with a reminder that having procedures operate on programs
is a basic part of computer science technology. For example, compilation refers to
taking any given program text written in some “high level” programming language
like Java, C++, Python, . . . , and then generating a program of low-level instruc-
tions that does the same thing but is targeted to run well on available hardware.
Similarly, interpreters or virtual machines are procedures that take a program text
designed to be run on one kind of computer and simulate it on another kind of com-
puter. Routine features of compilers involve “type-checking” programs to ensure
that certain kinds of run-time errors won’t happen, and “optimizing” the generated
programs so they run faster or use less memory.
The fundamental thing that just can’t be done by computation is a perfect job of
type-checking, optimizing, or any kind of analysis of the overall run time behavior
of programs. In this section, we’ll illustrate this with a basic example known as the
Halting Problem. The general Halting Problem for some programming language
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264 Chapter 8 Infinite Sets

is, given an arbitrary program, to determine whether the program will run forever if
it is not interrupted. If the program does not run forever, it is said to halt. Real pro-
grams may halt in many ways, for example, by returning some final value, aborting
with some kind of error, or by awaiting user input. But it’s easy to detect when any
given program will halt: just run it on a virtual machine and wait till it stops. The
problem comes when the given program does not halt—you may wind up waiting
indefinitely without realizing that the wait is fruitless. So how could you detect
that the program does not halt? We will use a diagonal argument to prove that if
an analysis program tries to recognize the non-halting programs, it is bound to give
wrong answers, or no answers, for an infinite number of the programs it is supposed
to be able to analyze!
To be precise about this, let’s call a programming procedure—written in your fa-
vorite programming language—C++, or Java, or Python—a string procedure when
it is applicable to strings in the set ASCII of strings over the 256 character ASCII
alphabet.
As a simple example, you might think about how to write a string procedure that
halts precisely when it is applied to a double letter string in ASCII , namely, a
string in which every character occurs twice in a row. For example, aaCC33, and
zz++ccBB are double letter strings, but aa;bb, b33, and AAAAA are not.
If the computation that happens when a procedure applied to a string eventually
comes to a halt, the procedure is said to recognize the string. In this context, a set of
strings a commonly called a (formal) language. We let lang.P / to be the language
recognized by procedure P :

lang.P / WWDfs 2 ASCII j P applied to s haltsg:

A language is called recognizable when it equals lang.P / for some string pro-
cedure P . For example, we’ve just agreed that the set of double letter strings is
recognizable.
There is no harm in assuming that every program can be written as a string in
ASCII ; they usually are. When a string s 2 ASCII is actually the ASCII descrip-
tion of some string procedure, we’ll refer to that string procedure as Ps . You can
think of Ps as the result of compiling s into something executable.2 It’s technically
helpful to treat every string in ASCII as a program for a string procedure. So when
a string s 2 ASCII doesn’t parse as a proper string procedure, we’ll define Ps to
2 The string s 2 ASCII and the procedure Ps have to be distinguished to avoid a type error:
you can’t apply a string to string. For example, let s be the string that you wrote as your program
to recognize the double letter strings. Applying s to a string argument, say aabbccdd, should
throw a type exception; what you need to do is compile s to the procedure Ps and then apply Ps to
aabbccdd.
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8.2. The Halting Problem 265

be some default string procedure—say one that never halts on anything it is applied
to.
Focusing just on string procedures, the general Halting Problem is to decide,
given strings s and t, whether or not the procedure Ps recognizes t. We’ll show
that the general problem can’t be solved by showing that a special case can’t be
solved, namely, whether or not Ps recognizes s.
Definition 8.2.1.

No-halt WWD fs j Ps applied to s does not haltg D fs … lang.Ps /g: (8.3)

We’re going to prove
Theorem 8.2.2. No-halt is not recognizable.
We’ll use an argument just like Cantor’s in the proof of Theorem 8.1.12.

Proof. By definition,
s 2 No-halt IFF s … lang.Ps /; (8.4)
for all strings s 2 ASCII .
Now suppose to the contrary that No-halt was recognizable. This means there is
some procedure Ps0 that recognizes No-halt, that is,

No-halt D lang.Ps0 / :

Combined with (8.4), we get

s 2 lang.Ps0 / iff s … lang.Ps / (8.5)

for all s 2 ASCII . Now letting s D s0 in (8.5) yields the immediate contradiction

s0 2 lang.Ps0 / iff s0 … lang.Ps0 / :

This contradiction implies that No-halt cannot be recognized by any string proce-
dure.

So that does it: it’s logically impossible for programs in any particular language
to solve just this special case of the general Halting Problem for programs in that
language. And having proved that it’s impossible to have a procedure that figures
out whether an arbitrary program halts, it’s easy to show that it’s impossible to have
a procedure that is a perfect recognizer for any overall run time property.3
3 The weasel word “overall” creeps in here to rule out some run time properties that are easy

to recognize because they depend only on part of the run time behavior. For example, the set of
programs that halt after executing at most 100 instructions is recognizable.
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266 Chapter 8 Infinite Sets

For example, most compilers do “static” type-checking at compile time to ensure
that programs won’t make run-time type errors. A program that type-checks is
guaranteed not to cause a run-time type-error. But since it’s impossible to recognize
perfectly when programs won’t cause type-errors, it follows that the type-checker
must be rejecting programs that really wouldn’t cause a type-error. The conclusion
is that no type-checker is perfect—you can always do better!
It’s a different story if we think about the practical possibility of writing pro-
gramming analyzers. The fact that it’s logically impossible to analyze perfectly
arbitrary programs does not mean that you can’t do a very good job analyzing in-
teresting programs that come up in practice. In fact, these “interesting” programs
are commonly intended to be analyzable in order to confirm that they do what
they’re supposed to do.
In the end, it’s not clear how much of a hurdle this theoretical limitation implies
in practice. But the theory does provide some perspective on claims about general
analysis methods for programs. The theory tells us that people who make such
claims either

are exaggerating the power (if any) of their methods, perhaps to make a sale
or get a grant, or

are trying to keep things simple by not going into technical limitations they’re
aware of, or

perhaps most commonly, are so excited about some useful practical successes
of their methods that they haven’t bothered to think about the limitations
which must be there.

So from now on, if you hear people making claims about having general program
analysis/verification/optimization methods, you’ll know they can’t be telling the
whole story.
One more important point: there’s no hope of getting around this by switching
programming languages. Our proof covered programs written in some given pro-
gramming language like Java, for example, and concluded that no Java program can
perfectly analyze all Java programs. Could there be a C++ analysis procedure that
successfully takes on all Java programs? After all, C++ does allow more intimate
manipulation of computer memory than Java does. But there is no loophole here:
it’s possible to write a virtual machine for C++ in Java, so if there were a C++ pro-
cedure that analyzed Java programs, the Java virtual machine would be able to do
it too, and that’s impossible. These logical limitations on the power of computation
apply no matter what kinds of programs or computers you use.
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8.3. The Logic of Sets 267

8.3 The Logic of Sets
8.3.1 Russell’s Paradox
Reasoning naively about sets turns out to be risky. In fact, one of the earliest at-
tempts to come up with precise axioms for sets in the late nineteenth century by
the logician Gotlob Frege, was shot down by a three line argument known as Rus-
sell’s Paradox4 which reasons in nearly the same way as the proof of Cantor’s
Theorem 8.1.12. This was an astonishing blow to efforts to provide an axiomatic
foundation for mathematics:

Russell’s Paradox

Let S be a variable ranging over all sets, and define

W WWD fS j S 62 S g:

So by definition,
S 2 W iff S 62 S;
for every set S . In particular, we can let S be W , and obtain the
contradictory result that

W 2 W iff W 62 W:

The simplest reasoning about sets crashes mathematics! Russell and his col-
league Whitehead spent years trying to develop a set theory that was not contra-
dictory, but would still do the job of serving as a solid logical foundation for all of
mathematics.
Actually, a way out of the paradox was clear to Russell and others at the time:
it’s unjustified to assume that W is a set. The step in the proof where we let S be
W has no justification, because S ranges over sets, and W might not be a set. In
fact, the paradox implies that W had better not be a set!
4 Bertrand Russell was a mathematician/logician at Cambridge University at the turn of the Twen-

tieth Century. He reported that when he felt too old to do mathematics, he began to study and write
about philosophy, and when he was no longer smart enough to do philosophy, he began writing about
politics. He was jailed as a conscientious objector during World War I. For his extensive philosophical
and political writing, he won a Nobel Prize for Literature.
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268 Chapter 8 Infinite Sets

But denying that W is a set means we must reject the very natural axiom that
every mathematically well-defined collection of sets is actually a set. The prob-
lem faced by Frege, Russell and their fellow logicians was how to specify which
well-defined collections are sets. Russell and his Cambridge University colleague
Whitehead immediately went to work on this problem. They spent a dozen years
developing a huge new axiom system in an even huger monograph called Prin-
cipia Mathematica, but for all intents and purposes, their approach failed. It was
so cumbersome no one ever used it, and it was subsumed by a much simpler, and
now widely accepted, axiomatization of set theory by the logicians Zermelo and
Fraenkel.

8.3.2 The ZFC Axioms for Sets
A formula of set theory5 is a predicate formula that only talks about membership
in sets. That is, a first-order formula of set theory is built using logical connectives
and quantifiers starting solely from expressions of the form “x 2 y.” The domain
of discourse is the collection of sets, and “x 2 y” is interpreted to mean that x and
y are variables that range over sets, and x is one of the elements in y.
Formulas of set theory are not even allowed to have the equality symbol “D,” but
sets are equal iff they have the same elements, so there is an easy way to express
equality of sets purely in terms of membership:

.x D y/ WWD 8z: .z 2 x IFF z 2 y/: (8.6)

Similarly, the subset symbol “” is not allowed in formulas of set theory, but we
can also express subset purely in terms of membership:

.x y/ WWD 8z: .z 2 x IMPLIES z 2 y/: (8.7)

So formulas using symbols “D; ,” in addition to “2” can be understood as
abbreviations for formulas only using “2.” We won’t worry about this distinction
between formulas and abbreviations for formulas—we’ll now just call them all
“formulas of set theory.” For example,

x D y IFF Œx y AND y x

is a formula of set theory that explains a basic connection between set equality and
set containment.
It’s generally agreed that essentially all of mathematics can be derived from a
few formulas of set theory, called the Axioms of Zermelo-Fraenkel Set Theory with
Choice (ZFC), using a few simple logical deduction rules.
5 Technically this is called a pure first-order formula of set theory
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8.3. The Logic of Sets 269

We’re not going to be studying the axioms of ZFC in this text, but we thought
you might like to see them—and while you’re at it, get some more practice reading
and writing quantified formulas:

Extensionality. Two sets are equal iff they are members of the same sets:

x D y IFF .8z: z 2 x IFF z 2 y/:

Pairing. For any two sets x and y, there is a set fx; yg with x and y as its only
elements:
8x; y9u8z: Œz 2 u IFF .z D x OR z D y/

Union. The union u of a collection z of sets is also a set:

8z9u8x: .x 2 u/ IFF .9y: x 2 y AND y 2 z/

Infinity. There is an infinite set. Specifically, there is a nonempty set x such that
for any set y 2 x, the set fyg is also a member of x.

Subset. Given any set x and any definable property of sets, there is a set y contain-
ing precisely those elements in x that have the property.

8x9y8z: z 2 y IFF Œz 2 x AND .z/

where .z/ is a formula of set theory.6

Power Set. All the subsets of a set form another set:

8x9p8u: u x IFF u 2 p:

Replacement. Suppose a formula of set theory defines the graph of a total func-
tion on a set s, that is,
8x 2 s 9y: .x; y/;
and
8x 2 s 8y; z: Œ.x; y/ AND .x; z/ IMPLIES y D z:
Then the image of s under that function is also a set t. Namely,

9t8y: y 2 t IFF Œ9x 2 s: .x; y/:
6 This axiom is more commonly called the Comprehension Axiom.
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270 Chapter 8 Infinite Sets

Foundation. The aim is to forbid any infinite sequence of sets of the form

2 xn 2 2 x1 2 x0

in which each set is a member of the next one. This can be captured by saying
every nonempty set has a “member-minimal” element. Namely, define

member-minimal.m; x/ WWD Œm 2 x AND 8y 2 x: y … m:

Then the Foundation Axiom7 is

8x: x ¤ ; IMPLIES 9m: member-minimal.m; x/:

Choice. Let s be a set of nonempty, disjoint sets. Then there is a set c consisting of
exactly one element from each set in s. The formula is given in Problem 8.30.

8.3.3 Avoiding Russell’s Paradox
These modern ZFC axioms for set theory are much simpler than the system Russell
and Whitehead first came up with to avoid paradox. In fact, the ZFC axioms are
as simple and intuitive as Frege’s original axioms, with one technical addition: the
Foundation axiom. Foundation captures the intuitive idea that sets must be built
up from “simpler” sets in certain standard ways. And in particular, Foundation
implies that no set is ever a member of itself. So the modern resolution of Russell’s
paradox goes as follows: since S 62 S for all sets S , it follows that W , defined
above, contains every set. This means W can’t be a set—or it would be a member
of itself.

8.4 Does All This Really Work?
So this is where mainstream mathematics stands today: there is a handful of ZFC
axioms from which virtually everything else in mathematics can be logically de-
rived. This sounds like a rosy situation, but there are several dark clouds, suggest-
ing that the essence of truth in mathematics is not completely resolved.

The ZFC axioms weren’t etched in stone by God. Instead, they were mostly
made up by Zermelo, who may have been a brilliant logician, but was also
a fallible human being—probably some days he forgot his house keys. So
7 This axiom is also called the Regularity Axiom.
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8.4. Does All This Really Work? 271

maybe Zermelo, just like Frege, didn’t get his axioms right and will be
shot down by some successor to Russell who will use his axioms to prove
a proposition P and its negation P . Then math as we understand it would be
broken—this may sound crazy, but it has happened before.
In fact, while there is broad agreement that the ZFC axioms are capable of
proving all of standard mathematics, the axioms have some further conse-
quences that sound paradoxical. For example, the Banach-Tarski Theorem
says that, as a consequence of the axiom of choice, a solid ball can be divided
into six pieces and then the pieces can be rigidly rearranged to give two solid
balls of the same size as the original!

Some basic questions about the nature of sets remain unresolved. For exam-
ple, Cantor raised the question whether there is a set whose size is strictly
between the smallest infinite set N (see Problem 8.8) and the strictly larger
set pow.N/? Cantor guessed not:
Cantor’s Contiuum Hypothesis: There is no set A such that

N strict A strict pow.N/:

The Continuum Hypothesis remains an open problem a century later. Its
difficulty arises from one of the deepest results in modern Set Theory—
discovered in part by Gödel in the 1930’s and Paul Cohen in the 1960’s—
namely, the ZFC axioms are not sufficient to settle the Continuum Hypoth-
esis: there are two collections of sets, each obeying the laws of ZFC, and in
one collection the Continuum Hypothesis is true, and in the other it is false.
Until a mathematician with a deep understanding of sets can extend ZFC with
persuasive new axioms, the Continuum Hypothesis will remain undecided.

But even if we use more or different axioms about sets, there are some un-
avoidable problems. In the 1930’s, Gödel proved that, assuming that an ax-
iom system like ZFC is consistent—meaning you can’t prove both P and P
for any proposition, P —then the very proposition that the system is consis-
tent (which is not too hard to express as a logical formula) cannot be proved
in the system. In other words, no consistent system is strong enough to verify
itself.

8.4.1 Large Infinities in Computer Science
If the romance of different-size infinities and continuum hypotheses doesn’t appeal
to you, not knowing about them is not going to limit you as a computer scientist.
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272 Chapter 8 Infinite Sets

These abstract issues about infinite sets rarely come up in mainstream mathemat-
ics, and they don’t come up at all in computer science, where the focus is generally
on “countable,” and often just finite, sets. In practice, only logicians and set the-
orists have to worry about collections that are “too big” to be sets. That’s part of
the reason that the 19th century mathematical community made jokes about “Can-
tor’s paradise” of obscure infinities. But the challenge of reasoning correctly about
this far-out stuff led directly to the profound discoveries about the logical limits of
computation described in Section 8.2, and that really is something every computer
scientist should understand.

Problems for Section 8.1
Practice Problems
Problem 8.1.
Show that the set f0; 1g of finite binary strings is countable.

Problem 8.2.
Describe an example of two uncountable sets A and B such that there is no bijec-
tion between A and B.

Problem 8.3.
Indicate which of the following assertions (there may be more than one) are equiv-
alent to
A strict N:

jAj is undefined.

A is countably infinite.

A is uncountable.

A is finite.

N surj A.

8n 2 N, jAj n.

8n 2 N, jAj n.
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8.4. Does All This Really Work? 273

9n 2 N: jAj n.

9n 2 N: jAj < n.

Problem 8.4.
Prove that if there is a total injective (Œ 1 out; 1 in) relation from S to N, then
S is countable.

Problem 8.5.
Prove that if S is an infinite set, then pow S is uncountable.

Problem 8.6.
Let A to be some infinite set and B to be some countable set. We know from
Lemma 8.1.7 that
A bij .A [ fb0 g/
for any element b0 2 B. An easy induction implies that

A bij .A [ fb0 ; b1 ; : : : ; bn g/ (8.8)

for any finite subset fb0 ; b1 ; : : : ; bn g B.
Students sometimes think that (8.8) shows that A bij .A [ B/. Now it’s true that
A bij .A [ B/ for all such A and B for any countable set B (Problem 8.13), but the
facts above do not prove it.
To explain this, let’s say that a predicate P .C / is finitely discontinuous when
P .A [ F / is true for every finite subset F B, but P .A [ B/ is false. The hole
in the claim that (8.8) implies A bij .A [ B/ is the assumption (without proof) that
the predicate
P0 .C / WWD ŒA bij C
is not finitely discontinuous. This assumption about P0 is correct, but it’s not com-
pletely obvious and takes some proving.
To illustrate this point, let A be the nonnegative integers and B be the nonneg-
ative rational numbers, and remember that both A and B are countably infinite.
Some of the predicates P .C / below are finitely discontinuous and some are not.
Indicate which is which.

1. C is finite.
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274 Chapter 8 Infinite Sets

2. C is countable.

3. C is uncountable.

4. C contains only finitely many non-integers.

5. C contains the rational number 2/3.

6. There is a maximum non-integer in C .

7. There is an > 0 such that any two elements of C are apart.

8. C is countable.

9. C is uncountable.

10. C has no infinite decreasing sequence c0 > c1 > .

11. Every nonempty subset of C has a minimum element.

12. C has a maximum element.

13. C has a minimum element.

Class Problems
Problem 8.7.
Show that the set N of finite sequences of nonnegative integers is countable.

Problem 8.8. (a) Several students felt the proof of Lemma 8.1.7 was worrisome,
if not circular. What do you think?

(b) Use the proof of Lemma 8.1.7 to show that if A is an infinite set, then A surj N,
that is, every infinite set is “at least as big as” the set of nonnegative integers.

Problem 8.9.
The rational numbers fill the space between integers, so a first thought is that there
must be more of them than the integers, but it’s not true. In this problem you’ll
show that there are the same number of positive rationals as positive integers. That
is, the positive rationals are countable.
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8.4. Does All This Really Work? 275

(a) Define a bijection between the set ZC of positive integers, and the set .ZC
ZC / of all pairs of positive integers:

.1; 1/; .1; 2/; .1; 3/; .1; 4/; .1; 5/; : : :
.2; 1/; .2; 2/; .2; 3/; .2; 4/; .2; 5/; : : :
.3; 1/; .3; 2/; .3; 3/; .3; 4/; .3; 5/; : : :
.4; 1/; .4; 2/; .4; 3/; .4; 4/; .4; 5/; : : :
.5; 1/; .5; 2/; .5; 3/; .5; 4/; .5; 5/; : : :
::
:

(b) Conclude that the set QC of all positive rational numbers is countable.

Problem 8.10.
This problem provides a proof of the [Schröder-Bernstein] Theorem:

If A inj B and B inj A, then A bij B. (8.9)

Since A inj B and B inj A, there are are total injective functions f W A ! B
and g W B ! A.
Assume for simplicity that A and B have no elements in common. Let’s picture
the elements of A arranged in a column, and likewise B arranged in a second col-
umn to the right, with left-to-right arrows connecting a to f .a/ for each a 2 A
and likewise right-to-left arrows for g. Since f and g are total functions, there is
exactly one arrow out of each element. Also, since f and g are injections, there is
at most one arrow into any element.
So starting at any element, there is a unique and unending path of arrows going
forwards (it might repeat). There is also a unique path of arrows going backwards,
which might be unending, or might end at an element that has no arrow into it.
These paths are completely separate: if two ran into each other, there would be two
arrows into the element where they ran together.
This divides all the elements into separate paths of four kinds:
(i) paths that are infinite in both directions,

(ii) paths that are infinite going forwards starting from some element of A.

(iii) paths that are infinite going forwards starting from some element of B.

(iv) paths that are unending but finite.
(a) What do the paths of the last type (iv) look like?
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276 Chapter 8 Infinite Sets

(b) Show that for each type of path, either

(i) the f -arrows define a bijection between the A and B elements on the path, or
(ii) the g-arrows define a bijection between B and A elements on the path, or
(iii) both sets of arrows define bijections.

For which kinds of paths do both sets of arrows define bijections?

(d) Justify the assumption that A and B are disjoint.

Problem 8.11. (a) Prove that if a nonempty set C is countable, then there is a total
surjective function f W N ! C .

(b) Conversely, suppose that N surj D, that is, there is a not necessarily total
surjective function f W ND. Prove that D is countable.

Problem 8.12. (a) For each of the following sets, indicate whether it is finite,
countably infinite, or uncountable.

(i) The set of even integers greater than 10100 .
(ii) The set of “pure” complex numbers of the form ri for nonzero real numbers r.
(iii) The powerset of the integer interval Œ10::1010 .
(iv) The complex numbers c such that 9m; n 2 Z: .m C nc/c D 0.
Let U be an uncountable set, C be a countably infinite subset of U, and D be
a countably infinite set.
(v) U [ D.
(vi) U \ C
(vii) U D

(b) Given examples of sets A and B such that

R strict A strict B:

Recall that A strict B means that A is not “as big as” B.
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8.4. Does All This Really Work? 277

Homework Problems
Problem 8.13.
Prove that if A is an infinite set and B is a countably infinite set that has no elements
in common with A, then
A bij .A [ B/:
Reminder: You may assume any of the results from class or text as long as you
state them explicitly.

Problem 8.14.
In this problem you will prove a fact that may surprise you—or make you even
more convinced that set theory is nonsense: the half-open unit interval is actually
the “same size” as the nonnegative quadrant of the real plane!8 Namely, there is a
bijection from .0; 1 to Œ0; 1/ Œ0; 1/.
(a) Describe a bijection from .0; 1 to Œ0; 1/.
Hint: 1=x almost works.

(b) An infinite sequence of the decimal digits f0; 1; : : : ; 9g will be called long if
it does not end with all 0’s. An equivalent way to say this is that a long sequence
is one that has infinitely many occurrences of nonzero digits. Let L be the set of
all such long sequences. Describe a bijection from L to the half-open real interval
.0; 1.
Hint: Put a decimal point at the beginning of the sequence.

(c) Describe a surjective function from L to L2 that involves alternating digits
from two long sequences. Hint: The surjection need not be total.

(d) Prove the following lemma and use it to conclude that there is a bijection from
L2 to .0; 12 .
Lemma 8.4.1. Let A and B be nonempty sets. If there is a bijection from A to B,
then there is also a bijection from A A to B B.

(e) Conclude from the previous parts that there is a surjection from .0; 1 to .0; 12 .
Then appeal to the Schröder-Bernstein Theorem to show that there is actually a
bijection from .0; 1 to .0; 12 .

(f) Complete the proof that there is a bijection from .0; 1 to Œ0; 1/2 .
8 The half-open unit interval .0; 1 is fr 2 R j 0 < r 1g. Similarly, Œ0; 1/ WWD fr 2 R j r 0g.
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278 Chapter 8 Infinite Sets

Exam Problems
Problem 8.15. (a) For each of the following sets, indicate whether it is finite,
countably infinite, or uncountable.

(i) The set of even integers greater than 10100 .
(ii) The set of “pure” complex numbers of the form ri for nonzero real numbers
r.
(iii) The powerset of the integer interval Œ10::1010 .
(iv) The complex numbers c such that c is the root of a quadratic with integer
coefficients, that is,
9m; n; p 2 Z; m ¤ 0: mc 2 C nc C p D 0:

Let U be an uncountable set, C be a countably infinite subset of U, and D be
a countably infinite set.
(v) U [ D.
(vi) U \ C
(vii) U D

(b) Give an example of sets A and B such that
R strict A strict B:

Problem 8.16.
Prove that if A0 ; A1 ; : : : ; An ; : : : is an infinite sequence of countable sets, then so
is
[1
An
nD0

Problem 8.17.
Let A and B be countably infinite sets:
A D fa0 ; a1 ; a2 ; a3 ; : : :g
B D fb0 ; b1 ; b2 ; b3 ; : : :g
Show that their product A B is also a countable set by showing how to list the
elements of AB. You need only show enough of the initial terms in your sequence
to make the pattern clear—a half dozen or so terms usually suffice.
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8.4. Does All This Really Work? 279

Problem 8.18.
Let f0; 1g be the set of finite binary sequences, f0; 1g! be the set of infinite bi-
nary sequences, and F be the set of sequences in f0; 1g! that contain only a finite
number of occurrences of 1’s.
(a) Describe a simple surjective function from f0; 1g to F .

(b) The set F WWD f0; 1g! F consists of all the infinite binary sequences with
infinitely many 1’s. Use the previous problem part to prove that F is uncountable.
Hint: We know that f0; 1g is countable and f0; 1g! is not.

Problem 8.19.
Let f0; 1g! be the set of infinite binary strings, and let B f0; 1g! be the set of
infinite binary strings containing infinitely many occurrences of 1’s. Prove that B
is uncountable. (We have already shown that f0; 1g! is uncountable.)
Hint: Define a suitable function from f0; 1g! to B.

Problem 8.20.
A real number is called quadratic when it is a root of a degree two polynomial with
integer coefficients. Explain why there are only countably many quadratic reals.

Problem 8.21.
Describe which of the following sets have bijections between them:

Z (integers); R (real numbers);
C (complex numbers); Q (rational numbers);
pow.Z/ (all subsets of integers); pow.;/;
pow.pow.;//; f0; 1g (finite binary sequences);
f0; 1g! (infinite binary sequences) fT; Fg (truth values)
pow.fT; Fg/; pow.f0; 1g! /

Problem 8.22.
Prove that the set ZC of all finite sequences of positive integers is countable.

Hint: If s 2 ZC , let sum(s) be the sum of the successive integers in s.
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280 Chapter 8 Infinite Sets

Problems for Section 8.2
Class Problems
Problem 8.23.
Let N! be the set of infinite sequences of nonnegative integers. For example, some
sequences of this kind are:

.0; 1; 2; 3; 4; : : : /;
.2; 3; 5; 7; 11; : : : /;
.3; 1; 4; 5; 9; : : : /:

Prove that this set of sequences is uncountable.

Problem 8.24.
There are lots of different sizes of infinite sets. For example, starting with the
infinite set N of nonnegative integers, we can build the infinite sequence of sets

N strict pow.N/ strict pow.pow.N// strict pow.pow.pow.N/// strict : : : :

where each set is “strictly smaller” than the next one by Theorem 8.1.12. Let
pown .N/ be the nth set in the sequence, and
1
[
U WWD pown .N/:
nD0

(a) Prove that
U surj pown .N/; (8.10)
for all n > 0.

(b) Prove that
pown .N/ strict U
for all n 2 N.
Now of course, we could take U; pow.U /; pow.pow.U //; : : : and keep on in this
way building still bigger infinities indefinitely.
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8.4. Does All This Really Work? 281

Homework Problems
Problem 8.25.
For any sets A and B, let ŒA ! B be the set of total functions from A to B. Prove
that if A is not empty and B has more than one element, then NOT.A surj ŒA !
B/.
Hint: Suppose that is a function from A to ŒA ! B mapping each element
a 2 A to a function a W A ! B. Pick any two elements of B; call them 0 and 1.
Then define (
0 if a .a/ D 1;
diag.a/ WWD
1 otherwise:

Problem 8.26.
String procedures are one-argument procedures that apply to strings over the ASCII
alphabet. If application of procedure P to string s results in a computation that
eventually halts, we say that P recognizes s. We define lang.P / to be the set of
strings or language recognized by P :

lang.P / WWDfs 2 ASCII j P recognizes sg:

A language is unrecognizable when it is not equal to lang.P / for any procedure P .
A string procedure declaration is a text s 2 ASCII that conforms to the gram-
matical rules for programs. The declaration defines a procedure Ps , which we can
think of as the result of compiling s into an executable object. If s 2 ASCII is not
a grammatically well-formed procedure declaration, we arbitrarily define Ps to be
the string procedure that fails to halt when applied to any string. Now every string
defines a string procedure, and every string procedure is Ps for some s 2 ASCII .
An easy diagonal argument in Section 8.2 showed that

No-halt WWD fs j Ps applied to s does not haltg D fs j s … lang.Ps /g

is not recognizable.
It may seem pretty weird to apply a procedure to its own declaration. Are there
any less weird examples of unrecognizable set? The answer is “many more.” In
this problem, we’ll show three more:

No-halt- WWD fs j Ps applied to does not haltg D fs j … lang.Ps /g;
Finite-halt WWD fs j lang.Ps / is finiteg;
Always-halt WWD fs j lang.Ps / D ASCII g:
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282 Chapter 8 Infinite Sets

Let’s begin by showing how we could use a recognizer for No-halt- to define a
recognizer for No-halt. That is, we will “reduce” the weird problem of recognizing
No-halt to the more understandable problem of recognizing No-halt-. Since there
is no recognizer for No-halt, it follows that there can’t be one for No-halt- either.
Here’s how this reduction would work: suppose we want to recognize when a
given string s is in No-halt. Revise s to be the declaration of a slightly modified
procedure Ps 0 which behaves as follows:
Ps 0 applied to argument t 2 ASCII , ignores t, and simulates Ps ap-
plied to s.
So, if Ps applied to s halts, then Ps 0 halts on every string it is applied to, and if Ps
applied to s does not halt, then Ps 0 does not halt on any string it is applied to. That
is,

s 2 No-halt IMPLIES lang.Ps 0 / D ;
IMPLIES … lang.Ps 0 /
IMPLIES s 0 2 No-halt-;
s … No-halt IMPLIES lang.Ps 0 / D ASCII
IMPLIES 2 lang.Ps 0 /
IMPLIES s 0 … No-halt-:

In short,
s 2 No-halt IFF s 0 2 No-halt-:
So to recognize when s 2 No-halt all you need to do is recognize when s 0 2
No-halt-. As already noted above (but we know that remark got by several students,
so we’re repeating the explanation), this means that if No-halt- was recognizable,
then No-halt would be as well. Since we know that No-halt is unrecognizable, then
No-halt- must also be unrecognizable, as claimed.
(a) Conclude that Finite-halt is unrecognizable.
Hint: Same s 0 .
Next, let’s see how a reduction of No-halt to Always-halt would work. Suppose we
want to recognize when a given string s is in No-halt. Revise s to be the declaration
of a slightly modified procedure Ps 00 which behaves as follows:
When Ps 00 is applied to argument t 2 ASCII , it simulates Ps applied
to s for up to jt j “steps” (executions of individual machine instruc-
tions). If Ps applied to s has not halted in jtj steps, then the application
of Ps 00 to t halts. If Ps applied to s has halted within jtj steps, then the
application of Ps 00 to t runs forever.
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8.4. Does All This Really Work? 283

(b) Conclude that Always-halt is unrecognizable.
Hint: Explain why
s 2 No-halt IFF s 00 2 Always-halt:

(c) Explain why Finite-halt is unrecognizable.
Hint: Same s 00 .
Note that it’s easy to recognize when Ps does halt on s: just simulate the appli-
cation of Ps to s until it halts. This shows that No-halt is recognizable. We’ve just
concluded that Finite-halt is nastier: neither it nor its complement is recognizable.

Problem 8.27.
There is a famous paradox about describing numbers which goes as follows:
There are only so many possible definitions of nonnegative integers that can be
written out in English using no more than 161 characters from the Roman alphabet,
punctuation symbols, and spaces. So there have to be an infinite number of nonneg-
ative integers that don’t have such short definitions. By the Well Ordering Principle,
there must be a least nonnegative integer n that has no such short definition. But
wait a minute,
“The least nonnegative integer that cannot be defined in English using
at most 161 characters from the Roman alphabet, punctuation symbols,
and spaces.”
is a definition of n that uses 161 characters (count ’em). So n can’t exist, and the
Well Ordering Principle is unsound!
Now this “paradox” doesn’t stand up to reason because it rests on the decidedly
murky concept of a “definition in English.” As usual, when you don’t know what
you’re talking about, reaching contradictory conclusions is to be expected.
But we can extract from this paradox a well-defined and interesting theorem
about definability in predicate logic. The method we use is essentialy the same as
the one used to prove Cantor’s Theorem 8.1.12, and it leads to many other important
results about the logical limits of mathematics and computer science. In particular,
we’ll present a simple and precise description of a set of binary strings that can’t
be described by ordinary logical formulas. In other words, we will give a precise
description of an undescribable set of strings, which sounds paradoxical, but won’t
be when we look at it more closely.
Let’s start by illustrating how a logical formula can describe the set of binary
strings that do not contain a 1:
NOTŒ9y: 9z: s D y1z : (no-1s)
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284 Chapter 8 Infinite Sets

So the strings s described by formula (no-1s) are exactly the strings consisting
solely of 0’s.
Formula (no-1s) is an example of a “string formula” of the kind we will use to
describe properties of binary strings. More precisely, an atomic string formula is a
formula, like “s D y1z” above, that is of the general form

“xy : : : z D uv : : : w 00

where x; y; : : : ; z; u; v; : : : ; w may be the constants 0,1, or may be variables rang-
ing over the set, f0; 1g , of finite binary strings. A string formula in general is
one like (no-1s), built up from atomic formulas using quantifiers and propositional
connectives.
When G.s/ is a string formula, we’ll use the notation desc.G/ for the set of
binary strings s that satisfy G. That is,

desc.G/ WWD fs 2 f0; 1g j G.s/g:

A set of binary strings is describable if it equals desc.G/ for some string formula
G. For example, the set 0 of finite strings of 0’s is describable because

desc.(no-1s)/ D 0 :

The next important idea comes from the observation that a string formula itself is
a syntactic object, consisting of a string of characters over some standard character
alphabet. Now coding characters of an alphabet into binary strings is a familiar
idea. For example, the characters of the ASCII alphabet have a standard coding
into the length eight binary strings. Once its individual characters are coded into
binary, a complete string formula can be coded into a binary string by concatenating
the binary codes of its consecutive characters—a very familiar idea to a computer
scientist.
Now suppose x is a binary string that codes some formula Gx . The details of
how we extract Gx from its code x don’t matter much—we only require that there
is some procedure to actually display the string formula Gx given its code x.
It’s technically convenient to treat every string as the code of a string formula,
so if x is not a binary string we would get from a string formula, we’ll arbitrarily
define Gx to be the formula (no-1s).
Now we have just the kind of situation where a Cantor-style diagonal argument
can be applied, namely, we’ll ask whether a string describes a property of itself !
That may sound like a mind-bender, but all we’re asking is whether

Gx .x/
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8.4. Does All This Really Work? 285

is true, or equivalently whether

x 2 desc.Gx /:

For example, using character-by-character translations of formulas into binary,
neither the string 0000 nor the string 10 would be the binary representation of a
formula, so our convention implies that

G0000 D G10 WWD formula (no-1s):

So
desc.G0000 / D desc.G10 / D 0 :
This means that

0000 2 desc.G0000 / and 10 … desc.G10 /:

Now we are in a position to give a precise mathematical description of an “un-
describable” set of binary strings, namely:

Theorem. Define

U WWD fx 2 f0; 1g j x … desc.Gx /g: (8.11)

The set U is not describable.

Use reasoning similar to Cantor’s Theorem 8.1.12 to prove this Theorem.
Hint: Suppose U D desc.GxU /.

Exam Problems
Problem 8.28.
Let f1; 2; 3g! be the set of infinite sequences containing only the numbers 1, 2, and
3. For example, some sequences of this kind are:

.1; 1; 1; 1:::/;
.2; 2; 2; 2:::/;
.3; 2; 1; 3:::/:

Prove that f1; 2; 3g! is uncountable.
Hint: One approach is to define a surjective function from f1; 2; 3g! to the power
set pow.N/.
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286 Chapter 8 Infinite Sets

Problems for Section 8.3
Class Problems
Problem 8.29.
Forming a pair .a; b/ of items a and b is a mathematical operation that we can
safely take for granted. But when we’re trying to show how all of mathematics can
be reduced to set theory, we need a way to represent the pair .a; b/ as a set.
(a) Explain why representing .a; b/ by fa; bg won’t work.

(b) Explain why representing .a; b/ by fa; fbgg won’t work either. Hint: What
pair does ff1g; f2gg represent?

(c) Define
pair.a; b/ WWD fa; fa; bgg:
Explain why representing .a; b/ as pair.a; b/ uniquely determines a and b. Hint:
Sets can’t be indirect members of themselves: a 2 a never holds for any set a, and
neither can a 2 b 2 a hold for any b.

Problem 8.30.
The axiom of choice says that if s is a set whose members are nonempty sets that
are pairwise disjoint—that is, no two sets in s have an element in common—then
there is a set c consisting of exactly one element from each set in s.
In formal logic, we could describe s with the formula,

pairwise-disjoint.s/WWD8x 2 s: x ¤ ; AND
8x; y 2 s: x ¤ y IMPLIES x \ y D ;:

Similarly we could describe c with the formula

choice-set.c; s/ WWD 8x 2 s: 9Šz: z 2 c \ x:

Here “9Š z:” is fairly standard notation for “there exists a unique z.”
Now we can give the formal definition:

Definition (Axiom of Choice).

8s: pairwise-disjoint.s/ IMPLIES 9c: choice-set.c; s/:
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8.4. Does All This Really Work? 287

The only issue here is that set theory is technically supposed to be expressed in
terms of pure formulas in the language of sets, which means formula that uses only
the membership relation 2 propositional connectives, the two quantifies 8 and 9,
and variables ranging over all sets. Verify that the axiom of choice can be expressed
as a pure formula, by explaining how to replace all impure subformulas above with
equivalent pure formulas.
For example, the formula x D y could be replaced with the pure formula 8z: z 2
x IFF z 2 y.

Problem 8.31.
Let R W A ! A be a binary relation on a set A. If a1 R a0 , we’ll say that a1 is “R-
smaller” than a0 . R is called well founded when there is no infinite “R-decreasing”
sequence:
R an R R a1 R a0 ; (8.12)
of elements ai 2 A.
For example, if A D N and R is the <-relation, then R is well founded because
if you keep counting down with nonnegative integers, you eventually get stuck at
zero:
0 < < n 1 < n:
But you can keep counting up forever, so the >-relation is not well founded:

> n > > 1 > 0:

Also, the -relation on N is not well founded because a constant sequence of, say,
2’s, gets -smaller forever:

2 2 2:

(a) If B is a subset of A, an element b 2 B is defined to be R-minimal in B iff
there is no R-smaller element in B. Prove that R W A ! A is well founded iff every
nonempty subset of A has an R-minimal element.
A logic formula of set theory has only predicates of the form “x 2 y” for vari-
ables x; y ranging over sets, along with quantifiers and propositional operations.
For example,
isempty.x/ WWD 8w: NOT.w 2 x/
is a formula of set theory that means that “x is empty.”
(b) Write a formula member-minimal.u; v/ of set theory that means that u is 2-
minimal in v.
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288 Chapter 8 Infinite Sets

(c) The Foundation axiom of set theory says that 2 is a well founded relation
on sets. Express the Foundation axiom as a formula of set theory. You may use
“member-minimal” and “isempty” in your formula as abbreviations for the formu-
las defined above.

(d) Explain why the Foundation axiom implies that no set is a member of itself.

Homework Problems
Problem 8.32.
In writing formulas, it is OK to use abbreviations introduced earlier (so it is now
legal to use “D” because we just defined it).
(a) Explain how to write a formula, Subsetn .x; y1 ; y2 ; : : : ; yn /, of set theory 9

that means x fy1 ; y2 ; : : : ; yn g.

(b) Now use the formula Subsetn to write a formula, Atmostn .x/, of set theory
that means that x has at most n elements.

(c) Explain how to write a formula Exactlyn of set theory that means that x has
exactly n elements. Your formula should only be about twice the length of the
formula Atmostn .

(d) The direct way to write a formula Dn .y1 ; : : : ; yn / of set theory that means
that y1 ; : : : ; yn are distinct elements is to write an AND of subformulas “yi ¤ yj ”
for 1 i < j n. Since there are n.n 1/=2 such subformulas, this approach
leads to a formula Dn whose length grows proportional to n2 . Describe how to
write such a formula Dn .y1 ; : : : ; yn / whose length only grows proportional to n.
Hint: Use Subsetn and Exactlyn .

Exam Problems
Problem 8.33. (a) Explain how to write a formula Members.p; a; b/ of set the-
ory10 that means p D fa; bg.
Hint: Say that everything in p is either a or b. It’s OK to use subformulas of the
form “x D y,” since we can regard “x D y” as an abbreviation for a genuine set
theory formula.
A pair .a; b/ is simply a sequence of length two whose first item is a and whose
second is b. Sequences are a basic mathematical data type we take for granted, but
when we’re trying to show how all of mathematics can be reduced to set theory, we
9 See Section 8.3.2.
10 See Section 8.3.2.
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8.4. Does All This Really Work? 289

need a way to represent the ordered pair .a; b/ as a set. One way that will work11
is to represent .a; b/ as
pair.a; b/ WWD fa; fa; bgg:
(b) Explain how to write a formula Pair.p; a; b/, of set theory 12 that means p D
pair.a; b/.
Hint: Now it’s OK to use subformulas of the form “Members.p; a; b/.”

Problems for Section 8.4
Homework Problems
Problem 8.34.
In this problem, structural induction and the Foundation Axiom of set theory pro-
vide simple proofs about some utterly infinite objects.

Definition. The class of ‘recursive set-like” objects, Recs, is defined recursively as
follows:
Base case: The empty set ; is a Recs.
Constructor step: If S is a nonempty set of Recs’s, then S is a Recs.

(a) Prove that Recs satisfies the Foundation Axiom: there is no infinite sequence
of Recs, ro ; r1 ; : : : ; rn 1 ; rn ; : : : such that

: : : rn 2 rn 1 2 : : : r1 2 r0 : (8.13)

Hint: Structural induction.

(b) Prove that every pure set is a Recs.13
Hint: Use the Foundation axiom.

(c) Every Recs R defines a special kind of two-person game of perfect informa-
tion called a uniform game. The initial “board position” of the game is R itself.
11 Some similar ways that don’t work are described in problem 8.29.
12 See Section 8.3.2.
13 A “pure” set is empty or is a set whose elements are all pure sets.
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290 Chapter 8 Infinite Sets

A player’s move consists of choosing any member R. The two players alternate
moves, with the player whose turn it is to move called the Next player. The Next
player’s move determines a game in which the other player, called the Previous
player, moves first.
The game is called “uniform” because the two players have the same objective: to
leave the other player stuck with no move to make. That is, whoever moves to the
empty set is a winner, because then the next player has no move.
Prove that in every uniform game, either the Previous player or the Next player has
a winning strategy.

Problem 8.35.
For any set x, define next.x/ to be the set consisting of all the elements of x, along
with x itself:
next.x/ WWD x [ fxg:
So by definition,
x 2 next.x/ and x next.x/: (8.14)
Now we give a recursive definition of a collection Ord of sets called ordinals that
provide a way to count infinite sets. Namely,

Definition.

; 2 Ord;
if 2 Ord; then next./ 2 Ord;
[
if S Ord; then 2 Ord:
2S

There is a method for proving things about ordinals that follows directly from
the way they are defined. Namely, let P .x/ be some property of sets. The Ordinal
Induction Rule says that to prove that P ./ is true for all ordinals , you need only
show two things

If P holds for all the members of next.x/, then it holds for next.x/, and

if P holds for all members of some set S , then it holds for their union.

That is:
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8.4. Does All This Really Work? 291

Rule. Ordinal Induction
8x: .8y 2 next.x/: P .y// IMPLIES S
P .next.x//;
8S: .8x 2 S: P .x// IMPLIES P . x2S x/
8 2 Ord: P ./

The intuitive justification for the Ordinal Induction Rule is similar to the justifi-
cation for strong induction. We will accept the soundness of the Ordinal Induction
Rule as a basic axiom.
(a) A set x is closed under membership if every element of x is also a subset of
x, that is
8y 2 x: y x:
Prove that every ordinal is closed under membership.

(b) A sequence
2 nC1 2 n 2 2 1 2 0 (8.15)
of ordinals i is called a member-decreasing sequence starting at 0 . Use Ordinal
Induction to prove that no ordinal starts an infinite member-decreasing sequence.14

14 Do not assume the Foundation Axiom of ZFC (Section 8.3.2) which says that there isn’t any set

that starts an infinite member-decreasing sequence. Even in versions of set theory in which the Foun-
dation Axiom does not hold, there cannot be any infinite member-decreasing sequence of ordinals.
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II Structures
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Introduction
The properties of the set of integers are the subject of Number Theory. This part
of the text starts with a chapter on this topic because the integers are a very famil-
iar mathematical structure that have lots of easy-to-state and interesting-to-prove
properties. This makes Number Theory a good place to start serious practice with
the methods of proof outlined in Part 1. Moreover, Number Theory has turned out
to have multiple applications in computer science. For example, most modern data
encryption methods are based on Number theory.
We study numbers as a “structure” that has multiple parts of different kinds. One
part is, of course, the set of all the integers. A second part is the collection of basic
integer operations: addition, multiplication, exponentiation,. . . . Other parts are the
important subsets of integers—like the prime numbers—out of which all integers
can be built using multiplication.
Structured objects more generally are fundamental in computer science. Whether
you are writing code, solving an optimization problem, or designing a network, you
will be dealing with structures.
Graphs, also known as networks, are a fundamental structure in computer sci-
ence. Graphs can model associations between pairs of objects; for example, two
exams that cannot be given at the same time, two people that like each other, or two
subroutines that can be run independently. In Chapter 10, we study directed graphs
which model one-way relationships such as being bigger than, loving (sadly, it’s
often not mutual), and being a prerequisite for. A highlight is the special case of
acyclic digraphs (DAGs) that correspond to a class of relations called partial or-
ders. Partial orders arise frequently in the study of scheduling and concurrency.
Digraphs as models for data communication and routing problems are the topic of
Chapter 11.
In Chapter 12 we focus on simple graphs that represent mutual or symmetric re-
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296 Part II Structures

lationships, such as being in conflict, being compatible, being independent, being
capable of running in parallel. Planar Graphs—simple graphs that can be drawn in
the plane—are examined in Chapter 13, the final chapter of Part II. The impossi-
bility of placing 50 geocentric satellites in orbit so that they uniformly blanket the
globe will be one of the conclusions reached in this chapter.
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9 Number Theory
Number theory is the study of the integers. Why anyone would want to study the
integers may not be obvious. First of all, what’s to know? There’s 0, there’s 1, 2,
3, and so on, and, oh yeah, -1, -2, . . . . Which one don’t you understand? What
practical value is there in it?
The mathematician G. H. Hardy delighted at its impracticality. He wrote:

[Number theorists] may be justified in rejoicing that there is one sci-
ence, at any rate, and that their own, whose very remoteness from or-
dinary human activities should keep it gentle and clean.

Hardy was especially concerned that number theory not be used in warfare; he
was a pacifist. You may applaud his sentiments, but he got it wrong: number theory
underlies modern cryptography, which is what makes secure online communication
possible. Secure communication is of course crucial in war—leaving poor Hardy
spinning in his grave. It’s also central to online commerce. Every time you buy a
book from Amazon, use a certificate to access a web page, or use a PayPal account,
you are relying on number theoretic algorithms.
Number theory also provides an excellent environment for us to practice and
apply the proof techniques that we developed in previous chapters. We’ll work out
properties of greatest common divisors (gcd’s) and use them to prove that integers
factor uniquely into primes. Then we’ll introduce modular arithmetic and work out
enough of its properties to explain the RSA public key crypto-system.
Since we’ll be focusing on properties of the integers, we’ll adopt the default
convention in this chapter that variables range over the set Z of integers.

9.1 Divisibility
The nature of number theory emerges as soon as we consider the divides relation.

Definition 9.1.1. a divides b (notation a j b) iff there is an integer k such that

ak D b:

The divides relation comes up so frequently that multiple synonyms for it are
used all the time. The following phrases all say the same thing:
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298 Chapter 9 Number Theory

a j b,

a divides b,

a is a divisor of b,

a is a factor of b,

b is divisible by a,

b is a multiple of a.

Some immediate consequences of Definition 9.1.1 are that for all n

n j 0; n j n; and ˙ 1 j n:

Also,
0 j n IMPLIES n D 0:
Dividing seems simple enough, but let’s play with this definition. The Pythagore-
ans, an ancient sect of mathematical mystics, said that a number is perfect if it
equals the sum of its positive integral divisors, excluding itself. For example,
6 D 1 C 2 C 3 and 28 D 1 C 2 C 4 C 7 C 14 are perfect numbers. On the
other hand, 10 is not perfect because 1 C 2 C 5 D 8, and 12 is not perfect because
1 C 2 C 3 C 4 C 6 D 16. Euclid characterized all the even perfect numbers around
300 BC (Problem 9.2). But is there an odd perfect number? More than two thou-
sand years later, we still don’t know! All numbers up to about 10300 have been
ruled out, but no one has proved that there isn’t an odd perfect number waiting just
over the horizon.
So a half-page into number theory, we’ve strayed past the outer limits of human
knowledge. This is pretty typical; number theory is full of questions that are easy to
pose, but incredibly difficult to answer. We’ll mention a few more such questions
in later sections.1

9.1.1 Facts about Divisibility
The following lemma collects some basic facts about divisibility.

Lemma 9.1.2.

1. If a j b and b j c, then a j c.
1 Don’t Panic—we’re going to stick to some relatively benign parts of number theory. These
super-hard unsolved problems rarely get put on problem sets.
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9.1. Divisibility 299

2. If a j b and a j c, then a j sb C t c for all s and t .

3. For all c ¤ 0, a j b if and only if ca j cb.

Proof. These facts all follow directly from Definition 9.1.1. To illustrate this, we’ll
prove just part 2:
Given that a j b, there is some k1 2 Z such that ak1 D b. Likewise, ak2 D c,
so
sb C t c D s.k1 a/ C t .k2 a/ D .sk1 C t k2 /a:
Therefore sb C t c D k3 a where k3 WWD .sk1 C t k2 /, which means that

a j sb C t c:

A number of the form sb C t c is called an integer linear combination of b and c,
or, since in this chapter we’re only talking about integers, just a linear combination.
So Lemma 9.1.2.2 can be rephrased as

If a divides b and c, then a divides every linear combination of b and c.

We’ll be making good use of linear combinations, so let’s get the general definition
on record:

Definition 9.1.3. An integer n is a linear combination of numbers b0 ; : : : ; bk iff

n D s0 b0 C s1 b1 C C sk bk

for some integers s0 ; : : : ; sk .

9.1.2 When Divisibility Goes Bad
As you learned in elementary school, if one number does not evenly divide another,
you get a “quotient” and a “remainder” left over. More precisely:

Theorem 9.1.4. [Division Theorem]2 Let n and d be integers such that d ¤ 0.
Then there exists a unique pair of integers q and r, such that

n D q d C r AND 0 r < jd j : (9.1)
2 This theorem is often called the “Division Algorithm,” but we prefer to call it a theorem since it
does not actually describe a division procedure for computing the quotient and remainder.
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300 Chapter 9 Number Theory

The number q is called the quotient and the number r is called the remainder of
n divided by d . We use the notation qcnt.n; d / for the quotient and rem.n; d / for
the remainder.
The absolute value notation jd j used above is probably familiar from introduc-
tory calculus, but for the record, let’s define it.

Definition 9.1.5. For any real number r, the absolute value jrj of r is:3
(
r if r 0;
jrj WWD
r if r < 0:

So by definition, the remainder rem.n; d / is nonnegative regardless of the sign
of n and d . For example, rem. 11; 7/ D 3, since 11 D . 2/ 7 C 3.
“Remainder” operations built into many programming languages can be a source
of confusion. For example, the expression “32 % 5” will be familiar to program-
mers in Java, C, and C++; it evaluates to rem.32; 5/ D 2 in all three languages.
On the other hand, these and other languages are inconsistent in how they treat re-
mainders like “32 % -5” or “-32 % 5” that involve negative numbers. So don’t be
distracted by your familiar programming language’s behavior on remainders, and
stick to the mathematical convention that remainders are nonnegative.
The remainder on division by d by definition is a number in the (integer) interval
from 0 to jd j 1. Such integer intervals come up so often that it is useful to have a
simple notation for them. For k n 2 Z,

.k::n/ WWD fi j k < i < ng;
.k::n WWD .k; n/ [ fng;
Œk::n/ WWD fkg [ .k; n/;
Œk::n WWD fkg [ .k; n/ [ fng D fi j k i ng:

9.1.3 Die Hard
Die Hard 3 is just a B-grade action movie, but we think it has an inner message:
everyone should learn at least a little number theory. In Section 6.2.3, we formal-
ized a state machine for the Die Hard jug-filling problem using 3 and 5 gallon jugs,
p
3 The absolute value of r could be defined as r 2 , which works because of the convention that
square root notation always refers to the nonnegative square root (see Problem 1.3). Absolute value
generalizes to complex numbers where it is called the norm. For a; b 2 R,
p
ja C bi j WWD a2 C b 2 :
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9.1. Divisibility 301

and also with 3 and 9 gallon jugs, and came to different conclusions about bomb
explosions. What’s going on in general? For example, how about getting 4 gallons
from 12- and 18-gallon jugs, getting 32 gallons with 899- and 1147-gallon jugs, or
getting 3 gallons into a jug using just 21- and 26-gallon jugs?
It would be nice if we could solve all these silly water jug questions at once. This
is where number theory comes in handy.

A Water Jug Invariant
Suppose that we have water jugs with capacities a and b with b a. Let’s carry
out some sample operations of the state machine and see what happens, assuming
the b-jug is big enough:

.0; 0/ ! .a; 0/ fill first jug
! .0; a/ pour first into second
! .a; a/ fill first jug
! .2a b; b/ pour first into second (assuming 2a b)
! .2a b; 0/ empty second jug
! .0; 2a b/ pour first into second
! .a; 2a b/ fill first
! .3a 2b; b/ pour first into second (assuming 3a 2b)

What leaps out is that at every step, the amount of water in each jug is a linear
combination of a and b. This is easy to prove by induction on the number of
transitions:
Lemma 9.1.6 (Water Jugs). In the Die Hard state machine of Section 6.2.3 with
jugs of sizes a and b, the amount of water in each jug is always a linear combination
of a and b.
Proof. The induction hypothesis P .n/ is the proposition that after n transitions, the
amount of water in each jug is a linear combination of a and b.
Base case (n D 0): P .0/ is true, because both jugs are initially empty, and 0 a C
0 b D 0.
Inductive step: Suppose the machine is in state .x; y/ after n steps, that is, the little
jug contains x gallons and the big one contains y gallons. There are two cases:

If we fill a jug from the fountain or empty a jug into the fountain, then that jug
is empty or full. The amount in the other jug remains a linear combination
of a and b. So P .n C 1/ holds.
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302 Chapter 9 Number Theory

Otherwise, we pour water from one jug to another until one is empty or the
other is full. By our assumption, the amount x and y in each jug is a linear
combination of a and b before we begin pouring. After pouring, one jug is
either empty (contains 0 gallons) or full (contains a or b gallons). Thus, the
other jug contains either x C y, x C y a or x C y b gallons, all of which
are linear combinations of a and b since x and y are. So P .n C 1/ holds in
this case as well.

Since P .n C 1/ holds in any case, this proves the inductive step, completing the
proof by induction.

So we have established that the jug problem has a preserved invariant, namely,
the amount of water in every jug is a linear combination of the capacities of the
jugs. Lemma 9.1.6 has an important corollary:

Corollary. In trying to get 4 gallons from 12- and 18-gallon jugs, and likewise to
get 32 gallons from 899- and 1147-gallon jugs,

Bruce will die!

Proof. By the Water Jugs Lemma 9.1.6, with 12- and 18-gallon jugs, the amount
in any jug is a linear combination of 12 and 18. This is always a multiple of 6 by
Lemma 9.1.2.2, so Bruce can’t get 4 gallons. Likewise, the amount in any jug using
899- and 1147-gallon jugs is a multiple of 31, so he can’t get 32 either.

But the Water Jugs Lemma doesn’t tell the complete story. For example, it leaves
open the question of getting 3 gallons into a jug using just 21- and 26-gallon jugs:
the only positive factor of both 21 and 26 is 1, and of course 1 divides 3, so the
Lemma neither rules out nor confirms the possibility of getting 3 gallons.
A bigger issue is that we’ve just managed to recast a pretty understandable ques-
tion about water jugs into a technical question about linear combinations. This
might not seem like a lot of progress. Fortunately, linear combinations are closely
related to something more familiar, greatest common divisors, and will help us
solve the general water jug problem.

9.2 The Greatest Common Divisor
A common divisor of a and b is a number that divides them both. The greatest
common divisor of a and b is written gcd.a; b/. For example, gcd.18; 24/ D 6.
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9.2. The Greatest Common Divisor 303

As long as a and b are not both 0, they will have a gcd. The gcd turns out
to be very valuable for reasoning about the relationship between a and b and for
reasoning about integers in general. We’ll be making lots of use of gcd’s in what
follows.
Some immediate consequences of the definition of gcd are that for n > 0,

gcd.n; n/ D n; gcd.n; 1/ D 1; gcd.n; 0/ D n;

where the last equality follows from the fact that everything is a divisor of 0.

9.2.1 Euclid’s Algorithm
The first thing to figure out is how to find gcd’s. A good way called Euclid’s
algorithm has been known for several thousand years. It is based on the following
elementary observation.
Lemma 9.2.1. For b ¤ 0,

gcd.a; b/ D gcd.b; rem.a; b//:

Proof. By the Division Theorem 9.1.4,

a D qb C r (9.2)

where r D rem.a; b/. So a is a linear combination of b and r, which implies that
any divisor of b and r is a divisor of a by Lemma 9.1.2.2. Likewise, r is a linear
combination a qb of a and b, so any divisor of a and b is a divisor of r. This
means that a and b have the same common divisors as b and r, and so they have
the same greatest common divisor.

Lemma 9.2.1 is useful for quickly computing the greatest common divisor of
two numbers. For example, we could compute the greatest common divisor of
1147 and 899 by repeatedly applying it:

gcd.1147; 899/ D gcd.899; rem.1147; 899//
„ ƒ‚ …
D248
D gcd .248; rem.899; 248/ D 155/
D gcd .155; rem.248; 155/ D 93/
D gcd .93; rem.155; 93/ D 62/
D gcd .62; rem.93; 62/ D 31/
D gcd .31; rem.62; 31/ D 0/
D 31
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304 Chapter 9 Number Theory

This calculation that gcd.1147; 899/ D 31 was how we figured out that with water
jugs of sizes 1147 and 899, Bruce dies trying to get 32 gallons.
On the other hand, applying Euclid’s algorithm to 26 and 21 gives

gcd.26; 21/ D gcd.21; 5/ D gcd.5; 1/ D 1;

so we can’t use the reasoning above to rule out Bruce getting 3 gallons into the big
jug. As a matter of fact, because the gcd here is 1, Bruce will be able to get any
number of gallons into the big jug up to its capacity. To explain this, we will need
a little more number theory.

Euclid’s Algorithm as a State Machine
Euclid’s algorithm can easily be formalized as a state machine. The set of states is
N2 and there is one transition rule:

.x; y/ ! .y; rem.x; y//; (9.3)

for y > 0. By Lemma 9.2.1, the gcd stays the same from one state to the next. That
means the predicate
gcd.x; y/ D gcd.a; b/
is a preserved invariant on the states .x; y/. This preserved invariant is, of course,
true in the start state .a; b/. So by the Invariant Principle, if y ever becomes 0, the
invariant will be true and so

x D gcd.x; 0/ D gcd.a; b/:

Namely, the value of x will be the desired gcd.
What’s more x and therefore also y, gets to be 0 pretty fast. To see why, note
that starting from .x; y/, two transitions leads to a state whose the first coordinate
is rem.x; y/, which is at most half the size of x.4 Since x starts off equal to a and
gets halved or smaller every two steps, it will reach its minimum value—which is
gcd.a; b/—after at most 2 log a transitions. After that, the algorithm takes at most
one more transition to terminate. In other words, Euclid’s algorithm terminates
after at most 1 C 2 log a transitions.5
4 In other words,
rem.x; y/ x=2 for 0 < y x: (9.4)
This is immediate if y x=2, since the remainder of x divided by y is less than y by definition. On
the other hand, if y > x=2, then rem.x; y/ D x y < x=2.
5 A tighter analysis shows that at most log .a/ transitions are possible where ' is the golden ratio
p '
.1 C 5/=2, see Problem 9.14.
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9.2. The Greatest Common Divisor 305

9.2.2 The Pulverizer
We will get a lot of mileage out of the following key fact:

Theorem 9.2.2. The greatest common divisor of a and b is a linear combination
of a and b. That is,
gcd.a; b/ D sa C t b;
for some integers s and t.6

We already know from Lemma 9.1.2.2 that every linear combination of a and b is
divisible by any common factor of a and b, so it is certainly divisible by the greatest
of these common divisors. Since any constant multiple of a linear combination is
also a linear combination, Theorem 9.2.2 implies that any multiple of the gcd is a
linear combination, giving:

Corollary 9.2.3. An integer is a linear combination of a and b iff it is a multiple of
gcd.a; b/.

We’ll prove Theorem 9.2.2 directly by explaining how to find s and t. This job
is tackled by a mathematical tool that dates back to sixth-century India, where it
was called kuttaka, which means “the Pulverizer.” Today, the Pulverizer is more
commonly known as the “Extended Euclidean Gcd Algorithm,” because it is so
close to Euclid’s algorithm.
For example, following Euclid’s algorithm, we can compute the gcd of 259
and 70 as follows:

gcd.259; 70/ D gcd.70; 49/ since rem.259; 70/ D 49
D gcd.49; 21/ since rem.70; 49/ D 21
D gcd.21; 7/ since rem.49; 21/ D 7
D gcd.7; 0/ since rem.21; 7/ D 0
D 7:

The Pulverizer goes through the same steps, but requires some extra bookkeeping
along the way: as we compute gcd.a; b/, we keep track of how to write each of
the remainders (49, 21, and 7, in the example) as a linear combination of a and b.
This is worthwhile, because our objective is to write the last nonzero remainder,
6 This result is often referred to as Bezout’s lemma, which is a misattribution since it was first

published in the West 150 years earlier by someone else, and was described a thousand years before
that by Indian mathematicians Aryabhata and Bhaskara.
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306 Chapter 9 Number Theory

which is the gcd, as such a linear combination. For our example, here is this extra
bookkeeping:

x y .rem.x; y// D x q y
259 70 49 D a 3 b
70 49 21 D b 1 49
D b 1 .a 3 b/
D 1aC4b
49 21 7 D 49 2 21
D .a 3 b/ 2 . 1 a C 4 b/
D 3 a 11 b
21 7 0

We began by initializing two variables, x D a and y D b. In the first two columns
above, we carried out Euclid’s algorithm. At each step, we computed rem.x; y/
which equals x qcnt.x; y/ y. Then, in this linear combination of x and y, we
replaced x and y by equivalent linear combinations of a and b, which we already
had computed. After simplifying, we were left with a linear combination of a and
b equal to rem.x; y/, as desired. The final solution is boxed.
This should make it pretty clear how and why the Pulverizer works. If you have
doubts, you may work through Problem 9.13, where the Pulverizer is formalized as
a state machine and then verified using an invariant that is an extension of the one
used for Euclid’s algorithm.
Since the Pulverizer requires only a little more computation than Euclid’s algo-
rithm, you can “pulverize” very large numbers very quickly by using this algorithm.
As we will soon see, its speed makes the Pulverizer a very useful tool in the field
of cryptography.
Now we can restate the Water Jugs Lemma 9.1.6 in terms of the greatest common
divisor:
Corollary 9.2.4. Suppose that we have water jugs with capacities a and b. Then
the amount of water in each jug is always a multiple of gcd.a; b/.
For example, there is no way to form 4 gallons using 3- and 6-gallon jugs, be-
cause 4 is not a multiple of gcd.3; 6/ D 3.

9.2.3 One Solution for All Water Jug Problems
Corollary 9.2.3 says that 3 can be written as a linear combination of 21 and 26,
since 3 is a multiple of gcd.21; 26/ D 1. So the Pulverizer will give us integers s
and t such that
3 D s 21 C t 26 (9.5)
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9.2. The Greatest Common Divisor 307

The coefficient s could be either positive or negative. However, we can readily
transform this linear combination into an equivalent linear combination

3 D s 0 21 C t 0 26 (9.6)

where the coefficient s 0 is positive. The trick is to notice that if in equation (9.5) we
increase s by 26 and decrease t by 21, then the value of the expression s 21 C t 26
is unchanged overall. Thus, by repeatedly increasing the value of s (by 26 at a
time) and decreasing the value of t (by 21 at a time), we get a linear combination
s 0 21 C t 0 26 D 3 where the coefficient s 0 is positive. (Of course t 0 must then be
negative; otherwise, this expression would be much greater than 3.)
Now we can form 3 gallons using jugs with capacities 21 and 26: We simply
repeat the following steps s 0 times:

1. Fill the 21-gallon jug.

2. Pour all the water in the 21-gallon jug into the 26-gallon jug. If at any time
the 26-gallon jug becomes full, empty it out, and continue pouring the 21-
gallon jug into the 26-gallon jug.

At the end of this process, we must have emptied the 26-gallon jug exactly t 0
times. Here’s why: we’ve taken s 0 21 gallons of water from the fountain, and
we’ve poured out some multiple of 26 gallons. If we emptied fewer than t 0 times,
then by (9.6), the big jug would be left with at least 3 C 26 gallons, which is more
than it can hold; if we emptied it more times, the big jug would be left containing
at most 3 26 gallons, which is nonsense. But once we have emptied the 26-gallon
jug exactly t 0 times, equation (9.6) implies that there are exactly 3 gallons left.
Remarkably, we don’t even need to know the coefficients s 0 and t 0 in order to
use this strategy! Instead of repeating the outer loop s 0 times, we could just repeat
until we obtain 3 gallons, since that must happen eventually. Of course, we have to
keep track of the amounts in the two jugs so we know when we’re done. Here’s the
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308 Chapter 9 Number Theory

solution using this approach starting with empty jugs, that is, at .0; 0/:
fill 21 pour 21 into 26
! .21; 0/ ! .0; 21/
fill 21 pour 21 to 26 empty 26 pour 21 to 26
! .21; 21/ ! .16; 26/ ! .16; 0/ ! .0; 16/
fill 21 pour 21 to 26 empty 26 pour 21 to 26
! .21; 16/ ! .11; 26/ ! .11; 0/ ! .0; 11/
fill 21 pour 21 to 26 empty 26 pour 21 to 26
! .21; 11/ ! .6; 26/ ! .6; 0/ ! .0; 6/
fill 21 pour 21 to 26 empty 26 pour 21 to 26
! .21; 6/ ! .1; 26/ ! .1; 0/ ! .0; 1/
fill 21 pour 21 to 26
! .21; 1/ ! .0; 22/
fill 21 pour 21 to 26 empty 26 pour 21 to 26
! .21; 22/ ! .17; 26/ ! .17; 0/ ! .0; 17/
fill 21 pour 21 to 26 empty 26 pour 21 to 26
! .21; 17/ ! .12; 26/ ! .12; 0/ ! .0; 12/
fill 21 pour 21 to 26 empty 26 pour 21 to 26
! .21; 12/ ! .7; 26/ ! .7; 0/ ! .0; 7/
fill 21 pour 21 to 26 empty 26 pour 21 to 26
! .21; 7/ ! .2; 26/ ! .2; 0/ ! .0; 2/
fill 21 pour 21 to 26
! .21; 2/ ! .0; 23/
fill 21 pour 21 to 26 empty 26 pour 21 to 26
! .21; 23/ ! .18; 26/ ! .18; 0/ ! .0; 18/
fill 21 pour 21 to 26 empty 26 pour 21 to 26
! .21; 18/ ! .13; 26/ ! .13; 0/ ! .0; 13/
fill 21 pour 21 to 26 empty 26 pour 21 to 26
! .21; 13/ ! .8; 26/ ! .8; 0/ ! .0; 8/
fill 21 pour 21 to 26 empty 26 pour 21 to 26
! .21; 8/ ! .3; 26/ ! .3; 0/ ! .0; 3/

The same approach works regardless of the jug capacities and even regardless of
the amount we’re trying to produce! Simply repeat these two steps until the desired
amount of water is obtained:

1. Fill the smaller jug.

2. Pour all the water in the smaller jug into the larger jug. If at any time the
larger jug becomes full, empty it out, and continue pouring the smaller jug
into the larger jug.

By the same reasoning as before, this method eventually generates every multiple—
up to the size of the larger jug—of the greatest common divisor of the jug capacities,
all the quantities we can possibly produce. No ingenuity is needed at all!
So now we have the complete water jug story:

Theorem 9.2.5. Suppose that we have water jugs with capacities a and b. For
any c 2 Œ0::a, it is possible to get c gallons in the size a jug iff c is a multiple of
gcd.a; b/.
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9.3. Prime Mysteries 309

9.2.4 Properties of the Greatest Common Divisor
It can help to have some basic gcd facts on hand:

Lemma 9.2.6.

a) gcd.ka; kb/ D k gcd.a; b/ for all k > 0.

b) .d j a AND d j b/ IFF d j gcd.a; b/.

c) If gcd.a; b/ D 1 and gcd.a; c/ D 1, then gcd.a; bc/ D 1.

d) If a j bc and gcd.a; b/ D 1, then a j c.

Showing how all these facts follow from Theorem 9.2.2 that gcd is a linear com-
bination is a good exercise (Problem 9.11).
These properties are also simple consequences of the fact that integers factor
into primes in a unique way (Theorem 9.4.1). But we’ll need some of these facts
to prove unique factorization in Section 9.4, so proving them by appeal to unique
factorization would be circular.

9.3 Prime Mysteries
Some of the greatest mysteries and insights in number theory concern properties of
prime numbers:

Definition 9.3.1. A prime is a number greater than 1 that is divisible only by itself
and 1. A number other than 0, 1, and 1 that is not a prime is called composite.7

Here are three famous mysteries:

Twin Prime Conjecture There are infinitely many primes p such that p C 2 is also
a prime.
In 1966, Chen showed that there are infinitely many primes p such that p C2
is the product of at most two primes. So the conjecture is known to be almost
true!

Conjectured Inefficiency of Factoring Given the product of two large primes n D
pq, there is no efficient procedure to recover the primes p and q. That is,
no polynomial time procedure (see Section 3.5) is guaranteed to find p and
7 So 0, 1, and 1 are the only integers that are neither prime nor composite.
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310 Chapter 9 Number Theory

q in a number of steps bounded by a polynomial in the length of the binary
representation of n (not n itself). The length of the binary representation at
most 1 C log2 n.
The best algorithm known is the “number field sieve,” which runs in time
proportional to:
1=3 2=3
e 1:9.ln n/ .ln ln n/ :
This number grows more rapidly than any polynomial in log n and is infea-
sible when n has 300 digits or more.
Efficient factoring is a mystery of particular importance in computer science,
as we’ll explain later in this chapter.

Goldbach’s Conjecture We’ve already mentioned Goldbach’s Conjecture 1.1.6 sev-
eral times: every even integer greater than two is equal to the sum of two
primes. For example, 4 D 2 C 2, 6 D 3 C 3, 8 D 3 C 5, etc.
In 1939, Schnirelman proved that every even number can be written as the
sum of not more than 300,000 primes, which was a start. Today, we know
that every even number is the sum of at most 6 primes.

Primes show up erratically in the sequence of integers. In fact, their distribution
seems almost random:

2; 3; 5; 7; 11; 13; 17; 19; 23; 29; 31; 37; 41; 43; : : : :

One of the great insights about primes is that their density among the integers has
a precise limit. Namely, let .n/ denote the number of primes up to n:

Definition 9.3.2.
.n/ WWD jfp 2 Œ2::n j p is primegj:

For example, .1/ D 0; .2/ D 1 and .10/ D 4, because 2, 3, 5, and 7 are the
primes less than or equal to 10. Step by step, grows erratically according to the
erratic spacing between successive primes, but its overall growth rate is known to
smooth out to be the same as the growth of the function n= ln n:

Theorem 9.3.3 (Prime Number Theorem).
.n/
lim D 1:
n!1 n= ln n
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9.4. The Fundamental Theorem of Arithmetic 311

Thus, primes gradually taper off. As a rule of thumb, about 1 integer out of every
ln n in the vicinity of n is a prime.
The Prime Number Theorem was conjectured by Legendre in 1798 and proved
a century later by de la Vallée Poussin and Hadamard in 1896. However, after his
death, a notebook of Gauss was found to contain the same conjecture, which he
apparently made in 1791 at age 15. (You have to feel sorry for all the otherwise
“great” mathematicians who had the misfortune of being contemporaries of Gauss.)
A proof of the Prime Number Theorem is beyond the scope of this text, but there
is a manageable proof (see Problem 9.22) of a related result that is sufficient for our
applications:

Theorem 9.3.4 (Chebyshev’s Theorem on Prime Density). For n > 1,
n
.n/ > :
3 ln n

9.4 The Fundamental Theorem of Arithmetic
There is an important fact about primes that you probably already know: every
positive integer number has a unique prime factorization. So every positive integer
can be built up from primes in exactly one way. These quirky prime numbers are
the building blocks for the integers.
Since the value of a product of numbers is the same if the numbers appear in a
different order, there usually isn’t a unique way to express a number as a product
of primes. For example, there are three ways to write 12 as a product of primes:

12 D 2 2 3 D 2 3 2 D 3 2 2:

What’s unique about the prime factorization of 12 is that any product of primes
equal to 12 will have exactly one 3 and two 2’s. This means that if we sort the
primes by size, then the product really will be unique.
Let’s state this more carefully. A sequence of numbers is weakly decreasing
when each number in the sequence is at least as big as the numbers after it. Note
that a sequence of just one number as well as a sequence of no numbers—the empty
sequence—is weakly decreasing by this definition.

Theorem 9.4.1. [Fundamental Theorem of Arithmetic] Every positive integer is a
product of a unique weakly decreasing sequence of primes.
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312 Chapter 9 Number Theory

A Prime for Google
In late 2004 a billboard appeared in various locations around the country:

first 10-digit prime found
. com
in consecutive digits of e
Substituting the correct number for the expression in curly-braces produced the
URL for a Google employment page. The idea was that Google was interested in
hiring the sort of people that could and would solve such a problem.
How hard is this problem? Would you have to look through thousands or millions
or billions of digits of e to find a 10-digit prime? The rule of thumb derived from
the Prime Number Theorem says that among 10-digit numbers, about 1 in

ln 1010 23

is prime. This suggests that the problem isn’t really so hard! Sure enough, the
first 10-digit prime in consecutive digits of e appears quite early:

e D2:718281828459045235360287471352662497757247093699959574966
9676277240766303535475945713821785251664274274663919320030
599218174135966290435729003342952605956307381323286279434 : : :
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9.4. The Fundamental Theorem of Arithmetic 313

For example, 75237393 is the product of the weakly decreasing sequence of
primes
23; 17; 17; 11; 7; 7; 7; 3;
and no other weakly decreasing sequence of primes will give 75237393.8
Notice that the theorem would be false if 1 were considered a prime; for example,
15 could be written as 5 3, or 5 3 1, or 5 3 1 1, . . . .
There is a certain wonder in unique factorization, especially in view of the prime
number mysteries we’ve already mentioned. It’s a mistake to take it for granted,
even if you’ve known it since you were in a crib. In fact, unique factorization
actually fails for many
p integer-like sets of numbers, such as the complex numbers
of the form n C m 5 for m; n 2 Z (see Problem 9.25).
The Fundamental Theorem is also called the Unique Factorization Theorem,
which is a more descriptive and less pretentious, name—but we really want to get
your attention to the importance and non-obviousness of unique factorization.

9.4.1 Proving Unique Factorization
The Fundamental Theorem is not hard to prove, but we’ll need a couple of prelim-
inary facts.

Lemma 9.4.2. If p is a prime and p j ab, then p j a or p j b.

Lemma 9.4.2 follows immediately from Unique Factorization: the primes in the
product ab are exactly the primes from a and from b. But proving the lemma this
way would be cheating: we’re going to need this lemma to prove Unique Factoriza-
tion, so it would be circular to assume it. Instead, we’ll use the properties of gcd’s
and linear combinations to give an easy, noncircular way to prove Lemma 9.4.2.

Proof. One case is if gcd.a; p/ D p. Then the claim holds, because a is a multiple
of p.
Otherwise, gcd.a; p/ ¤ p. In this case gcd.a; p/ must be 1, since 1 and p are
the only positive divisors of p. Now gcd.a; p/ is a linear combination of a and p,
so we have 1 D sa C tp for some s; t. Then b D s.ab/ C .t b/p, that is, b is a
linear combination of ab and p. Since p divides both ab and p, it also divides their
linear combination b.

A routine induction argument extends this statement to:
8 The “product” of just one number is defined to be that number, and the product of no numbers is
by convention defined to be 1. So each prime p is uniquely the product of the primes in the length-
one sequence consisting solely of p, and 1, which you will remember is not a prime, is uniquely the
product of the empty sequence.
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314 Chapter 9 Number Theory

Lemma 9.4.3. Let p be a prime. If p j a1 a2 an , then p divides some ai .
Now we’re ready to prove the Fundamental Theorem of Arithmetic.

Proof. Theorem 2.3.1 showed, using the Well Ordering Principle, that every posi-
tive integer can be expressed as a product of primes. So we just have to prove this
expression is unique. We will use Well Ordering to prove this too.
The proof is by contradiction: assume, contrary to the claim, that there exist
positive integers that can be written as products of primes in more than one way.
By the Well Ordering Principle, there is a smallest integer with this property. Call
this integer n, and let

n D p1 p2 pj ;
D q1 q2 qk ;

where both products are in weakly decreasing order and p1 q1 .
If q1 D p1 , then n=q1 would also be the product of different weakly decreasing
sequences of primes, namely,

p2 pj ;
q2 qk :

Since n=q1 < n, this can’t be true, so we conclude that p1 < q1 .
Since the pi ’s are weakly decreasing, all the pi ’s are less than q1 . But

q1 j n D p1 p2 pj ;

so Lemma 9.4.3 implies that q1 divides one of the pi ’s, which contradicts the fact
that q1 is bigger than all them.

9.5 Alan Turing
The man pictured in Figure 9.1 is Alan Turing, the most important figure in the
history of computer science. For decades, his fascinating life story was shrouded
by government secrecy, societal taboo, and even his own deceptions.
At age 24, Turing wrote a paper entitled On Computable Numbers, with an Ap-
plication to the Entscheidungsproblem. The crux of the paper was an elegant way
to model a computer in mathematical terms. This was a breakthrough, because it
allowed the tools of mathematics to be brought to bear on questions of computation.
For example, with his model in hand, Turing immediately proved that there exist
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9.5. Alan Turing 315

Figure 9.1 Alan Turing

problems that no computer can solve—no matter how ingenious the programmer.
Turing’s paper is all the more remarkable because he wrote it in 1936, a full decade
before any electronic computer actually existed.
The word “Entscheidungsproblem” in the title refers to one of the 28 mathemat-
ical problems posed by David Hilbert in 1900 as challenges to mathematicians of
the 20th century. Turing knocked that one off in the same paper. And perhaps
you’ve heard of the “Church-Turing thesis”? Same paper. So Turing was a brilliant
guy who generated lots of amazing ideas. But this lecture is about one of Turing’s
less-amazing ideas. It involved codes. It involved number theory. And it was sort
of stupid.
Let’s look back to the fall of 1937. Nazi Germany was rearming under Adolf
Hitler, world-shattering war looked imminent, and—like us —Alan Turing was
pondering the usefulness of number theory. He foresaw that preserving military
secrets would be vital in the coming conflict and proposed a way to encrypt com-
munications using number theory. This is an idea that has ricocheted up to our own
time. Today, number theory is the basis for numerous public-key cryptosystems,
digital signature schemes, cryptographic hash functions, and electronic payment
systems. Furthermore, military funding agencies are among the biggest investors
in cryptographic research. Sorry, Hardy!
Soon after devising his code, Turing disappeared from public view, and half a
century would pass before the world learned the full story of where he’d gone and
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316 Chapter 9 Number Theory

what he did there. We’ll come back to Turing’s life in a little while; for now, let’s
investigate the code Turing left behind. The details are uncertain, since he never
formally published the idea, so we’ll consider a couple of possibilities.

9.5.1 Turing’s Code (Version 1.0)
The first challenge is to translate a text message into an integer so we can perform
mathematical operations on it. This step is not intended to make a message harder
to read, so the details are not too important. Here is one approach: replace each
letter of the message with two digits (A D 01, B D 02, C D 03, etc.) and string all
the digits together to form one huge number. For example, the message “victory”
could be translated this way:
v i c t o r y
! 22 09 03 20 15 18 25
Turing’s code requires the message to be a prime number, so we may need to pad
the result with some more digits to make a prime. The Prime Number Theorem
indicates that padding with relatively few digits will work. In this case, appending
the digits 13 gives the number 2209032015182513, which is prime.
Here is how the encryption process works. In the description below, m is the
unencoded message (which we want to keep secret), m b is the encrypted message
(which the Nazis may intercept), and k is the key.

Beforehand The sender and receiver agree on a secret key, which is a large prime k.

Encryption The sender encrypts the message m by computing:

b Dmk
m

Decryption The receiver decrypts m
b by computing:
m
b
D m:
k

For example, suppose that the secret key is the prime number k D 22801763489
and the message m is “victory.” Then the encrypted message is:

b Dmk
m
D 2209032015182513 22801763489
D 50369825549820718594667857

There are a couple of basic questions to ask about Turing’s code.
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9.5. Alan Turing 317

1. How can the sender and receiver ensure that m and k are prime numbers, as
required?
The general problem of determining whether a large number is prime or com-
posite has been studied for centuries, and tests for primes that worked well
in practice were known even in Turing’s time. In the past few decades, very
fast primality tests have been found as described in the text box below.

Primality Testing
It’s easy ˘ see that an integer n is prime iff it is not divisible by any number from
p to
2 to n (see Problem 1.14). Of course this naive way to test if n is prime takes
p
more than n steps, which is exponential in the size of n measured by the number
of digits in the decimal or binary representation of n. Through the early 1970’s,
no prime testing procedure was known that would never blow up like this.
In 1974, Volker Strassen invented a simple, fast probabilistic primality test.
Strassens’s test gives the right answer when applied to any prime number, but
has some probability of giving a wrong answer on a nonprime number. However,
the probability of a wrong answer on any given number is so tiny that relying on
the answer is the best bet you’ll ever make.
Still, the theoretical possibility of a wrong answer was intellectually
bothersome—even if the probability of being wrong was a lot less than the prob-
ability of an undetectable computer hardware error leading to a wrong answer.
Finally in 2002, in a breakthrough paper beginning with a quote from Gauss em-
phasizing the importance and antiquity of primality testing, Manindra Agrawal,
Neeraj Kayal, and Nitin Saxena presented an amazing, thirteen line description of
a polynomial time primality test.
This definitively places primality testing way below the exponential effort ap-
parently needed for SAT and similar problems. The polynomial bound on the
Agrawal et al. test had degree 12, and subsequent research has reduced the de-
gree to 5, but this is still too large to be practical, and probabilistic primality tests
remain the method used in practice today. It’s plausible that the degree bound can
be reduced a bit more, but matching the speed of the known probabilistic tests
remains a daunting challenge.

2. Is Turing’s code secure?
The Nazis see only the encrypted message m b D m k, so recovering the
original message m requires factoring mb. Despite immense efforts, no really
efficient factoring algorithm has ever been found. It appears to be a funda-
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318 Chapter 9 Number Theory

mentally difficult problem. So, although a breakthrough someday can’t be
ruled out, the conjecture that there is no efficient way to factor is widely
accepted. In effect, Turing’s code puts to practical use his discovery that
there are limits to the power of computation. Thus, provided m and k are
sufficiently large, the Nazis seem to be out of luck!
This all sounds promising, but there is a major flaw in Turing’s code.

9.5.2 Breaking Turing’s Code (Version 1.0)
Let’s consider what happens when the sender transmits a second message using
Turing’s code and the same key. This gives the Nazis two encrypted messages to
look at:
c1 D m1 k
m and mc2 D m2 k
The greatest common divisor of the two encrypted messages, m c1 and m c2 , is the
secret key k. And, as we’ve seen, the gcd of two numbers can be computed very
efficiently. So after the second message is sent, the Nazis can recover the secret key
and read every message!
A mathematician as brilliant as Turing is not likely to have overlooked such a
glaring problem, and we can guess that he had a slightly different system in mind,
one based on modular arithmetic.

9.6 Modular Arithmetic
On the first page of his masterpiece on number theory, Disquisitiones Arithmeticae,
Gauss introduced the notion of “congruence.” Now, Gauss is another guy who
managed to cough up a half-decent idea every now and then, so let’s take a look
at this one. Gauss said that a is congruent to b modulo n iff n j .a b/. This is
written
a b .mod n/:
For example:
29 15 .mod 7/ because 7 j .29 15/:
It’s not useful to allow a modulus n 1, and so we will assume from now on
that moduli are greater than 1.
There is a close connection between congruences and remainders:
Lemma 9.6.1 (Remainder).
ab .mod n/ iff rem.a; n/ D rem.b; n/:
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9.6. Modular Arithmetic 319

Proof. By the Division Theorem 9.1.4, there exist unique pairs of integers q1 ; r1
and q2 ; r2 such that:

a D q1 n C r1
b D q2 n C r2 ;

where r1 ; r2 2 Œ0::n/. Subtracting the second equation from the first gives:

a b D .q1 q2 /n C .r1 r2 /;

where r1 r2 is in the interval . n; n/. Now a b .mod n/ if and only if n
divides the left-hand side of this equation. This is true if and only if n divides
the right-hand side, which holds if and only if r1 r2 is a multiple of n. But the
only multiple of n in . n; n/ is 0, so r1 r2 must in fact equal 0, that is, when
r1 WWD rem.a; n/ D r2 WWD rem.b; n/.

So we can also see that

29 15 .mod 7/ because rem.29; 7/ D 1 D rem.15; 7/:

Notice that even though “(mod 7)” appears on the end, the symbol isn’t any more
strongly associated with the 15 than with the 29. It would probably be clearer to
write 29 mod 7 15, for example, but the notation with the modulus at the end is
firmly entrenched, and we’ll just live with it.
The Remainder Lemma 9.6.1 explains why the congruence relation has proper-
ties like an equality relation. In particular, the following properties9 follow imme-
diately:

Lemma 9.6.2.

aa .mod n/ (reflexivity)
a b IFF b a .mod n/ (symmetry)
.a b AND b c/ IMPLIES a c .mod n/ (transitivity)

We’ll make frequent use of another immediate corollary of the Remainder Lemma 9.6.1:

Corollary 9.6.3.
a rem.a; n/ .mod n/
9 Binary relations with these properties are called equivalence relations, see Section 10.10.
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320 Chapter 9 Number Theory

Still another way to think about congruence modulo n is that it defines a partition
of the integers into n sets so that congruent numbers are all in the same set. For
example, suppose that we’re working modulo 3. Then we can partition the integers
into 3 sets as follows:
f :::; 6; 3; 0; 3; 6; 9; : : : g
f :::; 5; 2; 1; 4; 7; 10; : : : g
f :::; 4; 1; 2; 5; 8; 11; : : : g
according to whether their remainders on division by 3 are 0, 1, or 2. The upshot
is that when arithmetic is done modulo n, there are really only n different kinds
of numbers to worry about, because there are only n possible remainders. In this
sense, modular arithmetic is a simplification of ordinary arithmetic.
The next most useful fact about congruences is that they are preserved by addi-
tion and multiplication:
Lemma 9.6.4 (Congruence). If a b .mod n/ and c d .mod n/, then
aCc bCd .mod n/; (9.7)
ac bd .mod n/: (9.8)
Proof. Let’s start with 9.7. Since a b .mod n/, we have by definition that
n j .b a/ D .b C c/ .a C c/, so
aCc bCc .mod n/:
Since c d .mod n/, the same reasoning leads to
bCc bCd .mod n/:
Now transitivity (Lemma 9.6.2) gives
aCc bCd .mod n/:
The proof for 9.8 is virtually identical, using the fact that if n divides .b a/,
then it certainly also divides .bc ac/.

9.7 Remainder Arithmetic
The Congruence Lemma 9.6.1 says that two numbers are congruent iff their remain-
ders are equal, so we can understand congruences by working out arithmetic with
remainders. And if all we want is the remainder modulo n of a series of additions,
multiplications, subtractions applied to some numbers, we can take remainders at
every step so that the entire computation only involves number in the range Œ0::n/.
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9.7. Remainder Arithmetic 321

General Principle of Remainder Arithmetic
To find the remainder on division by n of the result of a series of additions and
multiplications, applied to some integers

replace each integer operand by its remainder on division by n,

keep each result of an addition or multiplication in the range Œ0::n/ by im-
mediately replacing any result outside that range by its remainder on divi-
sion by n.

For example, suppose we want to find

rem..444273456789 C 155558585555 /4036666666 ; 36/: (9.9)

This looks really daunting if you think about computing these large powers and
then taking remainders. For example, the decimal representation of 444273456789
has about 20 million digits, so we certainly don’t want to go that route. But re-
membering that integer exponents specify a series of multiplications, we follow the
General Principle and replace the numbers being multiplied by their remainders.
Since rem.44427; 36/ D 3; rem.15555858; 36/ D 6, and rem.403; 36/ D 7, we
find that (9.9) equals the remainder on division by 36 of

.33456789 C 65555 /76666666 : (9.10)

That’s a little better, but 33456789 has about a million digits in its decimal represen-
tation, so we still don’t want to compute that. But let’s look at the remainders of
the first few powers of 3:

rem.3; 36/ D 3
rem.32 ; 36/ D 9
rem.33 ; 36/ D 27
rem.34 ; 36/ D 9:

We got a repeat of the second step, rem.32 ; 36/ after just two more steps. This
means means that starting at 32 , the sequence of remainders of successive powers
of 3 will keep repeating every 2 steps. So a product of an odd number of at least
three 3’s will have the same remainder on division by 36 as a product of just three
3’s. Therefore,
rem.33456789 ; 36/ D rem.33 ; 36/ D 27:
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322 Chapter 9 Number Theory

What a win!
Powers of 6 are even easier because rem.62 ; 36/ D 0, so 0’s keep repeating after
the second step. Powers of 7 repeat after six steps, but on the fifth step you get a 1,
that is rem.76 ; 36/ D 1, so (9.10) successively simplifies to be the remainders of
the following terms:

.33456789 C 65555 /76666666
.33 C 62 65553 /.76 /1111111
.33 C 0 65553 /11111111
D 27:

Notice that it would be a disastrous blunder to replace an exponent by its re-
mainder. The general principle applies to numbers that are operands of plus and
times, whereas the exponent is a number that controls how many multiplications to
perform. Watch out for this.

9.7.1 The ring Zn
It’s time to be more precise about the general principle and why it works. To begin,
let’s introduce the notation Cn for doing an addition and then immediately taking
a remainder on division by n, as specified by the general principle; likewise for
multiplying:

i Cn j WWD rem.i C j; n/;
i n j WWD rem.ij; n/:

Now the General Principle is simply the repeated application of the following
lemma.

Lemma 9.7.1.

rem.i C j; n/ D rem.i; n/ Cn rem.j; n/; (9.11)
rem.ij; n/ D rem.i; n/ n rem.j; n/: (9.12)

Proof. By Corollary 9.6.3, i rem.i; n/ and j rem.j; n/, so by the Congru-
ence Lemma 9.6.4

i C j rem.i; n/ C rem.j; n/ .mod n/:

By Corollary 9.6.3 again, the remainders on each side of this congruence are equal,
which immediately gives (9.11). An identical proof applies to (9.12).
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9.8. Turing’s Code (Version 2.0) 323

The set of integers in the range Œ0::n/ together with the operations Cn and n is
referred to as Zn , the ring of integers modulo n. As a consequence of Lemma 9.7.1,
the familiar rules of arithmetic hold in Zn , for example:
.i n j / n k D i n .j n k/:
These subscript-n’s on arithmetic operations really clog things up, so instead
we’ll just write “(Zn )” on the side to get a simpler looking equation:
.i j / k D i .j k/ .Zn /:
In particular, all of the following equalities10 are true in Zn :
.i j / k D i .j k/ (associativity of );
.i C j / C k D i C .j C k/ (associativity of C);
1k Dk (identity for );
0Ck Dk (identity for C);
k C . k/ D 0 (inverse for C);
i Cj Dj Ci (commutativity of C)
i .j C k/ D .i j / C .i k/ (distributivity);
i j Dj i (commutativity of )

Associativity implies the familiar fact that it’s safe to omit the parentheses in
products:
k1 k2 km
comes out the same in Zn no matter how it is parenthesized.
The overall theme is that remainder arithmetic is a lot like ordinary arithmetic.
But there are a couple of exceptions we’re about to examine.

9.8 Turing’s Code (Version 2.0)
In 1940, France had fallen before Hitler’s army, and Britain stood alone against
the Nazis in western Europe. British resistance depended on a steady flow of sup-
10 A set with addition and multiplication operations that satisfy these equalities is known as a

commutative ring. In addition to Zn , the integers, rationals, reals, and polynomials with integer
coefficients are all examples of commutative rings. On the other hand, the set fT; Fg of truth values
with OR for addition and AND for multiplication is not a commutative ring because it fails to satisfy
one of these equalities. The n n matrices of integers are not a commutative ring because they fail
to satisfy another one of these equalities.
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324 Chapter 9 Number Theory

plies brought across the north Atlantic from the United States by convoys of ships.
These convoys were engaged in a cat-and-mouse game with German “U-boats”
—submarines—which prowled the Atlantic, trying to sink supply ships and starve
Britain into submission. The outcome of this struggle pivoted on a balance of in-
formation: could the Germans locate convoys better than the Allies could locate
U-boats, or vice versa?
Germany lost.
A critical reason behind Germany’s loss was not made public until 1974: Ger-
many’s naval code, Enigma, had been broken by the Polish Cipher Bureau,11 and
the secret had been turned over to the British a few weeks before the Nazi invasion
of Poland in 1939. Throughout much of the war, the Allies were able to route con-
voys around German submarines by listening in to German communications. The
British government didn’t explain how Enigma was broken until 1996. When the
story was finally released (by the US), it revealed that Alan Turing had joined the
secret British codebreaking effort at Bletchley Park in 1939, where he became the
lead developer of methods for rapid, bulk decryption of German Enigma messages.
Turing’s Enigma deciphering was an invaluable contribution to the Allied victory
over Hitler.
Governments are always tight-lipped about cryptography, but the half-century
of official silence about Turing’s role in breaking Enigma and saving Britain may
be related to some disturbing events after the war—more on that later. Let’s get
back to number theory and consider an alternative interpretation of Turing’s code.
Perhaps we had the basic idea right (multiply the message by the key), but erred in
using conventional arithmetic instead of modular arithmetic. Maybe this is what
Turing meant:

Beforehand The sender and receiver agree on a large number n, which may be
made public. (This will be the modulus for all our arithmetic.) As in Version
1.0, they also agree that some prime number k < n will be the secret key.

Encryption As in Version 1.0, the message m should be another prime in Œ0::n/.
The sender encrypts the message m to produce mb by computing mk, but this
time modulo n:
b WWD m k .Zn /
m (9.13)

Decryption (Uh-oh.)

The decryption step is a problem. We might hope to decrypt in the same way as
before by dividing the encrypted message m
b by the key k. The difficulty is that m
b
11 See http://en.wikipedia.org/wiki/Polish Cipher Bureau.
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9.9. Multiplicative Inverses and Cancelling 325

is the remainder when mk is divided by n. So dividing mb by k might not even give
us an integer!
This decoding difficulty can be overcome with a better understanding of when it
is ok to divide by k in modular arithmetic.

9.9 Multiplicative Inverses and Cancelling
The multiplicative inverse of a number x is another number x 1 such that
1
x x D 1:

From now on, when we say “inverse,” we mean multiplicative (not relational) in-
verse.
For example, over the rational numbers, 1=3 is, of course, an inverse of 3, since,
1
3 D 1:
3
In fact, with the sole exception of 0, every rational number n=m has an inverse,
namely, m=n. On the other hand, over the integers, only 1 and -1 have inverses.
Over the ring Zn , things get a little more complicated. For example, 2 is a multi-
plicative inverse of 8 in Z15 , since

2 8 D 1 .Z15 /:

On the other hand, 3 does not have a multiplicative inverse in Z15 . We can prove
this by contradiction: suppose there was an inverse j for 3, that is

1 D 3 j .Z15 /:

Then multiplying both sides of this equality by 5 leads directly to the contradiction
5 D 0:

5 D 5 .3 j /
D .5 3/ j
D 0 j D 0 .Z15 /:

So there can’t be any such inverse j .
So some numbers have inverses modulo 15 and others don’t. This may seem a
little unsettling at first, but there’s a simple explanation of what’s going on.
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326 Chapter 9 Number Theory

9.9.1 Relative Primality
Integers that have no prime factor in common are called relatively prime.12 This
is the same as having no common divisor (prime or not) greater than 1. It’s also
equivalent to saying gcd.a; b/ D 1.
For example, 8 and 15 are relatively prime, since gcd.8; 15/ D 1. On the other
hand, 3 and 15 are not relatively prime, since gcd.3; 15/ D 3 ¤ 1. This turns out
to explain why 8 has an inverse over Z15 and 3 does not.

Lemma 9.9.1. If k 2 Œ0::n/ is relatively prime to n, then k has an inverse in Zn .

Proof. If k is relatively prime to n, then gcd.n; k/ D 1 by definition of gcd. This
means we can use the Pulverizer from section 9.2.2 to find a linear combination of
n and k equal to 1:
sn C t k D 1:
So applying the General Principle of Remainder Arithmetic (Lemma 9.7.1), we get

.rem.s; n/ rem.n; n// C .rem.t; n/ rem.k; n// D 1 .Zn /:

But rem.n; n/ D 0, and rem.k; n/ D k since k 2 Œ0::n/, so we get

rem.t; n/ k D 1 .Zn /:

Thus, rem.t; n/ is a multiplicative inverse of k.

By the way, it’s nice to know that when they exist, inverses are unique. That is,

Lemma 9.9.2. If i and j are both inverses of k in Zn , then i D j .

Proof.
i D i 1 D i .k j / D .i k/ j D 1 j D j .Zn /:

So the proof of Lemma 9.9.1 shows that for any k relatively prime to n, the
inverse of k in Zn is simply the remainder of a coefficient we can easily find using
the Pulverizer.
Working with a prime modulus is attractive here because, like the rational and
real numbers, when p is prime, every nonzero number has an inverse in Zp . But
arithmetic modulo a composite is really only a little more painful than working
modulo a prime—though you may think this is like the doctor saying, “This is only
going to hurt a little,” before he jams a big needle in your arm.
12 Other texts call them coprime.
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9.9. Multiplicative Inverses and Cancelling 327

9.9.2 Cancellation
Another sense in which real numbers are nice is that it’s ok to cancel common
factors. In other words, if we know that t r D t s for real numbers r; s; t , then
as long as t ¤ 0, we can cancel the t’s and conclude that r D s. In general,
cancellation is not valid in Zn . For example,
3 10 D 3 5 .Z15 /; (9.14)
but cancelling the 3’s leads to the absurd conclusion that 10 equals 5.
The fact that multiplicative terms cannot be cancelled is the most significant way
in which Zn arithmetic differs from ordinary integer arithmetic.
Definition 9.9.3. A number k is cancellable in Zn iff
ka Dkb implies a D b .Zn /
for all a; b 2 Œ0::n/.
If a number is relatively prime to 15, it can be cancelled by multiplying by its
inverse. So cancelling works for numbers that have inverses:
Lemma 9.9.4. If k has an inverse in Zn , then it is cancellable.
But 3 is not relatively prime to 15, and that’s why it is not cancellable. More
generally, if k is not relatively prime to n, then we can show it isn’t cancellable in
Zn in the same way we showed that 3 is not cancellable in (9.14).
To summarize, we have
Theorem 9.9.5. The following are equivalent for k 2 Œ0::n/:
gcd.k; n/ D 1;
k has an inverse in Zn ;
k is cancellable in Zn :

9.9.3 Decrypting (Version 2.0)
Multiplicative inverses are the key to decryption in Turing’s code. Specifically,
we can recover the original message by multiplying the encoded message by the
Zn -inverse j of the key:
b j D .m k/ j D m .k j / D m 1 D m .Zn /:
m
So all we need to decrypt the message is to find an inverse of the secret key k, which
will be easy using the Pulverizer—providing k has an inverse. But k is positive and
less than the modulus n, so one simple way to ensure that k is relatively prime to
the modulus is to have n be a prime number.
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328 Chapter 9 Number Theory

9.9.4 Breaking Turing’s Code (Version 2.0)
The Germans didn’t bother to encrypt their weather reports with the highly-secure
Enigma system. After all, so what if the Allies learned that there was rain off the
south coast of Iceland? But amazingly, this practice provided the British with a
critical edge in the Atlantic naval battle during 1941.
The problem was that some of those weather reports had originally been trans-
mitted using Enigma from U-boats out in the Atlantic. Thus, the British obtained
both unencrypted reports and the same reports encrypted with Enigma. By com-
paring the two, the British were able to determine which key the Germans were
using that day and could read all other Enigma-encoded traffic. Today, this would
be called a known-plaintext attack.
Let’s see how a known-plaintext attack would work against Turing’s code. Sup-
pose that the Nazis know both the plain text m and its

b D m k .Zn /;
m
and since m is positive and less than the prime n, the Nazis can use the Pulverizer
to find the Zn -inverse j of m. Now

j m
b D j .m k/ D .j m/ k D 1 k D k .Zn /:

So by computing j m b D k .Zn /, the Nazis get the secret key and can then decrypt
any message!
This is a huge vulnerability, so Turing’s hypothetical Version 2.0 code has no
practical value. Fortunately, Turing got better at cryptography after devising this
code; his subsequent deciphering of Enigma messages surely saved thousands of
lives, if not the whole of Britain.

9.9.5 Turing Postscript
A few years after the war, Turing’s home was robbed. Detectives soon determined
that a former homosexual lover of Turing’s had conspired in the robbery. So they
arrested him—that is, they arrested Alan Turing—because at that time in Britain,
homosexuality was a crime punishable by up to two years in prison. Turing was
sentenced to a hormonal “treatment” for his homosexuality: he was given estrogen
injections. He began to develop breasts.
Three years later, Alan Turing, the founder of computer science, was dead. His
mother explained what happened in a biography of her own son. Despite her re-
peated warnings, Turing carried out chemistry experiments in his own home. Ap-
parently, her worst fear was realized: by working with potassium cyanide while
eating an apple, he poisoned himself.
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9.10. Euler’s Theorem 329

However, Turing remained a puzzle to the very end. His mother was a devout
woman who considered suicide a sin. And, other biographers have pointed out,
Turing had previously discussed committing suicide by eating a poisoned apple.
Evidently, Alan Turing, who founded computer science and saved his country, took
his own life in the end, and in just such a way that his mother could believe it was
an accident.
Turing’s last project before he disappeared from public view in 1939 involved the
construction of an elaborate mechanical device to test a mathematical conjecture
called the Riemann Hypothesis. This conjecture first appeared in a sketchy paper by
Bernhard Riemann in 1859 and is now one of the most famous unsolved problems
in mathematics.

9.10 Euler’s Theorem
The RSA cryptosystem examined in the next section, and other current schemes
for encoding secret messages, involve computing remainders of numbers raised to
large powers. A basic fact about remainders of powers follows from a theorem due
to Euler about congruences.

Definition 9.10.1. For n > 0, define13

.n/ WWD the number of integers in Œ0::n/, that are relatively prime to n.

This function is known as Euler’s function.14

¡For example, .7/ D 6 because all 6 positive numbers in Œ0::7/ are relatively
prime to the prime number 7. Only 0 is not relatively prime to 7. Also, .12/ D 4
since 1, 5, 7, and 11 are the only numbers in Œ0::12/ that are relatively prime to 12.
More generally, if p is prime, then .p/ D p 1 since every positive number in
Œ0::p/ is relatively prime to p. When n is composite, however, the function gets
a little complicated. We’ll get back to it in the next section.
Euler’s Theorem is traditionally stated in terms of congruence:

Theorem (Euler’s Theorem). If n and k are relatively prime, then

k .n/ 1 .mod n/: (9.15)
13 Since 0 is not relatively prime to anything, .n/ could equivalently be defined using the interval

.0::n/ instead of Œ0::n/.
14 Some texts call it Euler’s totient function.
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330 Chapter 9 Number Theory

The Riemann Hypothesis
The formula for the sum of an infinite geometric series says:
1
1 C x C x2 C x3 C D :
1 x
Substituting x D 21s , x D 1
3s , xD 1
5s , and so on for each prime number gives a
sequence of equations:
1 1 1 1
1C s
C 2s C 3s C D
2 2 2 1 1=2s
1 1 1 1
1 C s C 2s C 3s C D
3 3 3 1 1=3s
1 1 1 1
1 C s C 2s C 3s C D
5 5 5 1 1=5s
::
:

Multiplying together all the left-hand sides and all the right-hand sides gives:
1
X 1 Y 1
D :
ns 1 1=p s
nD1 p2primes

The sum on the left is obtained by multiplying out all the infinite series and ap-
plying the Fundamental Theorem of Arithmetic. For example, the term 1=300s
in the sum is obtained by multiplying 1=22s from the first equation by 1=3s in
the second and 1=52s in the third. Riemann noted that every prime appears in the
expression on the right. So he proposed to learn about the primes by studying
the equivalent, but simpler expression on the left. In particular, he regarded s as
a complex number and the left side as a function .s/. Riemann found that the
distribution of primes is related to values of s for which .s/ D 0, which led to
his famous conjecture:

Definition 9.9.6. The Riemann Hypothesis: Every nontrivial zero of the zeta
function .s/ lies on the line s D 1=2 C ci in the complex plane.

A proof would immediately imply, among other things, a strong form of the Prime
Number Theorem.
Researchers continue to work intensely to settle this conjecture, as they have for
over a century. It is another of the Millennium Problems whose solver will earn
$1,000,000 from the Clay Institute.
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9.10. Euler’s Theorem 331

Things get simpler when we rephrase Euler’s Theorem in terms of Zn .

Definition 9.10.2. Let Zn be the integers in .0::n/, that are relatively prime to n:15

Zn WWD fk 2 .0::n/ j gcd.k; n/ D 1g: (9.16)

Consequently,
.n/ D ˇZn ˇ :
ˇ ˇ

Theorem 9.10.3 (Euler’s Theorem for Zn ). For all k 2 Zn ,

k .n/ D 1 .Zn /: (9.17)

Theorem 9.10.3 will follow from two very easy lemmas.
Let’s start by observing that Zn is closed under multiplication in Zn :

Lemma 9.10.4. If j; k 2 Zn , then j n k 2 Zn .

There are lots of easy ways to prove this (see Problem 9.67).

Definition 9.10.5. For any element k and subset S of Zn , let

kS WWD fk n s j s 2 Sg:

Lemma 9.10.6. If k 2 Zn and S Zn , then

jkS j D jS j:

Proof. Since k 2 Zn , by Theorem 9.9.5 it is cancellable. Therefore,

Œks D k t .Zn / implies s D t:

So mulitplying by k in Zn maps all the elements of S to distinct elements of kS ,
which implies S and kS are the same size.

Corollary 9.10.7. If k 2 Zn , then

kZn D Zn :

Proof. A product of elements in Zn remains in Zn by Lemma 9.10.4. So if k 2 Zn ,
then kZn Zn . But by Lemma 9.10.6, kZn and Zn are the same size, so they must
be equal.

Now we can complete the proof of Euler’s Theorem 9.10.3 for Zn ):
15 Some other texts use the notation n for Zn .
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332 Chapter 9 Number Theory

Proof. Let
P WWD k1 k2 k.n/ .Zn /
be the product in Zn of all the numbers in Zn . Let

Q WWD .k k1 / .k k2 / .k k.n/ / .Zn /

for some k 2 Zn . Factoring out k’s immediately gives

Q D k .n/ P .Zn /:

But Q is the same as the product of the numbers in kZn , and kZn D Zn , so we
realize that Q is the product of the same numbers as P , just in a different order.
Altogether, we have
P D Q D k .n/ P .Zn /:
Furthermore, P 2 Zn by Lemma 9.10.4, and so it can be cancelled from both sides
of this equality, giving
1 D k .n/ .Zn /:

Euler’s theorem offers another way to find inverses modulo n: if k is relatively
prime to n, then k .n/ 1 is a Zn -inverse of k, and we can compute this power of
k efficiently using fast exponentiation. However, this approach requires computing
.n/. In the next section, we’ll show that computing .n/ is easy if we know the
prime factorization of n. But we know that finding the factors of n is generally hard
to do when n is large, and so the Pulverizer remains the best approach to computing
inverses modulo n.

Fermat’s Little Theorem
For the record, we mention a famous special case of Euler’s Theorem that was
known to Fermat a century earlier.

Corollary 9.10.8 (Fermat’s Little Theorem). Suppose p is a prime and k is not a
multiple of p. Then
k p 1 1 .mod p/:

9.10.1 Computing Euler’s Function
RSA works using arithmetic modulo the product of two large primes, so we begin
with an elementary explanation of how to compute .pq/ for primes p and q:
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9.10. Euler’s Theorem 333

Lemma 9.10.9.
.pq/ D .p 1/.q 1/
for primes p ¤ q.

Proof. Since p and q are prime, any number that is not relatively prime to pq must
be a multiple of p or a multiple of q. Among the pq numbers in Œ0::pq/, there are
precisely q multiples of p and p multiples of q. Since p and q are relatively prime,
the only number in Œ0::pq/ that is a multiple of both p and q is 0. Hence, there are
p C q 1 numbers in Œ0::pq/ that are not relatively prime to n. This means that

.pq/ D pq .p C q 1/
D .p 1/.q 1/;

as claimed.16

The following theorem provides a way to calculate .n/ for arbitrary n.

Theorem 9.10.10.

(a) If p is a prime, then .p k / D p k pk 1 for k 1.

(b) If a and b are relatively prime, then .ab/ D .a/.b/.

Here’s an example of using Theorem 9.10.10 to compute .300/:

.300/ D .22 3 52 /
D .22 / .3/ .52 / (by Theorem 9.10.10.(b))
2 1 1 0 2 1
D .2 2 /.3 3 /.5 5 / (by Theorem 9.10.10.(a))
D 80:

Note that Lemma 9.10.9 also follows as a special case of Theorem 9.10.10.(b),
since we know that .p/ D p 1 for any prime p.
To prove Theorem 9.10.10.(a), notice that every pth number among the p k num-
bers in Œ0::p k / is divisible by p, and only these are divisible by p. So 1=p of these
numbers are divisible by p and the remaining ones are not. That is,

.p k / D p k .1=p/p k D p k pk 1
:

We’ll leave a proof of Theorem 9.10.10.(b) to Problem 9.61.
As a consequence of Theorem 9.10.10, we have
16 This proof previews a kind of counting argument that we will explore more fully in Part III.
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334 Chapter 9 Number Theory

Corollary 9.10.11. For any number n, if p1 , p2 , . . . , pj are the (distinct) prime
factors of n, then

1 1 1
.n/ D n 1 1 1 :
p1 p2 pj
We’ll give another proof of Corollary 9.10.11 based on rules for counting in
Section 15.9.5.

9.11 RSA Public Key Encryption
Turing’s code did not work as he hoped. However, his essential idea—using num-
ber theory as the basis for cryptography—succeeded spectacularly in the decades
after his death.
In 1977, Ronald Rivest, Adi Shamir, and Leonard Adleman at MIT proposed a
highly secure cryptosystem, called RSA, based on number theory. The purpose of
the RSA scheme is to transmit secret messages over public communication chan-
nels. As with Turing’s codes, the messages transmitted are nonnegative integers of
some fixed size.
Moreover, RSA has a major advantage over traditional codes: the sender and
receiver of an encrypted message need not meet beforehand to agree on a secret key.
Rather, the receiver has both a private key, which they guard closely, and a public
key, which they distribute as widely as possible. A sender wishing to transmit a
secret message to the receiver encrypts their message using the receiver’s widely-
distributed public key. The receiver can then decrypt the received message using
their closely held private key. The use of such a public key cryptography system
allows you and Amazon, for example, to engage in a secure transaction without
meeting up beforehand in a dark alley to exchange a key.
Interestingly, RSA does not operate modulo a prime, as Turing’s hypothetical
Version 2.0 may have, but rather modulo the product of two large primes—typically
primes that are hundreds of digits long. Also, instead of encrypting by multiplica-
tion with a secret key, RSA exponentiates to a secret power—which is why Euler’s
Theorem is central to understanding RSA.
The scheme for RSA public key encryption appears in the box.
If the message m is relatively prime to n, then a simple application of Euler’s
Theorem implies that this way of decoding the encrypted message indeed repro-
duces the original unencrypted message. In fact, the decoding always works—even
in (the highly unlikely) case that m is not relatively prime to n. The details are
worked out in Problem 9.81.
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9.11. RSA Public Key Encryption 335

The RSA Cryptosystem
A Receiver who wants to be able to receive secret numerical messages creates a
private key, which they keep secret, and a public key, which they make publicly
available. Anyone with the public key can then be a Sender who can publicly
send secret messages to the Receiver—even if they have never communicated or
shared any information besides the public key.
Here is how they do it:

Beforehand The Receiver creates a public key and a private key as follows.

1. Generate two distinct primes, p and q. These are used to generate the
private key, and they must be kept hidden. (In current practice, p and
q are chosen to be hundreds of digits long.)
2. Let n WWD pq.
3. Select an integer e 2 Œ0::n/ such that gcd.e; .p 1/.q 1// D 1.
The public key is the pair .e; n/. This should be distributed widely.
4. Let the private key d 2 Œ0::n/ be the inverse of e in the ring
Z.p 1/.q 1/ . This private key can be found using the Pulverizer. The
private key d should be kept hidden!

Encoding To transmit a message m 2 Œ0::n/ to Receiver, a Sender uses the
public key to encrypt m into a numerical message

b WWD me .Zn /:
m

The Sender can then publicly transmit m
b to the Receiver.

Decoding The Receiver decrypts message m
b back to message m using the pri-
vate key:
mDm bd .Zn /:
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336 Chapter 9 Number Theory

Why is RSA thought to be secure? It would be easy to figure out the private
key d if you knew p and q—you could do it the same way the Receiver does using
the Pulverizer. But assuming the conjecture that it is hopelessly hard to factor a
number that is the product of two primes with hundreds of digits, an effort to factor
n is not going to break RSA.
Could there be another approach to reverse engineer the private key d from the
public key that did not involve factoring n? Not really. It turns out that given just
the private and the public keys, it is easy to factor n17 (a proof of this is sketched
in Problem 9.83). So if we are confident that factoring is hopelessly hard, then we
can be equally confident that finding the private key just from the public key will
be hopeless.
But even if we are confident that an RSA private key won’t be found, this doesn’t
rule out the possibility of decoding RSA messages in a way that sidesteps the pri-
vate key. It is an important unproven conjecture in cryptography that any way of
cracking RSA—not just by finding the secret key—would imply the ability to fac-
tor. This would be a much stronger theoretical assurance of RSA security than is
presently known.
But the real reason for confidence is that RSA has withstood all attacks by the
world’s most sophisticated cryptographers for nearly 40 years. Despite decades of
these attacks, no significant weakness has been found. That’s why the mathemat-
ical, financial, and intelligence communities are betting the family jewels on the
security of RSA encryption.
You can hope that with more studying of number theory, you will be the first to
figure out how to do factoring quickly and, among other things, break RSA. But
be further warned that even Gauss worked on factoring for years without a lot to
show for his efforts—and if you do figure it out, you might wind up meeting some
humorless fellows working for a Federal agency in charge of security. . . .

9.12 What has SAT got to do with it?
So why does society, or at least everybody’s secret codes, fall apart if there is an
efficient test for satisfiability (SAT), as we claimed in Section 3.5? To explain this,
remember that RSA can be managed computationally because multiplication of two
primes is fast, but factoring a product of two primes seems to be overwhelmingly
demanding.
17 In practice, for this reason, the public and private keys should be randomly chosen so that neither

is “too small.”
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9.13. References 337

Let’s begin with the observation from Section 3.2 that a digital circuit can be
described by a bunch of propositional formulas of about the same total size as the
circuit. So testing circuits for satisfiability is equivalent to the SAT problem for
propositional formulas (see Problem 3.22).
Now designing digital multiplication circuits is completely routine. We can eas-
ily build a digital “product checker” circuit out of AND, OR, and NOT gates with 1
output wire and 4n digital input wires. The first n inputs are for the binary repre-
sentation of an integer i , the next n inputs for the binary representation of an integer
j , and the remaining 2n inputs for the binary representation of an integer k. The
output of the circuit is 1 iff ij D k and i; j > 1. A straightforward design for such
a product checker uses proportional to n2 gates.
Now here’s how to factor any number m with a length 2n binary representation
using a SAT solver. First, fix the last 2n digital inputs—the ones for the binary
representation of k—so that k equals m.
Next, set the first of the n digital inputs for the representation of i to be 1. Do a
SAT test to see if there is a satisfying assignment of values for the remaining 2n 1
inputs used for the i and j representations. That is, see if the remaining inputs for
i and j can be filled in to cause the circuit to give output 1. If there is such an
assignment, fix the first i -input to be 1, otherwise fix it to be 0. So now we have set
the first i -input equal to the first digit of the binary representations of an i such that
ij D m.
Now do the same thing to fix the second of the n digital inputs for the represen-
tation of i , and then third, proceeding in this way through all the n inputs for the
number i . At this point, we have the complete n-bit binary representation of an
i > 1 such ij D m for some j > 1. In other words, we have found an integer i
that is a factor of m. We can now find j by dividing m by i .
So after n SAT tests, we have factored m. This means that if SAT for digital
circuits with 4n inputs and about n2 gates could be determined by a procedure
taking a number of steps bounded above by a degree d polynomial in n, then 2n
digit numbers can be factored in n times this many steps, that is, with a number of
steps bounded by a polynomial of degree d C 1 in n. So if SAT could be solved in
polynomial time, then so could factoring, and consequently RSA would be “easy”
to break.

9.13 References
[2], [42]
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338 Chapter 9 Number Theory

Problems for Section 9.1
Practice Problems
Problem 9.1.
Prove that a linear combination of linear combinations of integers a0 ; : : : ; an is a
linear combination of a0 ; : : : ; an .

Class Problems
Problem 9.2.
A number is perfect if it is equal to the sum of its positive divisors, other than itself.
For example, 6 is perfect, because 6 D 1 C 2 C 3. Similarly, 28 is perfect, because
28 D 1 C 2 C 4 C 7 C 14. Explain why 2k 1 .2k 1/ is perfect when 2k 1 is
prime.18

Problems for Section 9.2
Practice Problems
Problem 9.3.
Let

x WWD 21212121;
y WWD 12121212:

Use the Euclidean algorithm to find the GCD of x and y. Hint: Looks scary, but
it’s not.

Problem 9.4.
18 Euclid proved this 2300 years ago. About 250 years ago, Euler proved the
converse: every even perfect number is of this form (for a simple proof see
http://primes.utm.edu/notes/proofs/EvenPerfect.html). It is not known
if there are any odd perfect numbers at all. It is also not known if there are an infinite number of even
perfect numbers. One of the charms of number theory is that simple results like those given in this
problem lie at the brink of the unknown.
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9.13. References 339

Let

x WWD 1788 315 372 591000
22 /
y WWD 19.9 3712 533678 5929 :

(a) What is gcd.x; y/?

(b) What is lcm.x; y/? (“lcm” is least common multiple.)

Problem 9.5.
Show that there is an integer x such that

ax b .mod n/

iff
gcd.a; n/ j b:

Problem 9.6.
Prove that
gcd.a5 ; b 5 / D gcd.a; b/5
for all a; b 2 Z.

Class Problems
Problem 9.7.
Use the Euclidean Algorithm to prove that

gcd.13a C 8b; 5a C 3b/ D gcd.a; b/:

Problem 9.8.

(a) Use the Pulverizer to find integers x; y such that

x30 C y22 D gcd.30; 22/:

(b) Now find integers x 0 ; y 0 with 0 y 0 < 30 such that

x 0 30 C y 0 22 D gcd.30; 22/
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340 Chapter 9 Number Theory

Problem 9.9. (a) Use the Pulverizer to find gcd.84; 108/

(b) Find integers x, y with 0 y < 84 such that

x 84 C y 108 D gcd.84; 108/:

Problem 9.10.

Indicate true or false for the following statements about the greatest
common divisor, and provide counterexamples for those that are false.
(a) If gcd.a; b/ ¤ 1 and gcd.b; c/ ¤ 1, then gcd.a; c/ ¤ 1. true false

(b) If a j bc and gcd.a; b/ D 1, then a j c. true false

(d) gcd.ab; ac/ D a gcd.b; c/. true false

(e) gcd.1 C a; 1 C b/ D 1 C gcd.a; b/. true false

(f) If an integer linear combination of a and b equals 1, then so does some integer
linear combination of a and b 2 . true false

(g) If no integer linear combination of a and b equals 2, then neither does any
integer linear combination of a2 and b 2 . true false

Problem 9.11.
For nonzero integers a, b, prove the following properties of divisibility and GCD’S.
You may use Theorem 9.2.2 that gcd.a; b/ is an integer linear combination of a and
b. You may not appeal to uniqueness of prime factorization Theorem 9.4.1, because
some of these properties are needed to prove unique factorization.)
(a) Every common divisor of a and b divides gcd.a; b/.

(b) gcd.ka; kb/ D k gcd.a; b/ for all k > 0.

(d) If p j bc for some prime p then p j b or p j c.
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9.13. References 341

(e) Let m be the smallest integer linear combination of a and b that is positive.
Show that m D gcd.a; b/.

Homework Problems
Problem 9.12.
Here is a game you can analyze with number theory and always beat me. We start
with two distinct, positive integers written on a blackboard. Call them a and b.
Now we take turns. (I’ll let you decide who goes first.) On each turn, the player
must write a new positive integer on the board that is the difference of two numbers
that are already there. If a player cannot play, then they lose.
For example, suppose that 12 and 15 are on the board initially. Your first play
must be 3, which is 15 12. Then I might play 9, which is 12 3. Then you might
play 6, which is 15 9. Then I can’t play, so I lose.
(a) Show that every number on the board at the end of the game is a multiple of
gcd.a; b/.

(b) Show that every positive multiple of gcd.a; b/ up to max.a; b/ is on the board
at the end of the game.

Problem 9.13.
Define the Pulverizer State machine to have:

states WWD N6
start state WWD .a; b; 0; 1; 1; 0/ (where a b > 0)
transitions WWD .x; y; s; t; u; v/ !
.y; rem.x; y/; u sq; v t q; s; t / (for q D qcnt.x; y/; y > 0):

(a) Show that the following properties are preserved invariants of the Pulverizer
machine:

gcd.x; y/ D gcd.a; b/; (Inv1)
sa C t b D y; and (Inv2)
ua C vb D x: (Inv3)

(b) Conclude that the Pulverizer machine is partially correct.
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342 Chapter 9 Number Theory

Problem 9.14.
Prove that the smallest positive integers a b for which, starting in state .a; b/,
the Euclidean state machine will make n transitions are F .n C 1/ and F .n/, where
F .n/ is the nth Fibonacci number.
Hint: Induction.
a later chapter, we’ll show that F .n/ ' n where ' is the golden ratio
In p
.1 C 5/=2. This implies that the Euclidean algorithm halts after at most log' .a/
transitions. This is a somewhat smaller than the 2 log2 a bound derived from equa-
tion (9.4).

Problem 9.15.
Let’s extend the jug filling scenario of Section 9.1.3 to three jugs and a receptacle.
Suppose the jugs can hold a, b and c gallons of water, respectively.
The receptacle can be used to store an unlimited amount of water, but has no
measurement markings. Excess water can be dumped into the drain. Among the
possible moves are:

1. fill a bucket from the hose,

2. pour from the receptacle to a bucket until the bucket is full or the receptacle
is empty, whichever happens first,

3. empty a bucket to the drain,

4. empty a bucket to the receptacle, and

5. pour from one bucket to another until either the first is empty or the second
is full.

(a) Model this scenario with a state machine. (What are the states? How does a
state change in response to a move?)

(b) Prove that Bruce can get k 2 N gallons of water into the receptacle using the
above operations if gcd.a; b; c/ j k.
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9.13. References 343

Problem 9.16.
The Binary GCD state machine computes the GCD of integers a; b > 0 using only
division by 2 and subtraction, which makes it run very efficiently on hardware that
uses binary representation of numbers. In practice, it runs more quickly than the
more famous Euclidean algorithm described in Section 9.2.1.

statesWWDN3
start stateWWD.a; b; 1/
transitionsWWD if min.x; y/ > 0; then .x; y; e/ !
.x=2; y=2; 2e/ (if 2 j x and 2 j y) (i1)
.x=2; y; e/ (else if 2 j x) (i2)
.x; y=2; e/ (else if 2 j y) (i3)
.x y; y; e/ (else if x > y) (i4)
.y x; x; e/ (else if y > x) (i5)
.1; 0; ex/ (otherwise (x D y)): (i6)

(a) Use the Invariant Principle to prove that if this machine stops, that is, reaches
a state .x; y; e/ in which no transition is possible, then e D gcd.a; b/.

(b) Prove that rule (i1)

.x; y; e/ ! .x=2; y=2; 2e/

is never executed after any of the other rules is executed.

(c) Prove that the machine reaches a final state in at most 1 C 3.log a C log b/
transitions. (This is a coarse bound; you may be able to get a better one.)

Problem 9.17.
Extend the binary gcd procedure of Problem 9.16 to obtain a new pulverizer that
uses only division by 2 and subtraction.
Hint: After the binary gcd procedure has factored out 2’s, it starts computing the
gcd.a; b/ for numbers a; b at least one of which is odd. It does this by successively
updating a pair of numbers .x; y/ such that gcd.x; y/ D gcd.a; b/. Extend the
procedure to find and update coefficients ux ; vx ; uy ; vy such that

ux a C vx b D x and uy a C vy b D y:
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344 Chapter 9 Number Theory

To see how to update the coefficients when at least one of a and b is odd and
ua C vb is even, show that either u and v are both even, or else u b and v C a
are both even.

Problem 9.18.
For any set A of integers,

gcd.A/ WWD the greatest common divisor of the elements of A.

The following useful property of gcd’s of sets is easy to take for granted:

Theorem.
gcd.A [ B/ D gcd.gcd.A/; gcd.B//; (AuB)
for all finite sets A; B Z.

The theorem has an easy proof as a Corollary of the Unique Factorization The-
orem. In this problem we develop a proof by induction of Theorem (AuB) just
making repeated use of Lemma 9.2.6.b :

.d j a AND d j b/ IFF d j gcd.a; b/: (gcddiv)

The key to proving (AuB) will be generalizing (gcddiv) to finite sets.

Definition. For any subset A Z,

d j A WWD 8a 2 A: d j a: (divdef)

Lemma.
d j A IFF d j gcd.A/: (dAdgA)
for all d 2 Z and finite sets A Z.

(a) Prove that

gcd.a; gcd.b; c// D gcd.gcd.a; b/; c/ (gcd-associativity)

for all integers a; b; c.

From here on we write “a [ A” as an abbreviation for “fag [ A.”
(b) Prove that
d j .a [ b [ C / IFF d j .gcd.a; b/ [ C / (abCgcd)
for all a; b; d 2 Z, and C Z.
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9.13. References 345

Proof.

d j .a [ b [ C / IFF .d j a/ AND .d j b/ AND .d j C / (def (divdef) of divides)
IFF .d j gcd.a; b// AND .d j C / by (gcddiv)
IFF d j .gcd.a; b/ [ C / (def (divdef) of divides):

d j .a [ A/ IFF d j gcd.a; gcd.A//; (divauA)

for all integers a; d and finite sets A Z. Explain why this proves Lemma (dAdgA).

(d) Prove Theorem (AuB).

(e) Conclude that gcd.A/ is an integer linear combination of the elements in A.

Exam Problems
Problem 9.19.
Prove that gcd.mb C r; b/ D gcd.b; r/ for all integers m; b; r.

Problem 9.20.
The Stata Center’s delicate balance depends on two buckets of water hidden in a
secret room. The big bucket has a volume of 25 gallons, and the little bucket has a
volume of 10 gallons. If at any time a bucket contains exactly 13 gallons, the Stata
Center will collapse. There is an interactive display where tourists can remotely
fill and empty the buckets according to certain rules. We represent the buckets as a
state machine.
The state of the machine is a pair .b; l/, where b is the volume of water in big
bucket, and l is the volume of water in little bucket.
(a) We informally describe some of the legal operations tourists can perform be-
low. Represent each of the following operations as a transition of the state machine.
The first is done for you as an example.

1. Fill the big bucket.
.b; l/ ! .25; l/:

2. Empty the little bucket.
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346 Chapter 9 Number Theory

3. Pour the big bucket into the little bucket. You should have two cases defined
in terms of the state .b; l/: if all the water from the big bucket fits in the little
bucket, then pour all the water. If it doesn’t, pour until the little jar is full,
leaving some water remaining in the big jar.

(b) Use the Invariant Principle to show that, starting with empty buckets, the Stata
Center will never collapse. That is, the state .13; x/ in unreachable. (In verifying
your claim that the invariant is preserved, you may restrict to the representative
transitions of part (a).)

Problem 9.21.
Let
m D 29 524 74 117 ;
n D 23 722 11211 197 ;
p D 25 34 76042 1930 :
(a) What is the gcd.m; n; p/?

(b) What is the least common multiple lcm.m; n; p/?
Let k .n/ be the largest power of k that divides n, where k > 1. That is,
k .n/ WWD maxfi j k i divides ng:
If A is a nonempty set of nonnegative integers, define
k .A/ WWD fk .a/ j a 2 Ag:
(c) Express k .gcd.A// in terms of k .A/.

(d) Let p be a prime number. Express p .lcm.A// in terms of p .A/.

(e) Give an example of integers a; b where 6 .lcm.a; b// > max.6 .a/; 6 .b//.
Q Q
(f) Let A be the product of all the elements in A. Express p .n/. A/ in terms
of p .A/.

(g) Let B also be a nonempty set of nonnegative integers. Conclude that
gcd.A [ B/ D gcd.gcd.A/; gcd.B//: (9.18)
Hint: Consider p ./ of the left and right-hand sides of (9.18). You may assume
min.A [ B/ D min.min.A/; min.B//: (9.19)
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9.13. References 347

Problems for Section 9.3
Homework Problems
Problem 9.22.
TBA: Chebyshvev lower bound in prime density, based on Shoup pp.75–76

Problems for Section 9.4
Practice Problems
Problem 9.23.
Let p be a prime number and a1 ; : : : ; an integers. Prove the following Lemma by
induction:

Lemma.

If p divides a product a1 a2 an ; then p divides some ai : (*)

You may assume the case for n D 2 which was given by Lemma 9.4.2.
Be sure to clearly state and label your Induction Hypothesis, Base case(s), and
Induction step.

Class Problems
Problem 9.24. (a) Let m D 29 524 117 1712 and n D 23 722 11211 131 179 192 . What
is the gcd.m; n/? What is the least common multiple lcm.m; n/ of m and n? Verify
that
gcd.m; n/ lcm.m; n/ D mn: (9.20)

(b) Describe in general how to find the gcd.m; n/ and lcm.m; n/ from the prime
factorizations of m and n. Conclude that equation (9.20) holds for all positive
integers m; n.

Homework Problems
Problem 9.25. p
The set of complex
p numbers that are equal topm C n 5 for some integers m; n
is called ZŒ 5. It will turn out that in ZŒ 5, not all numbers have unique
factorizations.
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348 Chapter 9 Number Theory

p p p
A sum or product of numbers in ZŒ 5 is in ZŒ 5, and since ZŒ 5 is a
subset of the complex numbers, all the usual rules for addition and multiplication
are true for it. But some weird things do happen. For example, the prime 29 has
factors:
p
(a) Find x; y 2 ZŒ 5 such that xy D 29 and x ¤ ˙1 ¤ y.
p
On the other hand, the p number 3 is still a “prime” even pin ZŒ 5. More pre-
cisely, a number p p 2 ZŒ 5 is called irreducible over ZŒ 5 iff when xy D p
for some x; y 2 ZŒ 5, either x D ˙1 or y D ˙1.
p p p
Claim. The numbers 3; 2 C 5, and 2 5 are irreducible over ZŒ 5.

pIn particular, this Claim implies that the number 9 factors into irreducibles over
ZŒ 5 in two different ways:
p p
3 3 D 9 D .2 C 5/.2 5/:
p
So ZŒ 5 is an example of what is called a non-unique factorization domain.
To verify the Claim, we’ll appeal (without proof) to a familiar technical property
of complex numbers given in the following Lemma.
p
Definition. For p a complex number c D r C si where r; s 2 R and i is 1, the
norm jcj of c is r 2 C s 2 .

Lemma. For c; d 2 C,
jcd j D jcj jd j :
p
(b) Prove that jxj2 ¤ 3 for all x 2 ZŒ 5.
p
(c) Prove that if x 2 ZŒ 5 and jxj D 1, then x D ˙1.
p
(d) Prove that if jxyj D 3 for some x; y 2 ZŒ 5, then x D ˙1 or y D ˙1.
2
p
Hint: jzj 2 N for z 2 ZŒ 5.

(e) Complete the proof of the Claim.

Problems for Section 9.6
Practice Problems
Problem 9.26.
Prove that if a b .mod 14/ and a b .mod 5/, then a b .mod 70/.
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9.13. References 349

Class Problems
Problem 9.27. (a) Prove if n is not divisible by 3, then n2 1 .mod 3/.

(b) Show that if n is odd, then n2 1 .mod 8/.

Problem 9.28.
The values of polynomial p.n/ WWD n2 C n C 41 are prime for all the integers from
0 to 39 (see Section 1.1). Well, p didn’t work, but are there any other polynomials
whose values are always prime? No way! In fact, we’ll prove a much stronger
claim.

Definition. The set P of integer polynomials can be defined recursively:
Base cases:

the identity function IdZ .x/ WWD x is in P .

for any integer m the constant function cm .x/ WWD m is in P .

Constructor cases. If r; s 2 P , then r C s and r s 2 P .

(a) Using the recursive definition of integer polynomials given above, prove by
structural induction that for all q 2 P ,

j k .mod n/ IMPLIES q.j / q.k/ .mod n/;

for all integers j; k; n where n > 1.
Be sure to clearly state and label your Induction Hypothesis, Base case(s), and
Constructor step.

(b) We’ll say that q produces multiples if, for every integer greater than one in the
range of q, there are infinitely many different multiples of that integer in the range.
For example, if q.4/ D 7 and q produces multiples, then there are infinitely many
different multiples of 7 in the range of q, and of course, except for 7 itself, none of
these multiples is prime.
Prove that if q has positive degree and positive leading coefficient, then q produces
multiples. You may assume that every such polynomial is strictly increasing for
large arguments.
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350 Chapter 9 Number Theory

Part (b) implies that an integer polynomial with positive leading coefficient and
degree has infinitely many nonprimes in its range. This fact no longer holds true for
multivariate polynomials. An amazing consequence of Matiyasevich’s [32] solu-
tion to Hilbert’s Tenth Problem is that multivariate polynomials can be understood
as general purpose programs for generating sets of integers. If a set of nonnegative
integers can be generated by any program, then it equals the set of nonnegative
integers in the range of a multivariate integer polynomial! In particular, there is an
integer polynomial p.x1 ; : : : ; x7 / whose nonnegative values as x1 ; : : : ; x7 range
over N are precisely the set of all prime numbers!

Problems for Section 9.7
Practice Problems
Problem 9.29.
List the numbers of all statements below that are equivalent to

ab .mod n/;

where n > 1 and a and b are integers. Briefly explain your reasoning.

i) 2a 2b .mod n/

ii) 2a 2b .mod 2n/

iii) a3 b 3 .mod n/

iv) rem.a; n/ D rem.b; n/

v) rem.n; a/ D rem.n; b/

vi) gcd.a; n/ D gcd.b; n/

vii) gcd.n; a b/ D n

viii) .a b/ is a multiple of n

ix) 9k 2 Z: a D b C nk
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9.13. References 351

Problem 9.30.
What is remainder.3101 ; 21/?

Homework Problems
Problem 9.31.
Prove that congruence is preserved by arithmetic expressions. Namely, prove that
ab .mod n/; (9.21)
then
eval.e; a/ eval.e; b/ .mod n/; (9.22)
for all e 2 Aexp (see Section 7.4).

Problem 9.32.
A commutative ring is a set R of elements along with two binary operations ˚ and
˝ from R R to R. There is an element in R called the zero-element, 0, and
another element called the unit-element, 1. The operations in a commutative ring
satisfy the following ring axioms for r; s; t 2 R:
.r ˝ s/ ˝ t D r ˝ .s ˝ t / (associativity of ˝);
.r ˚ s/ ˚ t D r ˚ .s ˚ t / (associativity of ˚);
r ˚s Ds˚r (commutativity of ˚)
r ˝s Ds˝r (commutativity of ˝);
0˚r Dr (identity for ˚);
1˝r Dr (identity for ˝);
0 0
9r 2 R: r ˚ r D 0 (inverse for ˚);
r ˝ .s ˚ t / D .r ˝ s/ ˚ .r ˝ t / (distributivity):
(a) Show that the zero-element is unique, that is, show that if z 2 R has the
property that
z ˚ r D r; (9.23)
then z D 0.

(b) Show that additive inverses are unique, that is, show that
r ˚ r1 D 0 and (9.24)
r ˚ r2 D 0 (9.25)
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352 Chapter 9 Number Theory

implies r1 D r2 .

r ˝ r1 D 1
r ˝ r2 D 1

implies r1 D r2 .

Problem 9.33.
This problem will use elementary properties of congruences to prove that every
positive integer divides infinitely many Fibonacci numbers.
A function f W N ! N that satisifies

f .n/ D c1 f .n 1/ C c2 f .n 2/ C C cd f .n d/ (9.26)

for some ci 2 N and all n d is called a degree d linear-recursive.
A function f W N ! N has a degree d repeat modulo m at n and k when it
satisfies the following repeat congruences:

f .n/ f .k/ .mod m/;
f .n 1/ f .k 1/ .mod m/;
::
:
f .n .d 1// f .k .d 1// .mod m/:

for k > n d 1.
For the rest of this problem, assume linear-recursive functions and repeats are
degree d > 0.
(a) Prove that if a linear-recursive function has a repeat modulo m at n and k, then
it has one at n C 1 and k C 1.

(b) Prove that for all m > 1, every linear-recursive function repeats modulo m at
n and k for some n; k 2 Œd 1; d C md /.

(c) A linear-recursive function is reverse-linear if its d th coefficient cd D ˙1.
Prove that if a reverse-linear function repeats modulo m at n and k for some n d ,
then it repeats modulo m at n 1 and k 1.

(d) Conclude that every reverse-linear function must repeat modulo m at d 1
and .d 1/ C j for some j > 0.
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9.13. References 353

(e) Conclude that if f is an reverse-linear function and f .k/ D 0 for some k 2
Œ0; d /, then every positive integer is a divisor of f .n/ for infinitely many n.

(f) Conclude that every positive integer is a divisor of infinitely many Fibonacci
numbers.
Hint: Start the Fibonacci sequence with the values 0,1 instead of 1, 1.

Class Problems
Problem 9.34.
Find 5555
remainder 98763456789 999 67893414259 ; 14 : (9.27)

Problem 9.35.
The following properties of equivalence mod n follow directly from its definition
and simple properties of divisibility. See if you can prove them without looking up
the proofs in the text.

(a) If a b .mod n/, then ac bc .mod n/.

(b) If a b .mod n/ and b c .mod n/, then a c .mod n/.

(d) rem.a; n/ a .mod n/.

Problem 9.36. (a) Why is a number written in decimal evenly divisible by 9 if and
only if the sum of its digits is a multiple of 9? Hint: 10 1 .mod 9/.

(b) Take a big number, such as 37273761261. Sum the digits, where every other
one is negated:

3 C . 7/ C 2 C . 7/ C 3 C . 7/ C 6 C . 1/ C 2 C . 6/ C 1 D 11

Explain why the original number is a multiple of 11 if and only if this sum is a
multiple of 11.

Problem 9.37.
At one time, the Guinness Book of World Records reported that the “greatest human
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354 Chapter 9 Number Theory

calculator” was a guy who could compute 13th roots of 100-digit numbers that were
13th powers. What a curious choice of tasks. . . .
In this problem, we prove

n13 n .mod 10/ (9.28)

for all n.
(a) Explain why (9.28) does not follow immediately from Euler’s Theorem.

(b) Prove that
d 13 d .mod 10/ (9.29)
for 0 d < 10.

Problem 9.38. (a) Ten pirates find a chest filled with gold and silver coins. There
are twice as many silver coins in the chest as there are gold. They divide the gold
coins in such a way that the difference in the number of coins given to any two
pirates is not divisible by 10. They will only take the silver coins if it is possible
to divide them the same way. Is this possible, or will they have to leave the silver
behind? Prove your answer.

(b) There are also 3 sacks in the chest, containing 5, 49, and 51 rubies respec-
tively. The treasurer of the pirate ship is bored and decides to play a game with the
following rules:

He can merge any two piles together into one pile, and
he can divide a pile with an even number of rubies into two piles of equal size.

He makes one move every day, and he will finish the game when he has divided the
rubies into 105 piles of one. Is it possible for him to finish the game?

Exam Problems
Problem 9.39.
The sum of the digits of the base 10 representation of an integer is congruent mod-
ulo 9 to that integer. For example,

763 7 C 6 C 3 .mod 9/:

We can say that “9 is a good modulus for base 10.”
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9.13. References 355

More generally, we’ll say “k is a good modulus for base b” when, for any non-
negative integer n, the sum of the digits of the base b representation of n is congru-
ent to n modulo k. So 2 is not a good modulus for base 10 because

763 6 7 C 6 C 3 .mod 2/:

(a) What integers k > 1 are good moduli for base 10?

(b) Show that if b 1 .mod k/, then k is good for base b.

(d) Exactly which integers k > 1 are good moduli for base 106?

Problem 9.40.
We define the sequence of numbers
(
1; for n 3,
an D
an 1 C an 2 C an 3 C an 4; for n > 3.

Use strong induction to prove that remainder.an ; 3/ D 1 for all n 0.

Problems for Section 9.8
Exam Problems
Problem 9.41.

Definition. The set P of single variable integer polynomials can be defined recur-
sively:
Base cases:
the identity function, IdZ .x/ WWD x is in P .

for any integer m the constant function, cm .x/ WWD m is in P .
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356 Chapter 9 Number Theory

Constructor cases. If r; s 2 P , then r C s and r s 2 P .
Prove by structural induction that for all q 2 P ,

j k .mod n/ IMPLIES q.j / q.k/ .mod n/;

for all integers j; k; n where n > 1.
Be sure to clearly state and label your Induction Hypothesis, Base case(s), and
Constructor step.

Problems for Section 9.9
Practice Problems
Problem 9.42.

(a) Given inputs m; n 2 ZC , the Pulverizer will produce x; y 2 Z such that:

(b) Assume n > 1. Explain how to use the numbers x; y to find the inverse of m
modulo n when there is an inverse.

Problem 9.43.
What is the multiplicative inverse (mod 7) of 2? Reminder: by definition, your
answer must be an integer between 0 and 6.

Problem 9.44. (a) Find integer coefficients x, y such that 25xC32y D gcd.25; 32/.

(b) What is the inverse (mod 25) of 32?

Problem 9.45. (a) Use the Pulverizer to find integers s; t such that

40s C 7t D gcd.40; 7/:

(b) Adjust your answer to part (a) to find an inverse modulo 40 of 7 in Œ1; 40/.
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9.13. References 357

Class Problems
Problem 9.46.
Two nonparallel lines in the real plane intersect at a point. Algebraically, this means
that the equations

y D m1 x C b1
y D m2 x C b2

have a unique solution .x; y/, provided m1 ¤ m2 . This statement would be false if
we restricted x and y to the integers, since the two lines could cross at a noninteger
point:

However, an analogous statement holds if we work over the integers modulo a
prime p. Find a solution to the congruences

y m1 x C b1 .mod p/
y m2 x C b2 .mod p/

when m1 6 m2 .mod p/. Express your solution in the form x ‹ .mod p/ and
y ‹ .mod p/ where the ?’s denote expressions involving m1 , m2 , b1 and b2 . You
may find it helpful to solve the original equations over the reals first.

Problems for Section 9.10
Practice Problems
Problem 9.47.
Prove that k 2 Œ0; n/ has an inverse modulo n iff it has an inverse in Zn .
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358 Chapter 9 Number Theory

Problem 9.48.
What is rem.2479 ; 79/?
Hint: You should not need to do any actual multiplications!

Problem 9.49. (a) Prove that 2212001 has a multiplicative inverse modulo 175.

(b) What is the value of .175/, where is Euler’s function?

Problem 9.50.
How many numbers between 1 and 6042 (inclusive) are relatively prime to 3780?
Hint: 53 is a factor.

Problem 9.51.
How many numbers between 1 and 3780 (inclusive) are relatively prime to 3780?

Problem 9.52.

(a) What is the probability that an integer from 1 to 360 selected with uniform
probability is relatively prime to 360?

(b) What is the value of rem.798 ; 360/?

Class Problems
Problem 9.53.
Find the remainder of 261818181 divided by 297.
Hint: 1818181 D .180 10101/ C 1; use Euler’s theorem.

Problem 9.54.
77
Find the last digit of 77 .
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9.13. References 359

Problem 9.55.
Prove that n and n5 have the same last digit. For example:

25 D 32 795 D 3077056399

Problem 9.56.
Use Fermat’s theorem to find the inverse i of 13 modulo 23 with 1 i < 23.

Problem 9.57.
Let be Euler’s function.
(a) What is the value of .2/?

(b) What are three nonnegative integers k > 1 such that .k/ D 2?

(d) Briefly explain why .k/ D 2 for exactly three values of k.

Problem 9.58.
Suppose a; b are relatively prime and greater than 1. In this problem you will prove
the Chinese Remainder Theorem, which says that for all m; n, there is an x such
that

x m mod a; (9.30)
x n mod b: (9.31)

Moreover, x is unique up to congruence modulo ab, namely, if x 0 also satis-
fies (9.30) and (9.31), then
x 0 x mod ab:
(a) Prove that for any m; n, there is some x satisfying (9.30) and (9.31).
Hint: Let b 1 be an inverse of b modulo a and define ea WWD b 1 b. Define eb
similarly. Let x D mea C neb .

(b) Prove that

Œx 0 mod a AND x 0 mod b implies x 0 mod ab:
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360 Chapter 9 Number Theory

x x 0 mod a AND x x 0 mod b implies x x 0 mod ab:

(d) Conclude that the Chinese Remainder Theorem is true.

(e) What about the converse of the implication in part (c)?

Problem 9.59.
The order of k 2 Zn is the smallest positive m such that k m D 1 .Zn /.
(a) Prove that
k m D 1 .Zn / IMPLIES ord.k; n/ j m:

Hint: Take the remainder of m divided by the order.
Now suppose p > 2 is a prime of the form 2s C 1. For example, 21 C 1; 22 C
1; 24 C 1 are such primes.
(b) Conclude from part (a) that if 0 < k < p, then ord.k; p/ is a power of 2.

(c) Prove that ord.2; p/ D 2s and conclude that s is a power of 2.19
Hint: 2k 1 for k 2 Œ1::r is positive but too small to equal 0 .Zp /.

Homework Problems
Problem 9.60.
This problem is about finding square roots modulo a prime p.
(a) Prove that x 2 y 2 .mod p/ if and only if x y .mod p/ or x y
.mod p/. Hint: x 2 y 2 D .x C y/.x y/
An integer x is called a square root of n mod p when

x2 n .mod p/:

An integer with a square root is called a square mod p. For example, if n is con-
gruent to 0 or 1 mod p, then n is a square and it is it’s own square root.
So let’s assume that p is an odd prime and n 6 0 .mod p/. It turns out there is
a simple test we can perform to see if n is a square mod p:
k
19 Numbers of the form 22 C 1 are called Fermat numbers, so we can rephrase this conclusion as
saying that any prime of the form 2s C 1 must actually be a Fermat number. The Fermat numbers are
prime for k D 1; 2; 3; 4, but not for k D 5. In fact, it is not known if any Fermat number with k > 4
is prime.
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9.13. References 361

Euler’s Criterion

i. If n is a square modulo p, then n.p 1/=2 1 .mod p/.

ii. If n is not a square modulo p then n.p 1/=2 1 .mod p/.

(b) Prove Case (i) of Euler’s Criterion. Hint: Use Fermat’s theorem.

(d) Suppose that p 3 .mod 4/, and n is a square mod p. Find a simple expres-
sion in terms of n and p for a square root of n. Hint: Write p as p D 4k C 3 and
use Euler’s Criterion. You might have to multiply two sides of an equation by n at
one point.

Problem 9.61.
Suppose a; b are relatively prime integers greater than 1. In this problem you will
prove that Euler’s function is multiplicative, that is, that

.ab/ D .a/.b/:

The proof is an easy consequence of the Chinese Remainder Theorem (Problem 9.58).
(a) Conclude from the Chinese Remainder Theorem that the function f W Œ0::ab/ !
Œ0::a/ Œ0::b/ defined by

f .x/ WWD .rem.x; a/; rem.x; b//

is a bijection.

(b) For any positive integer k let Zk be the integers in Œ0::k/ that are relatively
prime to k. Prove that the function f from part (a) also defines a bijection from
Zab to Za Zb .

.ab/ D .a/.b/: (9.32)

(d) Prove Corollary 9.10.11: for any number n > 1, if p1 , p2 , . . . , pj are the
(distinct) prime factors of n, then

1 1 1
.n/ D n 1 1 1 :
p1 p2 pj
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362 Chapter 9 Number Theory

Problem 9.62.

Definition. Define the order of k over Zn to be

ord.k; n/ WWD minfm > 0 j k m D 1 .Zn /g:

If no positive power of k equals 1 in Zn , then ord.k; n/ WWD 1.
(a) Show that k 2 Zn iff k has finite order in Zn .

(b) Prove that for every k 2 Zn , the order of k over Zn divides .n/.
Hint: Let m D ord.k; n/. Consider the quotient and remainder of .n/ divided by
m.

Problem 9.63.
The general version of the Chinese Remainder Theorem (see Problem 9.58) extends
to more than two relatively prime moduli. Namely,

Theorem (General Chinese Remainder). Suppose a1 ; : : : ; ak are integers greater
than 1 and each is relatively prime to the others. Let n WWD a1 a2 ak . Then for
any integers m1 ; m2 ; : : : ; mk , there is a unique x 2 Œ0::n/ such that

x mi .mod ai /;

for 1 i k.

The proof is a routine induction on k using a fact that follows immediately from
unique factorization: if a number is relatively prime to some other numbers, then it
is relatively prime to their product.
The General Chinese Remainder Theorem is the basis for an efficient approach
to performing a long series of additions and multiplications on “large” numbers.
Namely, suppose n was large, but each of the factors ai was small enough to be
handled by cheap and available arithmetic hardware units. Suppose a calculation
requiring many additions and multiplications needs to be performed. To do a sin-
gle multiplication or addition of two large numbers x and y in the usual way in
this setting would involve breaking up the x and y into pieces small enough to be
handled by the arithmetic units, using the arithmetic units to perform additions and
multiplications on (many) pairs of small pieces, and then reassembling the pieces
into an answer. Moreover, the order in which these operations on pieces can be
performed is contrained by dependence among the pieces—because of “carries,”
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9.13. References 363

for example. And this process of breakup and reassembly has to be performed for
each addition and multiplication that needs to be performed on large numbers.
Explain how the General Chinese Remainder Theorem can be applied to per-
form a long series of additions and multiplications on “large” numbers much more
efficiently than the usual way described above.

Problem 9.64.
In this problem we’ll prove that for all integers a; m where m > 1,

am am .m/
.mod m/: (9.33)

Note that a and m need not be relatively prime.
Assume m D p1k1 pnkn for distinct primes, p1 ; : : : ; pn and positive integers
k1 ; : : : ; k n .
(a) Show that if pi does not divide a, then

a.m/ 1 .mod piki /:

(b) Show that if pi j a then

am .m/
0 .mod piki /: (9.34)

Problem 9.65.
The Generalized Postage Problem
Several other problems (2.7, 2.1, 5.32) work out which amounts of postage can
be formed using two stamps of given denominations. In this problem, we generalize
this to two stamps with arbitrary positive integer denominations a and b cents. Let’s
call an amount of postage that can be made from a and b cent stamps a makeable
amount.

Lemma. (Generalized Postage) If a and b are relatively prime positive integers,
then any integer greater than ab a b is makeable.
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364 Chapter 9 Number Theory

To prove the Lemma, consider the following array with a infinite rows:

0 a 2a 3a :::
b bCa b C 2a b C 3a :::
2b 2b C a 2b C 2a 2b C 3a :::
3b 3b C a 3b C 2a 3b C 3a :::
:: :: :: ::
: : : :
:::
.a 1/b .a 1/b C a .a 1/b C 2a .a 1/b C 3a : : :

Note that every element in this array is clearly makeable.
(a) Suppose that n is at least as large as, and also congruent mod a to, the first
element in some row of this array. Explain why n must appear in the array.

(b) Prove that every integer from 0 to a 1 is congruent modulo a to one of the
integers in the first column of this array.

(c) Complete the proof of the Generalized Postage Lemma by using parts (a)
and (b) to conclude that every integer n > ab a b appears in the array, and
hence is makeable.
Hint: Suppose n is congruent mod a to the first element in some row. Assume n is
less than that element, and then show that n ab a b.

(d) (Optional) What’s more, ab a b is not makeable. Prove it.

(e) Explain why the following even more general lemma follows directly from the
Generalized Lemma and part (d).
Lemma. (Generalized2 Postage) If m and n are positive integers and gWWDgcd.m; n/ >
1, then with m and n cent stamps, you can only make amounts of postage that
are multiples of g. You can actually make any amount of postage greater than
.mn=g/ m n that is a multiple of g, but you cannot make .mn=g/ m n cents
postage.

(f) Optional and possibly unknown. Suppose you have three denominations of
stamps, a; b; c and gcd.a; b; c/ D 1. Give a formula for the smallest number nabc
such that you can make every amount of postage nabc .

Exam Problems
Problem 9.66.
What is the remainder of 639601 divided by 220?
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9.13. References 365

Problem 9.67.
Prove that if k1 and k2 are relatively prime to n, then so is k1 n k2 ,
(a) . . . using the fact that k is relatively prime to n iff k has an inverse modulo n.
Hint: Recall that k1 k2 k1 n k2 .mod n/.

(b) . . . using the fact that k is relatively prime to n iff k is cancellable modulo n.

Problem 9.68.

Circle true or false for the statements below, and provide counterexamples for
those that are false. Variables, a; b; c; m; n range over the integers and m; n > 1.
(a) gcd.1 C a; 1 C b/ D 1 C gcd.a; b/. true false

(b) If a b .mod n/, then p.a/ p.b/ .mod n/
for any polynomial p.x/ with integer coefficients. true false

(d) gcd.an ; b n / D .gcd.a; b//n true false

(e) If gcd.a; b/ ¤ 1 and gcd.b; c/ ¤ 1, then gcd.a; c/ ¤ 1. true false

(f) If an integer linear combination of a and b equals 1,
then so does some integer linear combination of a2 and b 2 . true false

(g) If no integer linear combination of a and b equals 2,
then neither does any integer linear combination of a2 and b 2 . true false

(h) If ac bc .mod n/ and n does not divide c,
then a b .mod n/. true false

(i) Assuming a; b have inverses modulo n,
if a 1 b 1 .mod n/, then a b .mod n/. true false

(j) If ac bc .mod n/ and n does not divide c,
then a b .mod n/. true false
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366 Chapter 9 Number Theory

(k) If a b .mod .n// for a; b > 0, then c a c b .mod n/. true false

(l) If a b .mod nm/, then a b .mod n/. true false

(m) If gcd.m; n/ D 1, then
Œa b .mod m/ AND a b .mod n/ iff Œa b .mod mn/ true false

(n) If gcd.a; n/ D 1, then an 1 1 .mod n/ true false

(o) If a; b > 1, then
[a has a inverse mod b iff b has an inverse mod a]. true false

Problem 9.69.
Find an integer k > 1 such that n and nk agree in their last three digits whenever n
is divisible by neither 2 nor 5. Hint: Euler’s theorem.

Problem 9.70.

(a) Explain why . 12/482 has a multiplicative inverse modulo 175.

(b) What is the value of .175/, where is Euler’s function?

(c) Call a number from 0 to 174 powerful iff some positive power of the number
is congruent to 1 modulo 175. What is the probability that a random number from
0 to 174 is powerful?

(d) What is the remainder of . 12/482 divided by 175?

Problem 9.71. (a) Calculate the remainder of 3586 divided by 29.

(b) Part (a) implies that the remainder of 3586 divided by 29 is not equal to 1. So
there there must be a mistake in the following proof, where all the congruences are
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9.13. References 367

taken with modulus 29:

1 6 3586 (by part (a)) (9.35)
686 (since 35 6 .mod 29/) (9.36)
28
6 (since 86 28 .mod 29/) (9.37)
1 (by Fermat’s Little Theorem) (9.38)

Identify the exact line containing the mistake and explain the logical error.

Problem 9.72.
Indicate whether the following statements are true or false. For each of the false
statements, give counterexamples. All variables range over the integers, Z.
(a) For all a and b, there are x and y such that: ax C by D 1.

(b) gcd.mb C r; b/ D gcd.r; b/ for all m; r and b.

(d) For primes p ¤ q, .pq/ D .p 1/.q 1/, where is Euler’s totient function.

(e) If a and b are relatively prime to d , then

Œac bc mod d IMPLIES Œa b mod d :

Problem 9.73. (a) Show that if p j n for some prime p and integer n > 0, then
.p 1/ j .n/.

(b) Conclude that .n/ is even for all n > 2.

Problem 9.74. (a) Calculate the value of .6042/.
Hint: 53 is a factor of 6042.

(b) Consider an integer k > 0 that is relatively prime to 6042. Explain why
k 9361 k .mod 6042/.
Hint: Use your solution to part (a).
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368 Chapter 9 Number Theory

Problem 9.75.
Let
Sk D 1k C 2k C C p k ;
where p is an odd prime and k is a positive multiple of p 1. Find a 2 Œ0::p/ and
b 2 . p::0 such that
Sk a b .mod p/:

Problems for Section 9.11
Practice Problems
Problem 9.76.
Suppose a cracker knew how to factor the RSA modulus n into the product of
distinct primes p and q. Explain how the cracker could use the public key-pair
.e; n/ to find a private key-pair .d; n/ that would allow him to read any message
encrypted with the public key.

Problem 9.77.
Suppose the RSA modulus n D pq is the product of distinct 200 digit primes p and
q. A message m 2 Œ0::n/ is called dangerous if gcd.m; n/ D p, because such an m
can be used to factor n and so crack RSA. Circle the best estimate of the fraction
of messages in Œ0::n/ that are dangerous.

1 1 1 1 1 1
200 400 20010 10200 40010 10400

Problem 9.78.
Ben Bitdiddle decided to encrypt all his data using RSA. Unfortunately, he lost his
private key. He has been looking for it all night, and suddenly a genie emerges
from his lamp. He offers Ben a quantum computer that can perform exactly one
procedure on large numbers e; d; n. Which of the following procedures should Ben
choose to recover his data?

Find gcd.e; d /.

Find the prime factorization of n.
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9.13. References 369

Determine whether n is prime.
Find rem.e d ; n/.
Find the inverse of e modulo n (the inverse of e in Zn /.
Find the inverse of e modulo .n/.

Class Problems
Problem 9.79.
Let’s try out RSA!
(a) Go through the beforehand steps.

Choose primes p and q to be relatively small, say in the range 10–40. In
practice, p and q might contain hundreds of digits, but small numbers are
easier to handle with pencil and paper.
Try e D 3; 5; 7; : : : until you find something that works. Use Euclid’s algo-
rithm to compute the gcd.
Find d (using the Pulverizer).

When you’re done, put your public key on the board prominentally labelled “Public
Key.” This lets another team send you a message.

(b) Now send an encrypted message to another team using their public key. Select
your message m from the codebook below:

2 = Greetings and salutations!
3 = Yo, wassup?
4 = You guys are slow!
5 = All your base are belong to us.
6 = Someone on our team thinks someone on your team is kinda cute.
7 = You are the weakest link. Goodbye.

Problem 9.80. (a) Just as RSA would be trivial to crack knowing the factorization
into two primes of n in the public key, explain why RSA would also be trivial to
crack knowing .n/.
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370 Chapter 9 Number Theory

(b) Show that if you knew n, .n/, and that n was the product of two primes, then
you could easily factor n.

Problem 9.81.
A critical fact about RSA is, of course, that decrypting an encrypted message al-
ways gives back the original message m. Namely, if n D pq where p and q are
distinct primes, m 2 Œ0::pq/, and

d e 1 .mod .p 1/.q 1//;

then d
bd WWD me
m D m .Zn /: (9.39)
We’ll now prove this.
(a) Explain why (9.39) follows very simply from Euler’s theorem when m is rel-
atively prime to n.
All the rest of this problem is about removing the restriction that m be relatively
prime to n. That is, we aim to prove that equation (9.39) holds for all m 2 Œ0::n/.
It is important to realize that there is no practical reason to worry about—or to
bother to check for—this relative primality condition before sending a message m
using RSA. That’s because the whole RSA enterprise is predicated on the difficulty
of factoring. If an m ever came up that wasn’t relatively prime to n, then we could
factor n by computing gcd.m; n/. So believing in the security of RSA implies
believing that the liklihood of a message m turning up that was not relatively prime
to n is negligible.
But let’s be pure, impractical mathematicians and get rid of this technically un-
necessary relative primality side condition, even if it is harmless. One gain for
doing this is that statements about RSA will be simpler without the side condition.
More important, the proof below illustrates a useful general method of proving
things about a number n by proving them separately for the prime factors of n.
(b) Prove that if p is prime and a 1 .mod p 1/, then

ma D m .Zp /: (9.40)

(c) Give an elementary proof20 that if a b .mod pi / for distinct primes pi ,
then a b modulo the product of these primes.

(d) Note that (9.39) is a special case of
20 There is no need to appeal to the Chinese Remainder Theorem.
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9.13. References 371

Claim. If n is a product of distinct primes and a 1 .mod .n//, then

ma D m .Zn /:

Use the previous parts to prove the Claim.

Homework Problems
Problem 9.82.
Although RSA has successfully withstood cryptographic attacks for a more than a
quarter century, it is not known that breaking RSA would imply that factoring is
easy.
In this problem we will examine the Rabin cryptosystem that does have such
a security certification. Namely, if someone has the ability to break the Rabin
cryptosystem efficiently, then they also have the ability to factor numbers that are
products of two primes.
Why should that convince us that it is hard to break the cryptosystem efficiently?
Well, mathematicians have been trying to factor efficiently for centuries, and they
still haven’t figured out how to do it.
What is the Rabin cryptosystem? The public key will be a number N that is a
product of two very large primes p; q such that p q 3 .mod 4/. To send the
message m, send rem.m2 ; N /.21
The private key is the factorization of N , namely, the primes p; q. We need to
show that if the person being sent the message knows p; q, then they can decode
the message. On the other hand, if an eavesdropper who doesn’t know p; q listens
in, then we must show that they are very unlikely to figure out this message.
Say that s is a square modulo N if there is an m 2 Œ0; N / such that s m2
.mod N /. Such an m is a square root of s modulo N .
(a) What are the squares modulo 5? For each square in the interval Œ0; 5/, how
many square roots does it have?

(b) For each integer in Œ1; 15/ that is relatively prime to 15, how many square roots
(modulo 15) does it have? Note that all the square roots are also relatively prime to
15. We won’t go through why this is so here, but keep in mind that this is a general
phenomenon!

(c) Suppose that p is a prime such that p 3 .mod 4/. It turns out that squares
modulo p have exactly 2 square roots. First show that .p C 1/=4 is an integer.
21 We will see soon, that there are other numbers that would be encrypted by rem.m2 ; N /, so we’ll

have to disallow those other numbers as possible messages in order to make it possible to decode this
cryptosystem, but let’s ignore that for now.
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372 Chapter 9 Number Theory

Next figure out the two square roots of 1 modulo p. Then show that you can find a
“square root mod a prime p” of a number by raising the number to the .p C 1/=4th
power. That is, given s, to find m such that s m2 .mod p/, you can compute
rem.s .pC1/=4 ; p/.

(d) The Chinese Remainder Theorem (Problem 9.58) implies that if p; q are dis-
tinct primes, then s is a square modulo pq if and only if s is a square modulo p and
s is a square modulo q. In particular, if s x 2 .x 0 /2 .mod p/ where x ¤ x 0 ,
and likewise s y 2 .y 0 /2 .mod q/ then s has exactly four square roots modulo
N , namely,

s .xy/2 .x 0 y/2 .xy 0 /2 .x 0 y 0 /2 .mod pq/:

So, if you know p; q, then using the solution to part (c), you can efficiently find the
square roots of s! Thus, given the private key, decoding is easy.
But what if you don’t know p; q?
Let’s assume that the evil message interceptor claims to have a program that can
find all four square roots of any number modulo N . Show that he can actually
use this program to efficiently find the factorization of N . Thus, unless this evil
message interceptor is extremely smart and has figured out something that the rest
of the scientific community has been working on for years, it is very unlikely that
this efficient square root program exists!
Hint: Pick r arbitrarily from Œ1; N /. If gcd.N; r/ > 1, then you are done (why?)
so you can halt. Otherwise, use the program to find all four square roots of r, call
them r; r; r 0 ; r 0 . Note that r 2 r 02 .mod N /. How can you use these roots to
factor N ?

(e) If the evil message interceptor knows that the message is the encoding one of
two possible candidate messages (that is, either “meet at dome at dusk” or “meet at
dome at dawn”) and is just trying to figure out which of the two, then can he break
this cryptosystem?

Problem 9.83.
You’ve seen how the RSA encryption scheme works, but why is it hard to break?
In this problem, you will see that finding private keys is as hard as finding the
prime factorizations of integers. Since there is a general consensus in the crypto
community (enough to persuade many large financial institutions, for example)
that factoring numbers with a few hundred digits requires astronomical computing
resources, we can therefore be sure it will take the same kind of overwhelming
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9.13. References 373

effort to find RSA private keys of a few hundred digits. This means we can be
confident the private RSA keys are not somehow revealed by the public keys22 .
For this problem, assume that n D p q where p; q are both odd primes and that
e is the public key and d the private key of the RSA protocol.. Let c WWD e d 1.
(a) Show that .n/ divides c.

(b) Conclude that 4 divides c.

(c) Show that if gcd.r; n/ D 1, then r c 1 .mod n/:
A square root of m modulo n is an integer s 2 Œ0:n/ such that s 2 m .mod n/.
Here is a nice fact to know: when n is a product of two odd primes, then every
number m such that gcd.m; n/ D 1 has 4 square roots modulo n.
In particular, the number 1 has four square roots modulo n. The two trivial ones
are 1 and n 1 (which is 1 .mod n/). The other two are called the nontrivial
square roots of 1.
(d) Since you know c, then for any integer r you can also compute the remainder
y of r c=2 divided by n. So y 2 r c .mod n/. Now if r is relatively prime to n,
then y will be a square root of 1 modulo n by part (c).
Show that if y turns out to be a nontrivial root of 1 modulo n, then you can factor
n. Hint: From the fact that y 2 1 D .y C 1/.y 1/, show that y C 1 must be
divisible by exactly one of q and p.

(e) It turns out that at least half the positive integers r < n that are relatively
prime to n will yield y’s in part (d) that are nontrivial roots of 1. Conclude that if,
in addition to n and the public key e you also knew the private key d , then you can
be sure of being able to factor n.

Exam Problems
Problem 9.84.
Suppose Alice and Bob are using the RSA cryptosystem to send secure messages.
Each of them has a public key visible to everyone and a private key known only to
themselves, and using RSA in the usual way, they are able to send secret messages
to each other over public channels.
But a concern for Bob is how he knows that a message he gets is actually from
Alice—as opposed to some imposter claiming to be Alice. This concern can be met
by using RSA to add unforgeable “signatures” to messages. To send a message m
22 Thisis a very weak kind of “security” property, because it doesn’t even rule out the possibility
of deciphering RSA encoded messages by some method that did not require knowing the private key.
Nevertheless, over twenty years experience supports the security of RSA in practice.
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374 Chapter 9 Number Theory

to Bob with an unforgeable signature, Alice uses RSA encryption on her message
m, but instead using Bob’s public key to encrypt m, she uses her own private key
to obtain a message m1 . She then sends m1 as her “signed” message to Bob.
(a) Explain how Bob can read the original message m from Alice’s signed mes-
sage m1 . (Let .nA ; eA / be Alice’s public key and dA her private key. Assume
m 2 Œ0::nA /.)

(b) Briefly explain why Bob can be confident, assuming RSA is secure, that m1
came from Alice rather than some imposter.

(c) Notice that not only Bob, but anyone can use Alice’s public key to reconstruct
her message m from its signed version m1 . So how can Alice send a secret signed
message to Bob over public channels?
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10 Directed graphs & Partial Orders
Directed graphs, called digraphs for short, provide a handy way to represent how
things are connected together and how to get from one thing to another by following
those connections. They are usually pictured as a bunch of dots or circles with
arrows between some of the dots, as in Figure 10.1. The dots are called nodes or
vertices and the lines are called directed edges or arrows; the digraph in Figure 10.1
has 4 nodes and 6 directed edges.
Digraphs appear everywhere in computer science. For example, the digraph in
Figure 10.2 represents a communication net, a topic we’ll explore in depth in Chap-
ter 11. Figure 10.2 has three “in” nodes (pictured as little squares) representing
locations where packets may arrive at the net, the three “out” nodes representing
destination locations for packets, and the remaining six nodes (pictured with lit-
tle circles) represent switches. The 16 edges indicate paths that packets can take
through the router.
Another place digraphs emerge in computer science is in the hyperlink structure
of the World Wide Web. Letting the vertices x1 ; : : : ; xn correspond to web pages,
and using arrows to indicate when one page has a hyperlink to another, results in a
digraph like the one in Figure 10.3—although the graph of the real World Wide Web
would have n be a number in the billions and probably even the trillions. At first
glance, this graph wouldn’t seem to be very interesting. But in 1995, two students
at Stanford, Larry Page and Sergey Brin, ultimately became multibillionaires from
the realization of how useful the structure of this graph could be in building a search
engine. So pay attention to graph theory, and who knows what might happen!

a c

Figure 10.1 A 4-node directed graph with 6 edges.
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376 Chapter 10 Directed graphs & Partial Orders

in0 in1 in2

out0 out1 out2

Figure 10.2 A 6-switch packet routing digraph.

x3 x4

x2
x1 x5

x6
Figure 10.3 Links among Web Pages.
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10.1. Vertex Degrees 377

tail e head
u v

Figure 10.4 A directed edge e D hu ! vi. The edge e starts at the tail vertex u
and ends at the head vertex v.

Definition 10.0.1. A directed graph G consists of a nonempty set V .G/, called the
vertices of G, and a set E.G/, called the edges of G. An element of V .G/ is called
a vertex. A vertex is also called a node; the words “vertex” and “node” are used
interchangeably. An element of E.G/ is called a directed edge. A directed edge is
also called an “arrow” or simply an “edge.” A directed edge starts at some vertex
u called the tail of the edge, and ends at some vertex v called the head of the edge,
as in Figure 10.4. Such an edge can be represented by the ordered pair .u; v/. The
notation hu ! vi denotes this edge.
There is nothing new in Definition 10.0.1 except for a lot of vocabulary. For-
mally, a digraph G is the same as a binary relation on the set, V D V .G/—that is,
a digraph is just a binary relation whose domain and codomain are the same set V .
In fact, we’ve already referred to the arrows in a relation G as the “graph” of G.
For example, the divisibility relation on the integers in the interval Œ1::12 could be
pictured by the digraph in Figure 10.5.

10.1 Vertex Degrees
The in-degree of a vertex in a digraph is the number of arrows coming into it, and
similarly its out-degree is the number of arrows out of it. More precisely,
Definition 10.1.1. If G is a digraph and v 2 V .G/, then

indeg.v/ WWD jfe 2 E.G/ j head.e/ D vgj
outdeg.v/ WWD jfe 2 E.G/ j tail.e/ D vgj

An immediate consequence of this definition is
Lemma 10.1.2. X X
indeg.v/ D outdeg.v/:
v2V .G/ v2V .G/

Proof. Both sums are equal to jE.G/j.
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378 Chapter 10 Directed graphs & Partial Orders

4 2 8 10

12 6 1

3 9 11

Figure 10.5 The Digraph for Divisibility on f1; 2; : : : ; 12g.

10.2 Walks and Paths
Picturing digraphs with points and arrows makes it natural to talk about following
successive edges through the graph. For example, in the digraph of Figure 10.5,
you might start at vertex 1, successively follow the edges from vertex 1 to vertex
2, from 2 to 4, from 4 to 12, and then from 12 to 12 twice (or as many times as
you like). The sequence of edges followed in this way is called a walk through the
graph. A path is a walk which never visits a vertex more than once. So following
edges from 1 to 2 to 4 to 12 is a path, but it stops being a path if you go to 12 again.
The natural way to represent a walk is with the sequence of sucessive vertices it
went through, in this case:
1 2 4 12 12 12:
However, it is conventional to represent a walk by an alternating sequence of suc-
cessive vertices and edges, so this walk would formally be

1 h1 ! 2i 2 h2 ! 4i 4 h4 ! 12i 12 h12 ! 12i 12 h12 ! 12i 12: (10.1)

The redundancy of this definition is enough to make any computer scientist cringe,
but it does make it easy to talk about how many times vertices and edges occur on
the walk. Here is a formal definition:

Definition 10.2.1. A walk in a digraph is an alternating sequence of vertices and
edges that begins with a vertex, ends with a vertex, and such that for every edge
hu ! vi in the walk, vertex u is the element just before the edge, and vertex v is the
next element after the edge.
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10.2. Walks and Paths 379

So a walk v is a sequence of the form

v WWD v0 hv0 ! v1 i v1 hv1 ! v2 i v2 : : : hvk 1 ! vk i vk

where hvi ! vi C1 i 2 E.G/ for i 2 Œ0::k/. The walk is said to start at v0 , to end at
vk , and the length jvj of the walk is defined to be k.
The walk is a path iff all the vi ’s are different, that is, if i ¤ j , then vi ¤ vj .
A closed walk is a walk that begins and ends at the same vertex. A cycle is a
positive length closed walk whose vertices are distinct except for the beginning and
end vertices.
Note that a single vertex counts as a length zero path that begins and ends at itself.
It also is a closed walk, but does not count as a cycle, since cycles by definition
must have positive length. Length one cycles are possible when a node has an
arrow leading back to itself. The graph in Figure 10.1 has none, but every vertex in
the divisibility relation digraph of Figure 10.5 is in a length one cycle. Length one
cycles are sometimes called self-loops.
Although a walk is officially an alternating sequence of vertices and edges, it
is completely determined just by the sequence of successive vertices on it, or by
the sequence of edges on it. We will describe walks in these ways whenever it’s
convenient. For example, for the graph in Figure 10.1,
.a; b; d /, or simply abd , is a (vertex-sequence description of a) length two
path,

.ha ! bi ; hb ! d i/, or simply ha ! bi hb ! d i, is (an edge-sequence de-
scription of) the same length two path,

abcbd is a length four walk,

dcbcbd is a length five closed walk,

bdcb is a length three cycle,

hb ! ci hc ! bi is a length two cycle, and

hc ! bi hb ai ha ! d i is not a walk. A walk is not allowed to follow edges
in the wrong direction.
If you walk for a while, stop for a rest at some vertex, and then continue walking,
you have broken a walk into two parts. For example, stopping to rest after following
two edges in the walk (10.1) through the divisibility graph breaks the walk into the
first part of the walk
1 h1 ! 2i 2 h2 ! 4i 4 (10.2)
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380 Chapter 10 Directed graphs & Partial Orders

from 1 to 4, and the rest of the walk

4 h4 ! 12i 12 h12 ! 12i 12 h12 ! 12i 12: (10.3)

from 4 to 12, and we’ll say the whole walk (10.1) is the mergewalks (10.2) and (10.3).
In general, if a walk f ends with a vertex v and a walk r starts with the same vertex
v we’ll say that their merge f br is the walk that starts with f and continues with
r.1 Two walks can only be merged if the first walk ends at the same vertex v with
which the second one walk starts. Sometimes it’s useful to name the node v where
the walks merge; we’ll use the notation f bv r to describe the merge of a walk f that
ends at v with a walk r that begins at v.
A consequence of this definition is that
Lemma 10.2.2.
jfbrj D jfj C jrj:
In the next section we’ll get mileage out of walking this way.

10.2.1 Finding a Path
If you were trying to walk somewhere quickly, you’d know you were in trouble if
you came to the same place twice. This is actually a basic theorem of graph theory.
Theorem 10.2.3. The shortest walk from one vertex to another is a path.
Proof. If there is a walk from vertex u to another vertex v ¤ u, then by the Well
Ordering Principle, there must be a minimum length walk w from u to v. We claim
w is a path.
To prove the claim, suppose to the contrary that w is not a path, meaning that
some vertex x occurs twice on this walk. That is,

w D eb
x fb
xg

for some walks e; f; g where the length of f is positive. But then “deleting” f yields
a strictly shorter walk
ebxg
from u to v, contradicting the minimality of w.

Definition 10.2.4. The distance, dist .u; v/, in a graph from vertex u to vertex v is
the length of a shortest path from u to v.
1 It’s tempting to say the merge is the concatenation of the two walks, but that wouldn’t quite be

right because if the walks were concatenated, the vertex v would appear twice in a row where the
walks meet.
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10.3. Adjacency Matrices 381

As would be expected, this definition of distance satisfies:

Lemma 10.2.5. [The Triangle Inequality]

dist .u; v/ dist .u; x/ C dist .x; v/

for all vertices u; v; x with equality holding iff x is on a shortest path from u to v.

Of course, you might expect this property to be true, but distance has a technical
definition and its properties can’t be taken for granted. For example, unlike ordinary
distance in space, the distance from u to v is typically different from the distance
from v to u. So, let’s prove the Triangle Inequality

Proof. To prove the inequality, suppose f is a shortest path from u to x and r is
a shortest path from x to v. Then by Lemma 10.2.2, f b x r is a walk of length
dist .u; x/ C dist .x; v/ from u to v, so this sum is an upper bound on the length of
the shortest path from u to v by Theorem 10.2.3.
Proof of the “iff” is in Problem 10.3.

Finally, the relationship between walks and paths extends to closed walks and
cycles:

Lemma 10.2.6. The shortest positive length closed walk through a vertex is a cycle
through that vertex.

The proof of Lemma 10.2.6 is essentially the same as for Theorem 10.2.3; see
Problem 10.4.

10.3 Adjacency Matrices
If a graph G has n vertices v0 ; v1 ; : : : ; vn 1 , a useful way to represent it is with an
n n matrix of zeroes and ones called its adjacency matrix AG . The ij th entry of
the adjacency matrix, .AG /ij , is 1 if there is an edge from vertex vi to vertex vj
and 0 otherwise. That is,
( ˝ ˛
1 if vi ! vj 2 E.G/;
.AG /ij WWD
0 otherwise:
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382 Chapter 10 Directed graphs & Partial Orders

For example, let H be the 4-node graph shown in Figure 10.1. Its adjacency matrix
AH is the 4 4 matrix:

a b c d
a 0 1 0 1
AH D b 0 0 1 1
c 0 1 0 0
d 0 0 1 0

A payoff of this representation is that we can use matrix powers to count numbers
of walks between vertices. For example, there are two length two walks between
vertices a and c in the graph H :

a ha ! bi b hb ! ci c
a ha ! d i d hd ! ci c

and these are the only length two walks from a to c. Also, there is exactly one
length two walk from b to c and exactly one length two walk from c to c and from
d to b, and these are the only length two walks in H . It turns out we could have
read these counts from the entries in the matrix .AH /2 :

a b c d
a 0 0 2 1
.AH /2 D b 0 1 1 0
c 0 0 1 1
d 0 1 0 0

More generally, the matrix .AG /k provides a count of the number of length k
walks between vertices in any digraph G as we’ll now explain.

Definition 10.3.1. The length-k walk counting matrix for an n-vertex graph G is
the n n matrix C such that

Cuv WWD the number of length-k walks from u to v: (10.4)

Notice that the adjacency matrix AG is the length-1 walk counting matrix for G,
and that .AG /0 , which by convention is the identity matrix, is the length-0 walk
counting matrix.

Theorem 10.3.2. If C is the length-k walk counting matrix for a graph G, and D
is the length-m walk counting matrix, then CD is the length k C m walk counting
matrix for G.
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10.3. Adjacency Matrices 383

According to this theorem, the square .AG /2 of the adjacency matrix is the length
two walk counting matrix for G. Applying the theorem again to .AG /2 AG shows
that the length-3 walk counting matrix is .AG /3 . More generally, it follows by
induction that

Corollary 10.3.3. The length-k counting matrix of a digraph G is .AG /k , for all
k 2 N.

In other words, you can determine the number of length k walks between any
pair of vertices simply by computing the kth power of the adjacency matrix!
That may seem amazing, but the proof uncovers this simple relationship between
matrix multiplication and numbers of walks.

Proof of Theorem 10.3.2. Any length .kCm/ walk between vertices u and v begins
with a length k walk starting at u and ending at some vertex w followed by a length
m walk starting at w and ending at v. So the number of length .k C m/ walks from
u to v that go through w at the kth step equals the number Cuw of length k walks
from u to w, times the number Dw v of length m walks from w to v. We can get the
total number of length .k C m/ walks from u to v by summing, over all possible
vertices w, the number of such walks that go through w at the kth step. In other
words, X
#length .k C m/ walks from u to v D Cuw Dw v (10.5)
w2V .G/

But the right-hand side of (10.5) is precisely the definition of .CD/uv . Thus, CD
is indeed the length-.k C m/ walk counting matrix.

10.3.1 Shortest Paths
The relation between powers of the adjacency matrix and numbers of walks is
cool—to us math nerds at least—but a much more important problem is finding
shortest paths between pairs of nodes. For example, when you drive home for
vacation, you generally want to take the shortest-time route.
One simple way to find the lengths of all the shortest paths in an n-vertex graph G
is to compute the successive powers of AG one by one up to the n 1st, watching
for the first power at which each entry becomes positive. That’s because Theo-
rem 10.3.2 implies that the length of the shortest path, if any, between u and v,
that is, the distance from u to v, will be the smallest value k for which .AG /kuv is
nonzero, and if there is a shortest path, its length will be n 1. Refinements of
this idea lead to methods that find shortest paths in reasonably efficient ways. The
methods apply as well to weighted graphs, where edges are labelled with weights
or costs and the objective is to find least weight, cheapest paths. These refinements
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384 Chapter 10 Directed graphs & Partial Orders

are typically covered in introductory algorithm courses, and we won’t go into them
any further.

10.4 Walk Relations
A basic question about a digraph is whether there is a way to get from one particular
vertex to another. So for any digraph G we are interested in a binary relation G ,
called the walk relation on V .G/, where

u G v WWD there is a walk in G from u to v: (10.6)

Similarly, there is a positive walk relation

u G C v WWD there is a positive length walk in G from u to v: (10.7)

Definition 10.4.1. When there is a walk from vertex v to vertex w, we say that w
is reachable from v, or equivalently, that v is connected to w.

10.4.1 Composition of Relations
There is a simple way to extend composition of functions to composition of rela-
tions, and this gives another way to talk about walks and paths in digraphs.

Definition 10.4.2. Let R W B ! C and S W A ! B be binary relations. Then the
composition of R with S is the binary relation .R ı S / W A ! C defined by the
rule
a .R ı S / c WWD 9b 2 B: .a S b/ AND .b R c/: (10.8)
This agrees with the Definition 4.3.1 of composition in the special case when R
and S are functions.2

Remembering that a digraph is a binary relation on its vertices, it makes sense to
compose a digraph G with itself. Then if we let G n denote the composition of G
with itself n times, it’s easy to check (see Problem 10.11) that G n is the length-n
walk relation:

a Gn b iff there is a length n walk in G from a to b:
2 The reversal of the order of Rand S in (10.8) is not a typo. This is so that relational composition
generalizes function composition. The value of function f composed with function g at an argument
x is f .g.x//. So in the composition f ı g, the function g is applied first.
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10.5. Directed Acyclic Graphs & Scheduling 385

This even works for n D 0, with the usual convention that G 0 is the identity relation
IdV .G/ on the set of vertices.3 Since there is a walk iff there is a path, and every
path is of length at most jV .G/j 1, we now have4

G D G 0 [ G 1 [ G 2 [ : : : [ G jV .G/j 1
D .G [ G 0 /jV .G/j 1
: (10.9)

The final equality points to the use of repeated squaring as a way to compute G
with log n rather than n 1 compositions of relations.

10.5 Directed Acyclic Graphs & Scheduling
Some of the prerequisites of MIT computer science subjects are shown in Fig-
ure 10.6. An edge going from subject s to subject t indicates that s is listed in the
catalogue as a direct prerequisite of t. Of course, before you can take subject t ,
you have to take not only subject s, but also all the prerequisites of s, and any pre-
requisites of those prerequisites, and so on. We can state this precisely in terms of
the positive walk relation: if D is the direct prerequisite relation on subjects, then
subject u has to be completed before taking subject v iff u D C v.
Of course it would take forever to graduate if this direct prerequisite graph had
a positive length closed walk. We need to forbid such closed walks, which by
Lemma 10.2.6 is the same as forbidding cycles. So, the direct prerequisite graph
among subjects had better be acyclic:

Definition 10.5.1. A directed acyclic graph (DAG) is a directed graph with no
cycles.

DAGs have particular importance in computer science. They capture key con-
cepts used in analyzing task scheduling and concurrency control. When distributing
a program across multiple processors, we’re in trouble if one part of the program
needs an output that another part hasn’t generated yet! So let’s examine DAGs and
their connection to scheduling in more depth.
3 The identity relation IdA on a set A is the equality relation:

a IdA b iff a D b;

for a; b 2 A.
4 Equation (10.9) involves a harmless abuse of notation: we should have written

graph.G / D graph.G 0 / [ graph.G 1 / : : : :
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386 Chapter 10 Directed graphs & Partial Orders

New 6-3: SB in Computer Science and Engineering
Subjects
All subjects are 12 units
½+½ 6.UAT
6.UAT 6.UAP
6.UAP
66 units
units 66 units
units

Advanced
Advanced Undergraduate
Undergraduate Subjects
Subjects
2 AUS
AUS http://www.eecs.mit.edu/ug/newcurriculum/aus.html
http://www.eecs.mit.edu/ug/newcurriculum/aus.html
http://www.eecs.mit.edu/ug/newcurriculum/aus.html

1 Software
Software Lab
Lab
((http://www.eecs.mit.edu/ug/newcurriculum/verghese_6.005.html)
http://www.eecs.mit.edu/ug/newcurriculum/verghese_6.005.html)

3 6.033
6.033 6.034
6.034 6.046
6.046
Header comp
comp sys
sys AI
AI adv
adv algorithms
algorithms

3 6.005* 6.006*
Foundation 6.004
6.004 6.005* 6.006*
comp
comp architecture
architecture software
software algorithms
algorithms

2 6.01*
6.01* 6.02*
Introductory 6.02* coreq
(= 1 Institute Lab)
intro
intro EECS
EECS II intro
intro EECS
EECS IIII

18.06 or 18.03

2 coreq
18.06
18.06 18.03
18.03 6.042
6.042
Math
linear
linear algebra
algebra diff
diff eqs
eqs discrete
discrete math
math
(= 2 REST)

8.02
8.02 Elementary
Elementary
exposure
exposure toto programming
programming
June 2009 *new subject (high
(high school, IAP,
school, IAP, or
or 6.00)
6.00)

Figure 10.6 Subject prerequisites for MIT Computer Science (6-3) Majors.
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10.5. Directed Acyclic Graphs & Scheduling 387

left sock right sock underwear shirt

pants tie

left shoe right shoe belt

jacket

Figure 10.7 DAG describing which clothing items have to be put on before oth-
ers.

10.5.1 Scheduling
In a scheduling problem, there is a set of tasks, along with a set of constraints
specifying that starting certain tasks depends on other tasks being completed be-
forehand. We can map these sets to a digraph, with the tasks as the nodes and the
direct prerequisite constraints as the edges.
For example, the DAG in Figure 10.7 describes how a man might get dressed for
a formal occasion. As we describe above, vertices correspond to garments and the
edges specify which garments have to be put on before which others.
When faced with a set of prerequisites like this one, the most basic task is finding
an order in which to perform all the tasks, one at a time, while respecting the
dependency constraints. Ordering tasks in this way is known as topological sorting.
Definition 10.5.2. A topological sort of a finite DAG is a list of all the vertices
such that each vertex v appears earlier in the list than every other vertex reachable
from v.
There are many ways to get dressed one item at a time while obeying the con-
straints of Figure 10.7. We have listed two such topological sorts in Figure 10.8. In
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388 Chapter 10 Directed graphs & Partial Orders

underwear left sock
shirt shirt
pants tie
belt underwear
tie right sock
jacket pants
left sock right shoe
right sock belt
left shoe jacket
right shoe left shoe
(a) (b)

Figure 10.8 Two possible topological sorts of the prerequisites described in Fig-
ure 10.7
.

fact, we can prove that every finite DAG has a topological sort. You can think of
this as a mathematical proof that you can indeed get dressed in the morning.
Topological sorts for finite DAGs are easy to construct by starting from minimal
elements:

Definition 10.5.3. An vertex v of a DAG D is minimum iff every other vertex is
reachable from v.
A vertex v is minimal iff v is not reachable from any other vertex.

It can seem peculiar to use the words “minimum” and “minimal” to talk about
vertices that start paths. These words come from the perspective that a vertex is
“smaller” than any other vertex it connects to. We’ll explore this way of thinking
about DAGs in the next section, but for now we’ll use these terms because they are
conventional.
One peculiarity of this terminology is that a DAG may have no minimum element
but lots of minimal elements. In particular, the clothing example has four minimal
elements: leftsock, rightsock, underwear, and shirt.
To build an order for getting dressed, we pick one of these minimal elements—
say, shirt. Now there is a new set of minimal elements; the three elements we didn’t
chose as step 1 are still minimal, and once we have removed shirt, tie becomes
minimal as well. We pick another minimal element, continuing in this way until all
elements have been picked. The sequence of elements in the order they were picked
will be a topological sort. This is how the topological sorts above were constructed.
So our construction shows:
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10.5. Directed Acyclic Graphs & Scheduling 389

Theorem 10.5.4. Every finite DAG has a topological sort.
There are many other ways of constructing topological sorts. For example, in-
stead of starting from the minimal elements at the beginning of paths, we could
build a topological sort starting from maximal elements at the end of paths. In fact,
we could build a topological sort by picking vertices arbitrarily from a finite DAG
and simply inserting them into the list wherever they will fit.5

10.5.2 Parallel Task Scheduling
For task dependencies, topological sorting provides a way to execute tasks one after
another while respecting those dependencies. But what if we have the ability to
execute more than one task at the same time? For example, say tasks are programs,
the DAG indicates data dependence, and we have a parallel machine with lots of
processors instead of a sequential machine with only one. How should we schedule
the tasks? Our goal should be to minimize the total time to complete all the tasks.
For simplicity, let’s say all the tasks take the same amount of time and all the
processors are identical.
So given a finite set of tasks, how long does it take to do them all in an optimal
parallel schedule? We can use walk relations on acyclic graphs to analyze this
problem.
In the first unit of time, we should do all minimal items, so we would put on our
left sock, our right sock, our underwear, and our shirt.6 In the second unit of time,
we should put on our pants and our tie. Note that we cannot put on our left or right
shoe yet, since we have not yet put on our pants. In the third unit of time, we should
put on our left shoe, our right shoe, and our belt. Finally, in the last unit of time,
we can put on our jacket. This schedule is illustrated in Figure 10.9.
The total time to do these tasks is 4 units. We cannot do better than 4 units of
time because there is a sequence of 4 tasks that must each be done before the next.
We have to put on a shirt before pants, pants before a belt, and a belt before a jacket.
Such a sequence of items is known as a chain.
Definition 10.5.5. Two vertices in a DAG are comparable when one of them is
reachable from the other. A chain in a DAG is a set of vertices such that any two of
them are comparable. A vertex in a chain that is reachable from all other vertices
in the chain is called a maximum element of the chain. A finite chain is said to end
at its maximum element.
5 In fact, the DAG doesn’t even need to be finite, but you’ll be relieved to know that we have no
need to go into this.
6 Yes, we know that you can’t actually put on both socks at once, but imagine you are being dressed

by a bunch of robot processors and you are in a big hurry. Still not working for you? Ok, forget about
the clothes and imagine they are programs with the precedence constraints shown in Figure 10.7.
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390 Chapter 10 Directed graphs & Partial Orders

A1 left sock right sock underwear shirt

A2 pants tie

A3 left shoe right shoe belt

A4 jacket

Figure 10.9 A parallel schedule for the tasks-getting-dressed digraph in Fig-
ure 10.7. The tasks in Ai can be performed in step i for 1 i 4. A chain
of 4 tasks (the critical path in this example) is shown with bold edges.
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10.5. Directed Acyclic Graphs & Scheduling 391

The time it takes to schedule tasks, even with an unlimited number of processors,
is at least as large as the number of vertices in any chain. That’s because if we used
less time than the size of some chain, then two items from the chain would have to
be done at the same step, contradicting the precedence constraints. For this reason,
a largest chain is also known as a critical path. For example, Figure 10.9 shows
the critical path for the getting-dressed digraph.
In this example, we were able to schedule all the tasks with t steps, where t is
the size of the largest chain. A nice feature of DAGs is that this is always possible!
In other words, for any DAG, there is a legal parallel schedule that runs in t total
steps.
In general, a schedule for performing tasks specifies which tasks to do at suc-
cessive steps. Every task a has to be scheduled at some step, and all the tasks that
have to be completed before task a must be scheduled for an earlier step. Here’s a
rigorous definition of schedule.

Definition 10.5.6. A partition of a set A is a set of nonempty subsets of A called
the blocks7 of the partition, such that every element of A is in exactly one block.

For example, one possible partition of the set fa; b; c; d; eg into three blocks is

fa; cg fb; eg fd g:

Definition 10.5.7. A parallel schedule for a DAG D is a partition of V .D/ into
blocks A0 ; A1 ; : : : ; such that when j < k, no vertex in Aj is reachable from any
vertex in Ak . The block Ak is called the set of elements scheduled at step k, and the
time of the schedule is the number of blocks. The maximum number of elements
scheduled at any step is called the number of processors required by the schedule.

A largest chain ending at an element a is called a critical path to a, and the
number of elements less than a in the chain is called the depth of a. So in any
possible parallel schedule, there must be at least depth .a/ steps before task a can
be started. In particular, the minimal elements are precisely the elements with depth
0.
There is a very simple schedule that completes every task in its minimum num-
ber of steps: just use a “greedy” strategy of performing tasks as soon as possible.
Schedule all the elements of depth k at step k. That’s how we found the above
schedule for getting dressed.

7 We think it would be nicer to call them the parts of the partition, but “blocks” is the standard

terminology.
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392 Chapter 10 Directed graphs & Partial Orders

Theorem 10.5.8. A minimum time schedule for a finite DAG D consists of the sets
A0 ; A1 ; : : : ; where

Ak WWD fa 2 V .D/ j depth .a/ D kg:

We’ll leave to Problem 10.24 the proof that the sets Ak are a parallel schedule
according to Definition 10.5.7. We can summarize the story above in this way: with
an unlimited number of processors, the parallel time to complete all tasks is simply
the size of a critical path:

Corollary 10.5.9. Parallel time = size of critical path.

Things get more complex when the number of processors is bounded; see Prob-
lem 10.25 for an example.

10.5.3 Dilworth’s Lemma
Definition 10.5.10. An antichain in a DAG is a set of vertices such that no two ele-
ments in the set are comparable—no walk exists between any two different vertices
in the set.

Our conclusions about scheduling also tell us something about antichains.

Corollary 10.5.11. In a DAG D if the size of the largest chain is t , then V .D/ can
be partitioned into t antichains.

Proof. Let the antichains be the sets Ak WWD fa 2 V .D/ j depth .a/ D kg. It is an
easy exercise to verify that each Ak is an antichain (Problem 10.24).

Corollary 10.5.11 implies8 a famous result about acyclic digraphs:

Lemma 10.5.12 (Dilworth). For all t > 0, every DAG with n vertices must have
either a chain of size greater than t or an antichain of size at least n=t.

Proof. Assume that there is no chain of size greater than t . Let ` be the size of
the largest antichain. If we make a parallel schedule according to the proof of
Corollary 10.5.11, we create a number of antichains equal to the size of the largest
chain, which is less than or equal t . Each element belongs to exactly one antichain,
none of which are larger than `. So the total number of elements at most ` times
t—that is, `t n. Simple division implies that ` n=t.
8 Lemma 10.5.12 also follows from a more general result known as Dilworth’s Theorem, which

we will not discuss.
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10.6. Partial Orders 393

p
Corollary 10.5.13. Every DAG with n vertices has a chain of size greater than n
p
or an antichain of size at least n.
p
Proof. Set t D n in Lemma 10.5.12.

Example 10.5.14. When the man in our example is getting dressed, n D 10.
Try t D 3. There is a chain of size 4.
Try t D 4. There is no chain of size 5, but there is an antichain of size 4 10=4.

10.6 Partial Orders
After mapping the “direct prerequisite” relation onto a digraph, we were then able
to use the tools for understanding computer scientists’ graphs to make deductions
about something as mundane as getting dressed. This may or may not have im-
pressed you, but we can do better. In the introduction to this chapter, we mentioned
a useful fact that bears repeating: any digraph is formally the same as a binary
relation whose domain and codomain are its vertices. This means that any binary
relation whose domain is the same as its codomain can be translated into a digraph!
Talking about the edges of a binary relation or the image of a set under a digraph
may seem odd at first, but doing so will allow us to draw important connections
between different types of relations. For instance, we can apply Dilworth’s lemma
to the “direct prerequisite” relation for getting dressed, because the graph of that
relation was a DAG.
But how can we tell if a binary relation is a DAG? And once we know that a
relation is a DAG, what exactly can we conclude? In this section, we will abstract
some of the properties that a binary relation might have, and use those properties
to define classes of relations. In particular, we’ll explain this section’s title, partial
orders.

10.6.1 The Properties of the Walk Relation in DAGs
To begin, let’s talk about some features common to all digraphs. Since merging a
walk from u to v with a walk from v to w gives a walk from u to w, both the walk
and positive walk relations have a relational property called transitivity:

Definition 10.6.1. A binary relation R on a set A is transitive iff

.a R b AND b R c/ IMPLIES a R c

for every a; b; c 2 A.
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394 Chapter 10 Directed graphs & Partial Orders

So we have

Lemma 10.6.2. For any digraph G the walk relations G C and G are transitive.

Since there is a length zero walk from any vertex to itself, the walk relation has
another relational property called reflexivity:

Definition 10.6.3. A binary relation R on a set A is reflexive iff a R a for all a 2 A.

Now we have

Lemma 10.6.4. For any digraph G, the walk relation G is reflexive.

We know that a digraph is a DAG iff it has no positive length closed walks. Since
any vertex on a closed walk can serve as the beginning and end of the walk, saying
a graph is a DAG is the same as saying that there is no positive length path from
any vertex back to itself. This means that the positive walk relation of D C of a
DAG has a relational property called irreflexivity.

Definition 10.6.5. A binary relation R on a set A is irreflexive iff

NOT .a R a/

for all a 2 A.

So we have

Lemma 10.6.6. R is a DAG iff RC is irreflexive.

10.6.2 Strict Partial Orders
Here is where we begin to define interesting classes of relations:

Definition 10.6.7. A relation that is transitive and irreflexive is called a strict par-
tial order.

A simple connection between strict partial orders and DAGs now follows from
Lemma 10.6.6:

Theorem 10.6.8. A relation R is a strict partial order iff R is the positive walk
relation of a DAG.

Strict partial orders come up in many situations which on the face of it have
nothing to do with digraphs. For example, the less-than order < on numbers is a
strict partial order:
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10.6. Partial Orders 395

if x < y and y < z then x < z, so less-than is transitive, and

NOT.x < x/, so less-than is irreflexive.
The proper containment relation is also a partial order:
if A B and B C then A C , so containment is transitive, and

NOT.A A/, so proper containment is irreflexive.
If there are two vertices that are reachable from each other, then there is a posi-
tive length closed walk that starts at one vertex, goes to the other, and then comes
back. So DAGs are digraphs in which no two vertices are mutually reachable. This
corresponds to a relational property called asymmetry.
Definition 10.6.9. A binary relation R on a set A is asymmetric iff

a R b IMPLIES NOT.b R a/

for all a; b 2 A.
So we can also characterize DAGs in terms of asymmetry:
Corollary 10.6.10. A digraph D is a DAG iff D C is asymmetric.
Corollary 10.6.10 and Theorem 10.6.8 combine to give
Corollary 10.6.11. A binary relation R on a set A is a strict partial order iff it is
transitive and asymmetric.9
A strict partial order may be the positive walk relation of different DAGs. This
raises the question of finding a DAG with the smallest number of edges that deter-
mines a given strict partial order. For finite strict partial orders, the smallest such
DAG turns out to be unique and easy to find (see Problem 10.30).

10.6.3 Weak Partial Orders
The less-than-or-equal relation is at least as familiar as the less-than strict partial
order, and the ordinary containment relation is even more common than the
proper containment relation. These are examples of weak partial orders, which are
just strict partial orders with the additional condition that every element is related
to itself. To state this precisely, we have to relax the asymmetry property so it
does not apply when a vertex is compared to itself; this relaxed property is called
antisymmetry:
9 Some texts use this corollary to define strict partial orders.
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396 Chapter 10 Directed graphs & Partial Orders

Definition 10.6.12. A binary relation R on a set A, is antisymmetric iff, for all
a ¤ b 2 A,
a R b IMPLIES NOT.b R a/
Now we can give an axiomatic definition of weak partial orders that parallels the
definition of strict partial orders.
Definition 10.6.13. A binary relation on a set is a weak partial order iff it is tran-
sitive, reflexive, and antisymmetric.
The following lemma gives another characterization of weak partial orders that
follows directly from this definition.
Lemma 10.6.14. A relation R on a set A is a weak partial order iff there is a strict
partial order S on A such that

aRb iff .a S b OR a D b/;

for all a; b 2 A.
Since a length zero walk goes from a vertex to itself, this lemma combined with
Theorem 10.6.8 yields:
Corollary 10.6.15. A relation is a weak partial order iff it is the walk relation of a
DAG.
For weak partial orders in general, we often write an ordering-style symbol like
or v instead of a letter symbol like R.10 Likewise, we generally use or @ to
indicate a strict partial order.
Two more examples of partial orders are worth mentioning:
Example 10.6.16. Let A be some family of sets and define a R b iff a b. Then
R is a strict partial order.
Example 10.6.17. The divisibility relation is a weak partial order on the nonnega-
tive integers.
For practice with the definitions, you can check that two more examples are
vacuously partial orders on a set D: the identity relation IdD is a weak partial
order, and the empty relation—the relation with no arrows—is a strict partial order.
Note that some authors define “partial orders” to be what we call weak partial
orders. However, we’ll use the phrase “partial order” to mean a relation that may
be either a weak or strict partial order.
10 General relations are usually denoted by a letter like R instead of a cryptic squiggly symbol, so

is kind of like the musical performer/composer Prince, who redefined the spelling of his name to
be his own squiggly symbol. A few years ago he gave up and went back to the spelling “Prince.”
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10.7. Representing Partial Orders by Set Containment 397

10.7 Representing Partial Orders by Set Containment
Axioms can be a great way to abstract and reason about important properties of
objects, but it helps to have a clear picture of the things that satisfy the axioms.
DAGs provide one way to picture partial orders, but it also can help to picture them
in terms of other familiar mathematical objects. In this section, we’ll show that
every partial order can be pictured as a collection of sets related by containment.
That is, every partial order has the “same shape” as such a collection. The technical
word for “same shape” is “isomorphic.”

Definition 10.7.1. A binary relation R on a set A is isomorphic to a relation S on
a set B iff there is a relation-preserving bijection from A to B; that is, there is a
bijection f W A ! B such that for all a; a0 2 A,

a R a0 iff f .a/ S f .a0 /:

To picture a partial order on a set A as a collection of sets, we simply represent
each element A by the set of elements that are to that element, that is,

a ! fb 2 A j b ag:

For example, if is the divisibility relation on the set of integers f1; 3; 4; 6; 8; 12g,
then we represent each of these integers by the set of integers in A that divides it.
So

1 ! f1g
3 ! f1; 3g
4 ! f1; 4g
6 ! f1; 3; 6g
8 ! f1; 4; 8g
12 ! f1; 3; 4; 6; 12g

So, the fact that 3 j 12 corresponds to the fact that f1; 3g f1; 3; 4; 6; 12g.
In this way we have completely captured the weak partial order by the subset
relation on the corresponding sets. Formally, we have

Lemma 10.7.2. Let be a weak partial order on a set A. Then is isomorphic
to the subset relation on the collection of inverse images under the relation of
elements a 2 A.
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398 Chapter 10 Directed graphs & Partial Orders

We leave the proof to Problem 10.36. Essentially the same construction shows
that strict partial orders can be represented by sets under the proper subset relation,
(Problem 10.37). To summarize:

Theorem 10.7.3. Every weak partial order is isomorphic to the subset relation
on a collection of sets.
Every strict partial order is isomorphic to the proper subset relation on a
collection of sets.

10.8 Linear Orders
The familiar order relations on numbers have an important additional property:
given two different numbers, one will be bigger than the other. Partial orders with
this property are said to be linear orders. You can think of a linear order as one
where all the elements are lined up so that everyone knows exactly who is ahead
and who is behind them in the line.11

Definition 10.8.1. Let R be a binary relation on a set A and let a; b be elements
of A. Then a and b are comparable with respect to R iff Œa R b OR b R a.
A partial order for which every two different elements are comparable is called a
linear order.

So < and are linear orders on R. On the other hand, the subset relation is
not linear, since, for example, any two different finite sets of the same size will be
incomparable under . The prerequisite relation on Course 6 required subjects is
also not linear because, for example, neither 8.01 nor 6.042 is a prerequisite of the
other.

10.9 Product Orders
Taking the product of two relations is a useful way to construct new relations from
old ones.
11 Linear orders are often called “total” orders, but this terminology conflicts with the definition of

“total relation,” and it regularly confuses students.
Being a linear order is a much stronger condition than being a partial order that is a total relation.
For example, any weak partial order is a total relation but generally won’t be linear.
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10.10. Equivalence Relations 399

Definition 10.9.1. The product R1 R2 of relations R1 and R2 is defined to be
the relation with

domain.R1 R2 / WWD domain.R1 / domain.R2 /;
codomain.R1 R2 / WWD codomain.R1 / codomain.R2 /;
.a1 ; a2 / .R1 R2 / .b1 ; b2 / iff Œa1 R1 b1 and a2 R2 b2 :

It follows directly from the definitions that products preserve the properties of
transitivity, reflexivity, irreflexivity, and antisymmetry (see Problem 10.50). If R1
and R2 both have one of these properties, then so does R1 R2 . This implies that
if R1 and R2 are both partial orders, then so is R1 R2 .
Example 10.9.2. Define a relation Y on age-height pairs of being younger and
shorter. This is the relation on the set of pairs .y; h/ where y is a nonnegative
integer 2400 that we interpret as an age in months, and h is a nonnegative integer
120 describing height in inches. We define Y by the rule

.y1 ; h1 / Y .y2 ; h2 / iff y1 y2 AND h1 h2 :

That is, Y is the product of the -relation on ages and the -relation on heights.
Since both ages and heights are ordered numerically, the age-height relation Y is
a partial order. Now suppose we have a class of 101 students. Then we can apply
Dilworth’s lemma 10.5.12 to conclude that there is a chain of 11 students—that is,
11 students who get taller as they get older–or an antichain of 11 students—that is,
11 students who get taller as they get younger, which makes for an amusing in-class
demo.
On the other hand, the property of being a linear order is not preserved. For
example, the age-height relation Y is the product of two linear orders, but it is not
linear: the age 240 months, height 68 inches pair, (240,68), and the pair (228,72)
are incomparable under Y .

10.10 Equivalence Relations
Definition 10.10.1. A relation is an equivalence relation if it is reflexive, symmet-
ric, and transitive.

Congruence modulo n is an important example of an equivalence relation:

It is reflexive because x x .mod n/.
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400 Chapter 10 Directed graphs & Partial Orders

It is symmetric because x y .mod n/ implies y x .mod n/.

It is transitive because x y .mod n/ and y z .mod n/ imply that x z
.mod n/.

There is an even more well-known example of an equivalence relation: equality
itself.
Any total function defines an equivalence relation on its domain:

Definition 10.10.2. If f W A ! B is a total function, define a relation f by the
rule:
a f a0 IFF f .a/ D f .a0 /:

From its definition, f is reflexive, symmetric and transitive because these are
properties of equality. That is, f is an equivalence relation. This observation
gives another way to see that congruence modulo n is an equivalence relation:
the Remainder Lemma 9.6.1 implies that congruence modulo n is the same as r
where r.a/ is the remainder of a divided by n.
In fact, a relation is an equivalence relation iff it equals f for some total func-
tion f (see Problem 10.56). So equivalence relations could have been defined using
Definition 10.10.2.

10.10.1 Equivalence Classes
Equivalence relations are closely related to partitions because the images of ele-
ments under an equivalence relation are the blocks of a partition.

Definition 10.10.3. Given an equivalence relation R W A ! A, the equivalence
class ŒaR of an element a 2 A is the set of all elements of A related to a by R.
Namely,
ŒaR WWD fx 2 A j a R xg:

In other words, ŒaR is the image R.a/.
For example, suppose that A D Z and a R b means that a b .mod 5/. Then

Œ7R D f: : : ; 3; 2; 7; 12; 22; : : :g:

Notice that 7, 12, 17, etc., all have the same equivalence class; that is, Œ7R D
Œ12R D Œ17R D .
There is an exact correspondence between equivalence relations on A and parti-
tions of A. Namely, given any partition of a set, being in the same block is obviously
an equivalence relation. On the other hand we have:
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10.11. Summary of Relational Properties 401

Theorem 10.10.4. The equivalence classes of an equivalence relation on a set A
are the blocks of a partition of A.
We’ll leave the proof of Theorem 10.10.4 as a basic exercise in axiomatic rea-
soning (see Problem 10.55), but let’s look at an example. The congruent-mod-5
relation partitions the integers into five equivalence classes:
f: : : ; 5; 0; 5; 10; 15; 20; : : :g
f: : : ; 4; 1; 6; 11; 16; 21; : : :g
f: : : ; 3; 2; 7; 12; 17; 22; : : :g
f: : : ; 2; 3; 8; 13; 18; 23; : : :g
f: : : ; 1; 4; 9; 14; 19; 24; : : :g
In these terms, x y .mod 5/ is equivalent to the assertion that x and y are both
in the same block of this partition. For example, 6 16 .mod 5/, because they’re
both in the second block, but 2 6 9 .mod 5/ because 2 is in the third block while
9 is in the last block.
In social terms, if “likes” were an equivalence relation, then everyone would be
partitioned into cliques of friends who all like each other and no one else.

10.11 Summary of Relational Properties
A relation R W A ! A is the same as a digraph with vertices A.
Reflexivity R is reflexive when
8x 2 A: x R x:

Every vertex in R has a self-loop.
Irreflexivity R is irreflexive when
NOT Œ9x 2 A: x R x:

There are no self-loops in R.
Symmetry R is symmetric when
8x; y 2 A: x R y IMPLIES y R x:

If there is an edge from x to y in R, then there is an edge back from y to x
as well.
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402 Chapter 10 Directed graphs & Partial Orders

Asymmetry R is asymmetric when

8x; y 2 A: x R y IMPLIES NOT.y R x/:

There is at most one directed edge between any two vertices in R, and there
are no self-loops.

Antisymmetry R is antisymmetric when

8x ¤ y 2 A: x R y IMPLIES NOT.y R x/:

Equivalently,

8x; y 2 A: .x R y AND y R x/ IMPLIES x D y:

There is at most one directed edge between any two distinct vertices, but
there may be self-loops.

Transitivity R is transitive when

8x; y; z 2 A: .x R y AND y R z/ IMPLIES x R z:

If there is a positive length path from u to v, then there is an edge from u
to v.

Linear R is linear when

8x ¤ y 2 A: .x R y OR y R x/

Given any two vertices in R, there is an edge in one direction or the other
between them.

Strict Partial Order R is a strict partial order iff R is transitive and irreflexive iff
R is transitive and asymmetric iff it is the positive length walk relation of a
DAG.

Weak Partial Order R is a weak partial order iff R is transitive and anti-symmetric
and reflexive iff R is the walk relation of a DAG.

Equivalence Relation R is an equivalence relation iff R is reflexive, symmetric
and transitive iff R equals the in-the-same-block-relation for some partition
of domain.R/.
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10.11. Summary of Relational Properties 403

Problems for Section 10.1
Practice Problems
Problem 10.1.
Let S be a nonempty set of size n 2 ZC , and let f W S ! S be total function. Let
Df be the digraph with vertices S whose edges are fhs ! f .s/i j s 2 S g.
(a) What are the possible values of the out-degrees of vertices of Df ?

(b) What are the possible values of the in-degrees of the vertices?

Exam Problems
Problem 10.2.
The proof of the Handshaking Lemma 10.1.2 invoked the “obvious” fact that in any
finite digraph, the sum of the in-degrees of the vertices equals the number of arrows
in the graph. That is,

Claim. For any finite digraph G
X
indeg.v/ D j graph.G/j; (10.10)
v2V .G/

But this Claim might not be obvious to everyone. So prove it by induction on the
number j graph.G/j of arrows.

Problems for Section 10.2
Practice Problems
Problem 10.3.
Lemma 10.2.5 states that dist .u; v/ dist .u; x/ C dist .x; v/. It also states that
equality holds iff x is on a shortest path from u to v.
(a) Prove the “iff” statement from left to right.

(b) Prove the “iff” from right to left.
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404 Chapter 10 Directed graphs & Partial Orders

Class Problems
Problem 10.4. (a) Give an example of a digraph that has a closed walk including
two vertices but has no cycle including those vertices.

(b) Prove Lemma 10.2.6:
Lemma. The shortest positive length closed walk through a vertex is a cycle.

Problem 10.5.
A 3-bit string is a string made up of 3 characters, each a 0 or a 1. Suppose you’d
like to write out, in one string, all eight of the 3-bit strings in any convenient order.
For example, if you wrote out the 3-bit strings in the usual order starting with 000
001 010. . . , you could concatenate them together to get a length 3 8 D 24 string
that started 000001010. . . .
But you can get a shorter string containing all eight 3-bit strings by starting with
00010. . . . Now 000 is present as bits 1 through 3, and 001 is present as bits 2
through 4, and 010 is present as bits 3 through 5, . . . .
(a) Say a string is 3-good if it contains every 3-bit string as 3 consecutive bits
somewhere in it. Find a 3-good string of length 10, and explain why this is the
minimum length for any string that is 3-good.

(b) Explain how any walk that includes every edge in the graph shown in Fig-
ure 10.10 determines a string that is 3-good. Find the walk in this graph that deter-
mines your 3-good string from part (a).

(c) Explain why a walk in the graph of Figure 10.10 that includes every every
edge exactly once provides a minimum-length 3-good string.12

(d) Generalize the 2-bit graph to a k-bit digraph Bk for k 2, where V .Bk / WWD
f0; 1gk , and any walk through Bk that contains every edge exactly once determines
a minimum length .k C 1/-good bit-string.13
What is this minimum length?
Define the transitions of Bk . Verify that the in-degree of each vertex is the same as
its out-degree and that there is a positive path from any vertex to any other vertex
(including itself) of length at most k.
12 The 3-good strings explained here generalize to n-good strings for n 3. They were studied by
the great Dutch mathematician/logician Nicolaas de Bruijn, and are known as de Bruijn sequences.
de Bruijn died in February, 2012 at the age of 94.
13 Problem 10.7 explains why such “Eulerian” paths exist.
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10.11. Summary of Relational Properties 405

10 +0

11
+1

+0
+1
+0

00
01
+0 +1

Figure 10.10 The 2-bit graph.

Homework Problems
Problem 10.6. (a) Give an example of a digraph in which a vertex v is on a positive
even-length closed walk, but no vertex is on an even-length cycle.

(b) Give an example of a digraph in which a vertex v is on an odd-length closed
walk but not on an odd-length cycle.

Problem 10.7.
An Euler tour14 of a graph is a closed walk that includes every edge exactly once.
Such walks are named after the famous 17th century mathematician Leonhard Eu-
ler. (Same Euler as for the constant e 2:718 and the totient function —he did a
lot of stuff.)
So how do you tell in general whether a graph has an Euler tour? At first glance
this may seem like a daunting problem (the similar sounding problem of finding
a cycle that touches every vertex exactly once is one of those million dollar NP-
14 In some other texts, this is called an Euler circuit.
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406 Chapter 10 Directed graphs & Partial Orders

complete problems known as the Hamiltonian Cycle Problem)—but it turns out to
be easy.
(a) Show that if a graph has an Euler tour, then the in-degree of each vertex equals
its out-degree.
A digraph is weakly connected if there is a “path” between any two vertices that
may follow edges backwards or forwards.15 In the remaining parts, we’ll work out
the converse. Suppose a graph is weakly connected, and the in-degree of every
vertex equals its out-degree. We will show that the graph has an Euler tour.
A trail is a walk in which each edge occurs at most once.
(b) Suppose that a trail in a weakly connected graph does not include every edge.
Explain why there must be an edge not on the trail that starts or ends at a vertex on
the trail.
In the remaining parts, assume the graph is weakly connected, and the in-degree
of every vertex equals its out-degree. Let w be the longest trail in the graph.
(c) Show that if w is closed, then it must be an Euler tour.
Hint: part (b)

(d) Explain why all the edges starting at the end of w must be on w.

(e) Show that if w was not closed, then the in-degree of the end would be bigger
than its out-degree.
Hint: part (d)

(f) Conclude that if the in-degree of every vertex equals its out-degree in a finite,
weakly connected digraph, then the digraph has an Euler tour.

Problems for Section 10.3
Homework Problems
Problem 10.8.
The weight of a walk in a weighted graph is the sum of the weights of the successive
15 More precisely, a graph G is weakly connected iff there is a path from any vertex to any other

vertex in the graph H with
V .H / D V .G/; and
E.H / D E.G/ [ fhv ! ui j hu ! vi 2 E.G/g:

In other words H D G [ G 1.
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10.11. Summary of Relational Properties 407

edges in the walk. The minimum weight matrix for length k walks in an n-vertex
graph G is the n n matrix W such that for u; v 2 V .G/,
(
w if w is the minimum weight among length k walks from u to v;
Wuv WWD
1 if there is no length k walk from u to v:

The min+ product of two n n matrices W and M with entries in R [ f1g is the
n n matrix W M whose ij entry is
min+

.W V /ij WWD minfWi k C Vkj j 1 k ng :
min+

Prove the following theorem.

Theorem. If W is the minimum weight matrix for length k walks in a weighted
graph G, and V is the minimum weight matrix for length m walks, then W V is
min+
the minimum weight matrix for length k C m walks.

Problems for Section 10.4
Practice Problems
Problem 10.9.
Let

A WWD f1; 2; 3g
B WWD f4; 5; 6g
R WWD f.1; 4/; .1; 5/; .2; 5/; .3; 6/g
S WWD f.4; 5/; .4; 6/; .5; 4/g:

Note that R is a relation from A to B and S is a relation from B to B.
List the pairs in each of the relations below.
(a) S ı R.

(b) S ı S .

408 Chapter 10 Directed graphs & Partial Orders

Problem 10.10.
In a round-robin tournament, every two distinct players play against each other
just once. For a round-robin tournament with no tied games, a record of who beat
whom can be described with a tournament digraph, where the vertices correspond
to players and there is an edge hx ! yi iff x beat y in their game.
A ranking is a path that includes all the players. So in a ranking, each player won
the game against the next lowest ranked player, but may very well have lost their
games against much lower ranked players—whoever does the ranking may have a
lot of room to play favorites.
(a) Give an example of a tournament digraph with more than one ranking.

(b) Prove that if a tournament digraph is a DAG, then it has at most one ranking.

(c) Prove that every finite tournament digraph has a ranking.
Optional
(d) Prove that the greater-than relation > on the rational numbers Q is a DAG and
a tournament graph that has no ranking.

Homework Problems
Problem 10.11.
Let R be a binary relation on a set A. Regarding R as a digraph, let W .n/ denote
the length-n walk relation in the digraph R, that is,

a W .n/ b WWD there is a length n walk from a to b in R:

(a) Prove that
W .n/ ı W .m/ D W .mCn/ (10.11)
for all m; n 2 N, where ı denotes relational composition.

(b) Let Rn be the composition of R with itself n times for n 0. So R0 WWD IdA ,
and RnC1 WWD R ı Rn .
Conclude that
Rn D W .n/ (10.12)
for all n 2 N.

10.11. Summary of Relational Properties 409

where RC is the positive length walk relation determined by R on the set A.

Problem 10.12.
We can represent a relation S between two sets A D fa1 ; : : : ; an g and B D
fb1 ; : : : ; bm g as an n m matrix MS of zeroes and ones, with the elements of
MS defined by the rule

MS .i; j / D 1 IFF ai S bj :

If we represent relations as matrices this way, then we can compute the com-
position of two relations R and S by a “boolean” matrix multiplication ˝ of their
matrices. Boolean matrix multiplication is the same as matrix multiplication except
that addition is replaced by OR, multiplication is replaced by AND, and 0 and 1 are
used as the Boolean values False and True. Namely, suppose R W B ! C is a bi-
nary relation with C D fc1 ; : : : ; cp g. So MR is an m p matrix. Then MS ˝ MR
is an n p matrix defined by the rule:

ŒMS ˝ MR .i; j / WWD ORm
kD1 ŒMS .i; k/ AND MR .k; j /: (10.13)

Prove that the matrix representation MRıS of R ı S equals MS ˝ MR (note the
reversal of R and S).

Problem 10.13.
Suppose that there are n chickens in a farmyard. Chickens are rather aggressive
birds that tend to establish dominance in relationships by pecking; hence the term
“pecking order.” In particular, for each pair of distinct chickens, either the first
pecks the second or the second pecks the first, but not both. We say that chicken u
virtually pecks chicken v if either:
Chicken u directly pecks chicken v, or

Chicken u pecks some other chicken w who in turn pecks chicken v.
A chicken that virtually pecks every other chicken is called a king chicken.
We can model this situation with a chicken digraph whose vertices are chickens
with an edge from chicken u to chicken v precisely when u pecks v. In the graph
in Figure 10.11, three of the four chickens are kings. Chicken c is not a king in
this example since it does not peck chicken b and it does not peck any chicken that
pecks chicken b. Chicken a is a king since it pecks chicken d , who in turn pecks
chickens b and c.
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410 Chapter 10 Directed graphs & Partial Orders

a b
king king

king not a king
d c

Figure 10.11 A 4-chicken tournament in which chickens a, b and d are kings.
.

In general, a tournament digraph is a digraph with exactly one edge between
each pair of distinct vertices.
(a) Define a 10-chicken tournament graph with a king chicken that has outdegree
1.

(b) Describe a 5-chicken tournament graph in which every player is a king.

(c) Prove
Theorem (King Chicken Theorem). Any chicken with maximum outdegree in a
tournament is a king.
The King Chicken Theorem means that if the player with the most victories is
defeated by another player x, then at least he/she defeats some third player that
defeats x. In this sense, the player with the most victories has some sort of bragging
rights over every other player. Unfortunately, as Figure 10.11 illustrates, there can
be many other players with such bragging rights, even some with fewer victories.

Problems for Section 10.5
Practice Problems
Problem 10.14.
What is the size of the longest chain that is guaranteed to exist in any partially
ordered set of n elements? What about the largest antichain?

Problem 10.15.
Let fA; :::; H g be a set of tasks that we must complete. The following DAG de-
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10.11. Summary of Relational Properties 411

scribes which tasks must be done before others, where there is an arrow from S to
T iff S must be done before T .

(a) Write the longest chain.

(b) Write the longest antichain.

(c) If we allow parallel scheduling, and each task takes 1 minute to complete,
what is the minimum amount of time needed to complete all tasks?

Problem 10.16.
Describe a sequence consisting of the integers from 1 to 10,000 in some order so
that there is no increasing or decreasing subsequence of size 101.

Problem 10.17.
What is the smallest number of partially ordered tasks for which there can be more
than one minimum time schedule, if there are unlimited number of processors?
Explain your answer.

Problem 10.18.
The following DAG describes the prerequisites among tasks f1; : : : ; 9g.
(a) If each task takes unit time to complete, what is the minimum parallel time to
complete all the tasks? Briefly explain.
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412 Chapter 10 Directed graphs & Partial Orders

3 5 7

2 4 6

(b) What is the minimum parallel time if no more than two tasks can be completed
in parallel? Briefly explain.

Problem 10.19.
The following DAG describes the prerequisites among tasks f1; : : : ; 9g.

8 9

3 5 7

2 4 6

1
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10.11. Summary of Relational Properties 413

(a) If each task takes unit time to complete, what is the minimum parallel time to
complete all the tasks? Briefly explain.

(b) What is the minimum parallel time if no more than two tasks can be completed
in parallel? Briefly explain.

Class Problems
Problem 10.20.
The table below lists some prerequisite information for some subjects in the MIT
Computer Science program (in 2006). This defines an indirect prerequisite relation
that is a DAG with these subjects as vertices.

18:01 ! 6:042 18:01 ! 18:02
18:01 ! 18:03 6:046 ! 6:840
8:01 ! 8:02 6:001 ! 6:034
6:042 ! 6:046 18:03; 8:02 ! 6:002
6:001; 6:002 ! 6:003 6:001; 6:002 ! 6:004
6:004 ! 6:033 6:033 ! 6:857

(a) Explain why exactly six terms are required to finish all these subjects, if you
can take as many subjects as you want per term. Using a greedy subject selection
strategy, you should take as many subjects as possible each term. Exhibit your
complete class schedule each term using a greedy strategy.

(b) In the second term of the greedy schedule, you took five subjects including
18.03. Identify a set of five subjects not including 18.03 such that it would be
possible to take them in any one term (using some nongreedy schedule). Can you
figure out how many such sets there are?

(d) Suppose that you want to take all of the subjects, but can handle only two per
term. Exactly how many terms are required to graduate? Explain why.

(e) What if you could take three subjects per term?
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414 Chapter 10 Directed graphs & Partial Orders

Problem 10.21.
A pair of Math for Computer Science Teaching Assistants, Lisa and Annie, have
decided to devote some of their spare time this term to establishing dominion over
the entire galaxy. Recognizing this as an ambitious project, they worked out the
following table of tasks on the back of Annie’s copy of the lecture notes.

1. Devise a logo and cool imperial theme music - 8 days.

2. Build a fleet of Hyperwarp Stardestroyers out of eating paraphernalia swiped
from Lobdell - 18 days.

3. Seize control of the United Nations - 9 days, after task #1.

4. Get shots for Lisa’s cat, Tailspin - 11 days, after task #1.

5. Open a Starbucks chain for the army to get their caffeine - 10 days, after
task #3.

6. Train an army of elite interstellar warriors by dragging people to see The
Phantom Menace dozens of times - 4 days, after tasks #3, #4, and #5.

7. Launch the fleet of Stardestroyers, crush all sentient alien species, and es-
tablish a Galactic Empire - 6 days, after tasks #2 and #6.

8. Defeat Microsoft - 8 days, after tasks #2 and #6.

We picture this information in Figure 10.12 below by drawing a point for each
task, and labelling it with the name and weight of the task. An edge between
two points indicates that the task for the higher point must be completed before
beginning the task for the lower one.
(a) Give some valid order in which the tasks might be completed.
Lisa and Annie want to complete all these tasks in the shortest possible time.
However, they have agreed on some constraining work rules.

Only one person can be assigned to a particular task; they cannot work to-
gether on a single task.

Once a person is assigned to a task, that person must work exclusively on
the assignment until it is completed. So, for example, Lisa cannot work on
building a fleet for a few days, run to get shots for Tailspin, and then return
to building the fleet.
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10.11. Summary of Relational Properties 415

devise logo build fleet
u8 u 18
A E
A E
A E
A E
seize control u9 A uget shots
E
11
B E
B E
E
B
B E
open chain u B
E
10 QQ B
E
Q B E
Q E
Q B 4
army B uP
QQ B E
train QPP E
Q PP E
Q PP E
Q PP
Q PP E
u
Q PP Pu EE defeat
6Q Microsoft
launch fleet 8

Figure 10.12 Graph representing the task precedence constraints.

(b) Lisa and Annie want to know how long conquering the galaxy will take. Annie
suggests dividing the total number of days of work by the number of workers, which
is two. What lower bound on the time to conquer the galaxy does this give, and why
might the actual time required be greater?

(c) Lisa proposes a different method for determining the duration of their project.
She suggests looking at the duration of the critical path, the most time-consuming
sequence of tasks such that each depends on the one before. What lower bound
does this give, and why might it also be too low?

(d) What is the minimum number of days that Lisa and Annie need to conquer the
galaxy? No proof is required.

Problem 10.22.
Answer the following questions about the powerset pow.f1; 2; 3; 4g/ partially or-
dered by the strict subset relation .
(a) Give an example of a maximum length chain.
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416 Chapter 10 Directed graphs & Partial Orders

(b) Give an example of an antchain of size 6.

(d) Suppose the partial order describes scheduling constraints on 16 tasks. That
is, if
A B f1; 2; 3; 4g;
then A has to be completed before B starts.16 What is the minimum number of
processors needed to complete all the tasks in minimum parallel time?
Prove it.

(e) What is the length of a minimum time 3-processor schedule?
Prove it.

Homework Problems
Problem 10.23.
The following operations can be applied to any digraph, G:
1. Delete an edge that is in a cycle.

2. Delete edge hu ! vi if there is a path from vertex u to vertex v that does not
include hu ! vi.

3. Add edge hu ! vi if there is no path in either direction between vertex u and
vertex v.
The procedure of repeating these operations until none of them are applicable can
be modeled as a state machine. The start state is G, and the states are all possible
digraphs with the same vertices as G.
(a) Let G be the graph with vertices f1; 2; 3; 4g and edges

fh1 ! 2i ; h2 ! 3i ; h3 ! 4i ; h3 ! 2i ; h1 ! 4ig

What are the possible final states reachable from G?
A line graph is a graph whose edges are all on one path. All the final graphs in
part (a) are line graphs.
(b) Prove that if the procedure terminates with a digraph H then H is a line graph
with the same vertices as G.
Hint: Show that if H is not a line graph, then some operation must be applicable.
16 As usual, we assume each task requires one time unit to complete.
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10.11. Summary of Relational Properties 417

(d) Prove that if G is a DAG and the procedure terminates, then the walk relation
of the final line graph is a topological sort of G.
Hint: Verify that the predicate
P .u; v/ WWD there is a directed path from u to v
is a preserved invariant of the procedure, for any two vertices u; v of a DAG.

(e) Prove that if G is finite, then the procedure terminates.
Hint: Let s be the number of cycles, e be the number of edges, and p be the number
of pairs of vertices with a directed path (in either direction) between them. Note
that p n2 where n is the number of vertices of G. Find coefficients a; b; c such
that as C bp C e C c is nonnegative integer valued and decreases at each transition.

Problem 10.24.
Let be a strict partial order on a set A and let
Ak WWD fa j depth .a/ D kg
where k 2 N.
(a) Prove that A0 ; A1 ; : : : is a parallel schedule for according to Definition 10.5.7.

(b) Prove that Ak is an antichain.

Problem 10.25.
We want to schedule n tasks with prerequisite constraints among the tasks defined
by a DAG.
(a) Explain why any schedule that requires only p processors must take time at
least dn=pe.

(b) Let Dn;t be the DAG with n elements that consists of a chain of t 1 elements,
with the bottom element in the chain being a prerequisite of all the remaining ele-
ments as in the following figure:
What is the minimum time schedule for Dn;t ? Explain why it is unique. How many
processors does it require?

(c) Write a simple formula M.n; t; p/ for the minimum time of a p-processor
schedule to complete Dn;t .
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418 Chapter 10 Directed graphs & Partial Orders

t-1

...
...

n - (t - 1)

(d) Show that every partial order with n vertices and maximum chain size t has a
p-processor schedule that runs in time M.n; t; p/.
Hint: Use induction on t.

Problems for Section 10.6
Practice Problems
Problem 10.26.
In this DAG (Figure 10.13) for the divisibility relation on f1; : : : ; 12g, there is
an upward path from a to b iff ajb. If 24 was added as a vertex, what is the
minimum number of edges that must be added to the DAG to represent divisibility
on f1; : : : ; 12; 24g? What are those edges?

Problem 10.27. (a) Prove that every strict partial order is a DAG.

(b) Give an example of a DAG that is not a strict partial order.

10.11. Summary of Relational Properties 419

9 4

6 10

11 3 2 5 7

Figure 10.13

Class Problems
Problem 10.28. (a) What are the maximal and minimal elements, if any, of the
power set pow.f1; : : : ; ng/, where n is a positive integer, under the empty relation?

(b) What are the maximal and minimal elements, if any, of the set N of all non-
negative integers under divisibility? Is there a minimum or maximum element?

(d) Describe a partially ordered set that has no minimal or maximal elements.

(e) Describe a partially ordered set that has a unique minimal element, but no
minimum element. Hint: It will have to be infinite.

Problem 10.29.
The proper subset relation defines a strict partial order on the subsets of Œ1::6,
that is, on pow.Œ1::6/.
(a) What is the size of a maximal chain in this partial order? Describe one.

(b) Describe the largest antichain you can find in this partial order.

(d) Answer the previous part for the partial order on the set pow Œ1::6 ;.

Problem 10.30.
If a and b are distinct nodes of a digraph, then a is said to cover b if there is an
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420 Chapter 10 Directed graphs & Partial Orders

edge from a to b and every path from a to b includes this edge. If a covers b, the
edge from a to b is called a covering edge.
(a) What are the covering edges in the DAG in Figure 10.14?

(b) Let covering .D/ be the subgraph of D consisting of only the covering edges.
Suppose D is a finite DAG. Explain why covering .D/ has the same positive walk
relation as D.
Hint: Consider longest paths between a pair of vertices.

(d) Conclude that covering .D/ is the unique DAG with the smallest number of
edges among all digraphs with the same positive walk relation as D.
The following examples show that the above results don’t work in general for
digraphs with cycles.
(e) Describe two graphs with vertices f1; 2g which have the same set of covering
edges, but not the same positive walk relation (Hint: Self-loops.)

(f) (i) The complete digraph without self-loops on vertices 1; 2; 3 has directed
edges in each direction between every two distinct vertices. What are its
covering edges?
(ii) What are the covering edges of the graph with vertices 1; 2; 3 and edges
h1 ! 2i ; h2 ! 3i ; h3 ! 1i?
(iii) What about their positive walk relations?

Problems for Section 10.6
Exam Problems
Problem 10.31.
Prove that for any nonempty set D, there is a unique binary relation on D that is
both asymmetric and symmetric.

Problem 10.32.
Let D be a set of size n > 0. Shown that there are exactly 2n binary relations on
D that are both symmetric and antisymmetric.
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10.11. Summary of Relational Properties 421

2
4

6
1
3

Figure 10.14 DAG with edges not needed in paths

Homework Problems
Problem 10.33.
Prove that if R is a transitive binary relation on a set A then R D RC .

Class Problems
Problem 10.34.
Let R be a binary relation on a set D. Each of the following equalities and contain-
ments expresses the fact that R has one of the basic relational properties: reflexive,
irreflexive, symmetric, asymmetric, antisymmetric, transitive. Identify which prop-
erty is expressed by each of these formulas and explain your reasoning.
(a) R \ IdD D ;

(b) R R 1

(d) IdD R

(e) R ı R R

(f) R \ R 1 D;

(g) R \ R 1 IdD
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422 Chapter 10 Directed graphs & Partial Orders

Problems for Section 10.7
Class Problems
Problem 10.35.

Direct Prerequisites Subject
18.01 6.042
18.01 18.02
18.01 18.03
8.01 8.02
8.01 6.01
6.042 6.046
18.02, 18.03, 8.02, 6.01 6.02
6.01, 6.042 6.006
6.01 6.034
6.02 6.004

(a) For the above table of MIT subject prerequisites, draw a diagram showing the
subject numbers with a line going down to every subject from each of its (direct)
prerequisites.

(b) Give an example of a collection of sets partially ordered by the proper subset
relation that is isomorphic to (“same shape as”) the prerequisite relation among
MIT subjects from part (a).

(c) Explain why the empty relation is a strict partial order and describe a collection
of sets partially ordered by the proper subset relation that is isomorphic to the empty
relation on five elements—that is, the relation under which none of the five elements
is related to anything.

(d) Describe a simple collection of sets partially ordered by the proper subset re-
lation that is isomorphic to the ”properly contains” relation on pow f1; 2; 3; 4g.

Problem 10.36.
This problem asks for a proof of Lemma 10.7.2 showing that every weak partial
order can be represented by (is isomorphic to) a collection of sets partially ordered
under set inclusion (). Namely,
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10.11. Summary of Relational Properties 423

Lemma. Let be a weak partial order on a set A. For any element a 2 A, let
L.a/ WWD fb 2 A j b ag;
L WWD fL.a/ j a 2 Ag:
Then the function L W A ! L is an isomorphism from the relation on A, to the
subset relation on L.
(a) Prove that the function L W A ! L is a bijection.

(b) Complete the proof by showing that
ab iff L.a/ L.b/ (10.14)
for all a; b 2 A.

Homework Problems
Problem 10.37.
Every partial order is isomorphic to a collection of sets under the subset relation
(see Section 10.7). In particular, if R is a strict partial order on a set A and a 2 A,
define
L.a/ WWD fag [ fx 2 A j x R ag: (10.15)
Then
aRb iff L.a/ L.b/ (10.16)
holds for all a; b 2 A.
(a) Carefully prove statement (10.16), starting from the definitions of strict partial
order and the strict subset relation .

(b) Prove that if L.a/ D L.b/ then a D b.

(c) Give an example showing that the conclusion of part (b) would not hold if the
definition of L.a/ in equation (10.15) had omitted the expression “fag[.”

Problems for Section 10.8
Practice Problems
Problem 10.38.
For each of the binary relations below, state whether it is a strict partial order, a
weak partial order, or neither. If it is not a partial order, indicate which of the
axioms for partial order it violates.
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424 Chapter 10 Directed graphs & Partial Orders

(a) The superset relation, on the power set pow f1; 2; 3; 4; 5g.

(b) The relation between any two nonnegative integers a, b given by a b
.mod 8/.

(d) The relation ’beats’ on Rock, Paper and Scissor (for those who don’t know the
game “Rock, Paper, Scissors:” Rock beats Scissors, Scissors beats Paper and Paper
beats Rock).

(e) The empty relation on the set of real numbers.

(f) The identity relation on the set of integers.

Problem 10.39. (a) Verify that the divisibility relation on the set of nonnegative
integers is a weak partial order.

(b) What about the divisibility relation on the set of integers?

Problem 10.40.
Prove directly from the definitions (without appealing to DAG properties) that if a
binary relation R on a set A is transitive and irreflexive, then it is asymmetric.

Class Problems
Problem 10.41.
Show that the set of nonnegative integers partially ordered under the divides rela-
tion. . .
(a) . . . has a minimum element.

(b) . . . has a maximum element.

(d) . . . has an infinite antichain.

(e) What are the minimal elements of divisibility on the integers greater than 1?
What are the maximal elements?
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10.11. Summary of Relational Properties 425

Problem 10.42.
How many binary relations are there on the set f0; 1g?
How many are there that are transitive?, . . . asymmetric?, . . . reflexive?, . . . irreflexive?,
. . . strict partial orders?, . . . weak partial orders?
Hint: There are easier ways to find these numbers than listing all the relations
and checking which properties each one has.

Problem 10.43.
Prove that if R is a partial order, then so is R 1.

Problem 10.44. (a) Indicate which of the following relations below are equiva-
lence relations, (Eq), strict partial orders (SPO), weak partial orders (WPO). For
the partial orders, also indicate whether it is linear (Lin).
If a relation is none of the above, indicate whether it is transitive (Tr), symmetric
(Sym), or asymmetric (Asym).

(i) The relation a D b C 1 between integers a, b,
(ii) The superset relation on the power set of the integers.
(iii) The empty relation on the set of rationals.
(iv) The divides relation on the nonegative integers N.
(v) The divides relation on all the integers Z.
(vi) The divides relation on the positive powers of 4.
(vii) The relatively prime relation on the nonnegative integers.
(viii) The relation “has the same prime factors” on the integers.

(b) A set of functions f; g W D ! R can be partially ordered by the relation,
where
Œf g WWD 8d 2 D: f .d / g.d /:
Let L be the set of functions f W R ! R of the form

f .x/ D ax C b

for constants a; b 2 R.
Describe an infinite chain and an infinite anti-chain in L.
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426 Chapter 10 Directed graphs & Partial Orders

Problem 10.45.
In an n-player round-robin tournament, every pair of distinct players compete in a
single game. Assume that every game has a winner—there are no ties. The results
of such a tournament can then be represented with a tournament digraph where the
vertices correspond to players and there is an edge hx ! yi iff x beat y in their
game.
(a) Explain why a tournament digraph cannot have cycles of length one or two.

(b) Is the “beats” relation for a tournament graph always/sometimes/never:

asymmetric?
reflexive?
irreflexive?
transitive?

Explain.

Homework Problems
Problem 10.46.
Let R and S be transitive binary relations on the same set A. Which of the following
new relations must also be transitive? For each part, justify your answer with a brief
argument if the new relation is transitive and a counterexample if it is not.
(a) R 1

(b) R \ S

(d) R ı S

Exam Problems
Problem 10.47.
Suppose the precedence constraints on a set of 32 unit time tasks was isomorphic
to the powerset, pow.f1; 2; 3; 4; 5g/ under the strict subset relation .
For example, the task corresponding to the set f2; 4g must be completed be-
fore the task corresponding to the set f1; 2; 4g because f2; 4g f1; 2; 4g; the task
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10.11. Summary of Relational Properties 427

corresponding to the empty set must be scheduled first because ; S for every
nonempty set S f1; 2; 3; 4; 5g.
(a) What is the minimum parallel time to complete these tasks?

(b) Describe a maximum size antichain in this partial order.

(c) Briefly explain why the minimum number of processors required to complete
these tasks in minimum parallel time is equal to the size of the maximum antichain.

Problem 10.48.
Let R be a weak partial order on a set A. Suppose C is a finite chain.17
(a) Prove that C has a maximum element. Hint: Induction on the size of C .

(b) Conclude that there is a unique sequence of all the elements of C that is strictly
increasing.
Hint: Induction on the size of C , using part (a).

Problems for Section 10.9
Practice Problems
Problem 10.49.
Verify that if either of R1 or R2 is irreflexive, then so is R1 R2 .

Class Problems
Problem 10.50.
Let R1 , R2 be binary relations on the same set A. A relational property is preserved
under product, if R1 R2 has the property whenever both R1 and R2 have the
property.
(a) Verify that each of the following properties are preserved under product.

1. reflexivity,
2. antisymmetry,
3. transitivity.
17 Aset C is a chain when it is nonempty, and all elements c; d 2 C are comparable. Elements c
and d are comparable iff Œc R d OR d R c.
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428 Chapter 10 Directed graphs & Partial Orders

(b) Verify that if R1 and R2 are partial orders and at least one of them is strict,
then R1 R2 is a strict partial order.

Problem 10.51.
A partial order on a set A is well founded when every non-empty subset of A has a
minimal element. For example, the less-than relation on a well ordered set of real
numbers (see 2.4) is a linear order that is well founded.
Prove that if R and S are well founded partial orders, then so is their product
R S.

Homework Problems
Problem 10.52.
Let S be a sequence of n different numbers. A subsequence of S is a sequence that
can be obtained by deleting elements of S .
For example, if S is
.6; 4; 7; 9; 1; 2; 5; 3; 8/;
then 647 and 7253 are both subsequences of S (for readability, we have dropped the
parentheses and commas in sequences, so 647 abbreviates .6; 4; 7/, for example).
An increasing subsequence of S is a subsequence of whose successive elements
get larger. For example, 1238 is an increasing subsequence of S . Decreasing sub-
sequences are defined similarly; 641 is a decreasing subsequence of S .
(a) List all the maximum-length increasing subsequences of S, and all the maximum-
length decreasing subsequences.
Now let A be the set of numbers in S . (So A is the integers Œ1::9 for the example
above.) There are two straightforward linear orders for A. The first is numerical
order where A is ordered by the < relation. The second is to order the elements by
which comes first in S ; call this order <S . So for the example above, we would
have
6 <S 4 <S 7 <S 9 <S 1 <S 2 <S 5 <S 3 <S 8
Let be the product relation of the linear orders <s and <. That is, is defined
by the rule
a a0 WWD a < a0 AND a <S a0 :
So is a partial order on A (Section 10.9).
(b) Draw a diagram of the partial order on A. What are the maximal and mini-
mal elements?
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10.11. Summary of Relational Properties 429

(d) Prove that every sequence S of length n has an increasing subsequence of
p p
length greater than n or a decreasing subsequence of length at least n.

Problems for Section 10.10
Practice Problems
Problem 10.53.
For each of the following relations, decide whether it is reflexive, whether it is
symmetric, whether it is transitive, and whether it is an equivalence relation.
(a) f.a; b/ j a and b are the same ageg

(b) f.a; b/ j a and b have the same parentsg

Problem 10.54.
For each of the binary relations below, state whether it is a strict partial order, a
weak partial order, an equivalence relation, or none of these. If it is a partial order,
state whether it is a linear order. If it is none, indicate which of the axioms for
partial-order and equivalence relations it violates.
(a) The superset relation on the power set pow f1; 2; 3; 4; 5g.

(b) The relation between any two nonnegative integers a and b such that a b
.mod 8/.

(d) The relation between propositional formulas G and H such that ŒG IFF H is
valid.

(e) The relation ‘beats’ on Rock, Paper, and Scissors (for those who don’t know
the game Rock, Paper, Scissors, Rock beats Scissors, Scissors beats Paper, and
Paper beats Rock).

(f) The empty relation on the set of real numbers.
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430 Chapter 10 Directed graphs & Partial Orders

(g) The identity relation on the set of integers.

(h) The divisibility relation on the integers Z.

Class Problems
Problem 10.55.
Prove Theorem 10.10.4: The equivalence classes of an equivalence relation form a
partition of the domain.
Namely, let R be an equivalence relation on a set A and define the equivalence
class of an element a 2 A to be

ŒaR WWD fb 2 A j a R bg:

That is, ŒaR D R.a/.
(a) Prove that every block is nonempty and every element of A is in some block.

(b) Prove that if ŒaR \ ŒbR ¤ ;, then a R b. Conclude that the sets ŒaR for
a 2 A are a partition of A.

Problem 10.56.
For any total function f W A ! B define a relation f by the rule:

a f a0 iff f .a/ D f .a0 /: (10.17)

(a) Sketch a proof that f is an equivalence relation on A.

(b) Prove that every equivalence relation R on a set A is equal to f for the
function f W A ! pow.A/ defined as

f .a/ WWD fa0 2 A j a R a0 g:

That is, f .a/ D R.a/.

Problem 10.57.
Let R be a binary relation on a set D. Each of the following formulas expresses the
fact that R has a familiar relational property such as reflexivity, asymmetry, tran-
sitivity. Predicate formulas have roman numerals i.,ii.,. . . , and relational formulas
(equalities and containments) are labelled with letters (a),(b),. . . .
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10.11. Summary of Relational Properties 431

Next to each of the relational formulas, write the roman numerals of all the pred-
icate formulas equivalent to it. It is not necessary to name the property expressed,
but you can get partial credit if you do. For example, part (a) gets the label “i.” It
expresses irreflexivity.

i. 8d: NOT.d R d/

ii. 8d: dRd

iii. 8c; d: c R d IFF d R c

iv. 8c; d: c R d IMPLIES d R c

v. 8c; d: c R d IMPLIES NOT.d R c/

vi. 8c ¤ d: c R d IMPLIES NOT.d R c/

vii. 8c ¤ d: c R d IFF NOT.d R c/

viii. 8b; c; d: .b R c AND c R d / IMPLIES b R d

ix. 8b; d: Œ9c: .b R c AND c R d / IMPLIES b R d

x. 8b; d: b R d IMPLIES Œ9c: .b R c AND c R d /

(a) R \ IdD D ; i.

(b) R R 1

(d) IdD R

(e) R ı R R

(f) R R ı R

(g) R \ R 1 IdD

(h) R R 1

(i) R \ IdR D R 1 \ IdR

(j) R \ R 1 D;
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432 Chapter 10 Directed graphs & Partial Orders

Homework Problems
Problem 10.58.
Let R1 and R2 be two equivalence relations on a set A. Prove or give a counterex-
ample to the claims that the following are also equivalence relations:
(a) R1 \ R2 .

(b) R1 [ R2 .

Problem 10.59.
Prove that for any nonempty set D, there is a unique binary relation on D that is
both a weak partial order and also an equivalence relation.

Exam Problems
Problem 10.60.
Let A be a nonempty set.
(a) Describe a single relation on A that is both an equivalence relation and a weak
partial order on A.

(b) Prove that the relation of part (a) is the only relation on A with these properties.
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11 Communication Networks
Modeling communication networks is an important application of digraphs in com-
puter science. In this such models, vertices represent computers, processors, and
switches; edges will represent wires, fiber, or other transmission lines through
which data flows. For some communication networks, like the internet, the cor-
responding graph is enormous and largely chaotic. Highly structured networks, by
contrast, find application in telephone switching systems and the communication
hardware inside parallel computers. In this chapter, we’ll look at some of the nicest
and most commonly used structured networks.

11.1 Routing
The kinds of communication networks we consider aim to transmit packets of data
between computers, processors, telephones, or other devices. The term packet
refers to some roughly fixed-size quantity of data—256 bytes or 4096 bytes or
whatever.

11.1.1 Complete Binary Tree
Let’s start with a complete binary tree. Figure 11.1 is an example with 4 inputs and
4 outputs.
In this diagram, and many that follow, the squares represent terminals—sources
and destinations for packets of data. The circles represent switches, which direct
packets through the network. A switch receives packets on incoming edges and
relays them forward along the outgoing edges. Thus, you can imagine a data
packet hopping through the network from an input terminal, through a sequence
of switches joined by directed edges, to an output terminal.
In a tree there is a unique path between every pair of vertices, so there is only
one way to route a packet of data from an input terminal to an output. For example,
the route of a packet traveling from input 1 to output 3 is shown in bold.

11.1.2 Routing Problems
Communication networks are supposed to get packets from inputs to outputs, with
each packet entering the network at its own input switch and arriving at its own
output switch. We’re going to consider several different communication network
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434 Chapter 11 Communication Networks

IN OUT IN OUT IN OUT IN OUT
0 0 1 1 2 2 3 3

Figure 11.1 Binary Tree net with 4 inputs and outputs

designs, where each network has N inputs and N outputs; for convenience, we’ll
assume N is a power of two.
Which input is supposed to go where is specified by a permutation of Œ0::N 1.
So a permutation defines a routing problem: get a packet that starts at input i to
output .i /. A routing that solves a routing problem is a set P of paths from
each input to its specified output. That is, P is a set of paths Pi where Pi goes
from input i to output .i / for i 2 Œ0::N 1.

11.2 Routing Measures
11.2.1 Network Diameter
The delay between the time that a packets arrives at an input and arrives at its
designated output is a critical issue in communication networks. Generally, this
delay is proportional to the length of the path a packet follows. Assuming it takes
one time unit to travel across a wire, the delay of a packet will be the number of
wires it crosses going from input to output.
Packets are usually routed from input to output by the shortest path possible.
With a shortest-path routing, the worst-case delay is the distance between the input
and output that are farthest apart. This is called the diameter of the network. In
other words, the diameter of a network1 is the maximum length of any shortest
1 Theusual definition of diameter for a general graph (simple or directed) is the largest distance
between any two vertices, but in the context of a communication network we’re only interested in the
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11.2. Routing Measures 435

path between an input and an output. For example, in the complete binary tree
above, the distance from input 1 to output 3 is six. No input and output are farther
apart than this, so the diameter of this tree is also six.
More broadly, the diameter of a complete binary tree with N inputs and outputs
is 2 log N C2. This is quite good, because the logarithm function grows very slowly.
We could connect up 210 D 1024 inputs and outputs using a complete binary tree
and the worst input-output delay for any packet would be 2 log.210 / C 2 D 22.

Switch Size
One way to reduce the diameter of a network is to use larger switches. For example,
in the complete binary tree, most of the switches have three incoming edges and
three outgoing edges, which makes them 3 3 switches. If we had 4 4 switches,
then we could construct a complete ternary tree with an even smaller diameter. In
principle, we could even connect up all the inputs and outputs via a single monster
N N switch.
Of course this isn’t very productive. Using an N N switch would just conceal
the original network design problem inside this abstract switch. Eventually, we’ll
have to design the internals of the monster switch using simpler components, and
then we’re right back where we started. So, the challenge in designing a commu-
nication network is figuring out how to get the functionality of an N N switch
using fixed size, elementary devices, like 3 3 switches.

11.2.2 Switch Count
Another goal in designing a communication network is to use as few switches as
possible. In a complete binary tree, there is one “root” switch at the top, and the
number of switches doubles at successive rows, so the number of switches in an
N -input complete binary tree is 1 C 2 C 4 C 8 C C N . So the total number of
switches is 2N 1 by the formula for geometric sums (Problem 5.4). This is nearly
the best possible with 3 3 switches.

11.2.3 Network Latency
We’ll sometimes be choosing routings through a network that optimize some quan-
tity besides delay. For example, in the next section we’ll be trying to minimize
packet congestion. When we’re not minimizing delay, shortest routings are not al-
ways the best, and in general, the delay of a packet will depend on how it is routed.
For any routing, the most delayed packet will be the one that follows the longest
path in the routing. The length of the longest path in a routing is called its latency.
distance between inputs and outputs, not between arbitrary pairs of vertices.
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436 Chapter 11 Communication Networks

IN OUT IN OUT IN OUT IN OUT IN OUT IN OUT IN OUT IN OUT
0 0 1 1 2 2 3 3 0 0 1 1 2 2 3 3

Figure 11.2 Two Routings in the Binary Tree Net

The latency of a network depends on what’s being optimized. It is measured by
assuming that optimal routings are always chosen in getting inputs to their specified
outputs. That is, for each routing problem , we choose an optimal routing that
solves . Then network latency is defined to be the largest routing latency among
these optimal routings. Network latency will equal network diameter if routings
are always chosen to optimize delay, but it may be significantly larger if routings
are chosen to optimize something else.
For the networks we consider below, paths from input to output are uniquely
determined (in the case of the tree) or all paths are the same length, so network
latency will always equal network diameter.

11.2.4 Congestion
The complete binary tree has a fatal drawback: the root switch is a bottleneck. At
best, this switch must handle right and vice-versa. Passing all these packets through
a single switch could take a long time. At worst, if this switch fails, the network is
broken into two equal-sized pieces.
It’s true that if the routing problem is given by the identity permutation, Id.i / WWD
i , then there is an easy routing P that solves the problem: let Pi be the path from
input i up through one switch and back down to output i . On the other hand, if the
problem was given by .i / WWD .N 1/ i , then in any solution Q for , each path
Qi beginning at input i must eventually loop all the way up through the root switch
and then travel back down to output .N 1/ i . These two situations are illustrated
in Figure 11.2. We can distinguish between a “good” set of paths and a “bad” set
based on congestion. The congestion of a routing P is equal to the largest number
of paths in P that pass through a single switch. For example, the congestion of the
routing on the left is 1, since at most 1 path passes through each switch. However,
the congestion of the routing on the right is 4, since 4 paths pass through the root
switch (and the two switches directly below the root). Generally, lower congestion
is better since packets can be delayed at an overloaded switch.
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11.3. Network Designs 437

in0

in1

in2

in3

out0 out1 out2 out3

Figure 11.3 Two-dimensional Array with N D 4.

By extending the notion of congestion to networks, we can also distinguish be-
tween “good” and “bad” networks with respect to bottleneck problems. For each
routing problem for the network, we assume a routing is chosen that optimizes
congestion, that is, that has the minimum congestion among all routings that solve
. Then the largest congestion that will ever be suffered by a switch will be the
maximum congestion among these optimal routings. This “maximin” congestion
is called the congestion of the network.
So for the complete binary tree, the worst permutation would be .i / WWD .N
1/ i . Then in every possible solution for , every packet would have to follow
a path passing through the root switch. Thus, the max congestion of the complete
binary tree is N —which is horrible!
Let’s tally the results of our analysis so far:

network diameter switch size # switches congestion
complete binary tree 2 log N C 2 33 2N 1 N

11.3 Network Designs
11.3.1 2-D Array
Communication networks can also be designed as2-dimensional arrays or grids. A
2-D array with four inputs and outputs is shown in Figure 11.3.
The diameter in this example is 8, which is the number of edges between input 0
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438 Chapter 11 Communication Networks

and output 3. More generally, the diameter of an array with N inputs and outputs is
2N , which is much worse than the diameter of 2 log N C 2 in the complete binary
tree. But we get something in exchange: replacing a complete binary tree with an
array almost eliminates congestion.

Theorem 11.3.1. The congestion of an N -input array is 2.

Proof. First, we show that the congestion is at most 2. Let be any permutation.
Define a solution P for to be the set of paths, Pi , where Pi goes to the right from
input i to column .i / and then goes down to output .i /. Thus, the switch in row
i and column j transmits at most two packets: the packet originating at input i and
the packet destined for output j .
Next, we show that the congestion is at least 2. This follows because in any
routing problem , where .0/ D 0 and .N 1/ D N 1, two packets must
pass through the lower left switch.

As with the tree, the network latency when minimizing congestion is the same
as the diameter. That’s because all the paths between a given input and output are
the same length.
Now we can record the characteristics of the 2-D array.

network diameter switch size # switches congestion
complete binary tree 2 log N C 2 33 2N 1 N
2-D array 2N 22 N2 2

The crucial entry here is the number of switches, which is N 2 . This is a major
defect of the 2-D array; a network of size N D 1000 would require a million
2 2 switches! Still, for applications where N is small, the simplicity and low
congestion of the array make it an attractive choice.

11.3.2 Butterfly
The Holy Grail of switching networks would combine the best properties of the
complete binary tree (low diameter, few switches) and of the array (low conges-
tion). The butterfly is a widely-used compromise between the two.
A good way to understand butterfly networks is as a recursive data type. The
recursive definition works better if we define just the switches and their connec-
tions, omitting the terminals. So we recursively define Fn to be the switches and
connections of the butterfly net with N WWD 2n input and output switches.
The base case is F1 with 2 input switches and 2 output switches connected as in
Figure 11.4.
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11.3. Network Designs 439

À
2 inputs 2 outputs

ND21
Figure 11.4 F1 , the Butterfly Net with N D 21 .

⎧ Fn
2n ⎨
⎩ 2n+1
n 1 outputs
t t
⎧ Fn
2n ⎨
⎩
new inputs
Fn+1

Figure 11.5 Butterfly Net FnC1 with 2nC1 inputs from two Fn ’s.

In the constructor step, we construct FnC1 out of two Fn nets connected to a
new set of 2nC1 input switches, as shown in as in Figure 11.5. That is, the i th
and 2n C i th new input switches are each connected to the same two switches, the
i th input switches of each of two Fn components for i D 1; : : : ; 2n . The output
switches of FnC1 are simply the output switches of each of the Fn copies.
So FnC1 is laid out in columns of height 2nC1 by adding one more column of
switches to the columns in Fn . Since the construction starts with two columns
when n D 1, the FnC1 switches are arrayed in n C 1 columns. The total number of
switches is the height of the columns times the number 2nC1 .n C 1/ of columns.
Remembering that n D log N , we conclude that the Butterfly Net with N inputs
has N.log N C 1/ switches.
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440 Chapter 11 Communication Networks

Since every path in FnC1 from an input switch to an output is length-n C 1 the
diameter of the Butterfly net with 2nC1 inputs is this length plus two because of
the two edges connecting to the terminals (square boxes)—one edge from input
terminal to input switch (circle) and one from output switch to output terminal.
There is an easy recursive procedure to route a packet through the Butterfly Net.
In the base case, there is only one way to route a packet from one of the two inputs
to one of the two outputs. Now suppose we want to route a packet from an input
switch to an output switch in FnC1 . If the output switch is in the “top” copy of Fn ,
then the first step in the route must be from the input switch to the unique switch
it is connected to in the top copy; the rest of the route is determined by recursively
routing the rest of the way in the top copy of Fn . Likewise, if the output switch is in
the “bottom” copy of Fn , then the first step in the route must be to the switch in the
bottom copy, and the rest of the route is determined by recursively routing in the
bottom copy of Fn . In fact, this argument shows that the routing is unique: there is
exactly one path in the Butterfly Net from each input to each output, which implies
that the network latency when minimizing congestion p is the same as the diameter.
The congestion
p of the butterfly network is about
p N . More precisely, the con-
gestion is N if N is an even power of 2 and N=2 if N is an odd power of 2. A
simple proof of this appears in Problem 11.8.
Let’s add the butterfly data to our comparison table:

network diameter switch size # switches congestion
complete binary tree 2 log N C 2 33 2N 1 N
2-D array 2N 22 N2 p 2p
butterfly log N C 2 22 N.log.N / C 1/ N or N=2

The butterfly has lower congestion than the complete binary tree. It also uses fewer
switches and has lower diameter than the array. However, the butterfly does not
capture the best qualities of each network, but rather is a compromise somewhere
between the two. Our quest for the Holy Grail of routing networks goes on.

11.3.3 Beneš Network
In the 1960’s, a researcher at Bell Labs named Václav E. Beneš had a remarkable
idea. He obtained a marvelous communication network with congestion 1 by plac-
ing two butterflies back-to-back. This amounts to recursively growing Beneš nets
by adding both inputs and outputs at each stage. Now we recursively define Bn
to be the switches and connections (without the terminals) of the Beneš net with
N WWD 2n input and output switches.
The base case B1 with 2 input switches and 2 output switches is exactly the same
as F1 in Figure 11.4.
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11.3. Network Designs 441

À À
2n Bn

2nC1

2n Bn

new inputs BnC1 new outputs

Figure 11.6 Beneš Net BnC1 with 2nC1 inputs from two Bn ’s.

In the constructor step, we construct BnC1 out of two Bn nets connected to a
new set of 2nC1 input switches and also a new set of 2nC1 output switches. This is
illustrated in Figure 11.6.
The i th and 2n C i th new input switches are each connected to the same two
switches: the i th input switches of each of two Bn components for i D 1; : : : ; 2n ,
exactly as in the Butterfly net. In addition, the i th and 2n C i th new output switches
are connected to the same two switches, namely, to the i th output switches of each
of two Bn components.
Now, BnC1 is laid out in columns of height 2nC1 by adding two more columns
of switches to the columns in Bn . So, the BnC1 switches are arrayed in 2.n C 1/
columns. The total number of switches is the number of columns times the height
2.n C 1/2nC1 of the columns.
All paths in BnC1 from an input switch to an output are length 2.n C 1/ 1, and
the diameter of the Beneš net with 2nC1 inputs is this length plus two because of
the two edges connecting to the terminals.
So Beneš has doubled the number of switches and the diameter, but by doing so
he has completely eliminated congestion problems! The proof of this fact relies on
a clever induction argument that we’ll come to in a moment. Let’s first see how the
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442 Chapter 11 Communication Networks

Beneš network stacks up:

network diameter switch size # switches congestion
complete binary tree 2 log N C 2 33 2N 1 N
2-D array 2N 22 N2 p 2p
butterfly log N C 2 22 N.log.N / C 1/ N or N=2
Beneš 2 log N C 1 22 2N log N 1

The Beneš network has small size and diameter, and it completely eliminates con-
gestion. The Holy Grail of routing networks is in hand!

Theorem 11.3.2. The congestion of the N -input Beneš network is 1.

Proof. By induction on n where N D 2n . So the induction hypothesis is

P .n/ WWD the congestion of Bn is 1:

Base case (n D 1): B1 D F1 is shown in Figure 11.4. The unique routings in F1
have congestion 1.
Inductive step: We assume that the congestion of an N D 2n -input Beneš network
is 1 and prove that the congestion of a 2N -input Beneš network is also 1.
Digression. Time out! Let’s work through an example, develop some intuition,
and then complete the proof. In the Beneš network shown in Figure 11.7 with
N D 8 inputs and outputs, the two 4-input/output subnetworks are in dashed boxes.
By the inductive assumption, the subnetworks can each route an arbitrary per-
mutation with congestion 1. So if we can guide packets safely through just the first
and last levels, then we can rely on induction for the rest! Let’s see how this works
in an example. Consider the following permutation routing problem:

.0/ D 1 .4/ D 3
.1/ D 5 .5/ D 6
.2/ D 4 .6/ D 0
.3/ D 7 .7/ D 2

We can route each packet to its destination through either the upper subnetwork
or the lower subnetwork. However, the choice for one packet may constrain the
choice for another. For example, we cannot route both packet 0 and packet 4
through the same network, since that would cause two packets to collide at a sin-
gle switch, resulting in congestion. Rather, one packet must go through the upper
network and the other through the lower network. Similarly, packets 1 and 5, 2 and
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11.3. Network Designs 443

in0 out0

in1 out1

in2 out2

in3 out3

in4 out4

in5 out5

in6 out6

in7 out7

Figure 11.7 Beneš net B3 .

6, and 3 and 7 must be routed through different networks. Let’s record these con-
straints in a graph. The vertices are the 8 packets. If two packets must pass through
different networks, then there is an edge between them. Thus, our constraint graph
looks like this:
1 5
0 2

4 6
7 3

Notice that at most one edge is incident to each vertex.
The output side of the network imposes some further constraints. For example,
the packet destined for output 0 (which is packet 6) and the packet destined for
output 4 (which is packet 2) cannot both pass through the same network; that would
require both packets to arrive from the same switch. Similarly, the packets destined
for outputs 1 and 5, 2 and 6, and 3 and 7 must also pass through different switches.
We can record these additional constraints in our graph with gray edges:
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444 Chapter 11 Communication Networks

1 5
0 2

4 6
7 3

Notice that at most one new edge is incident to each vertex. The two lines drawn
between vertices 2 and 6 reflect the two different reasons why these packets must
be routed through different networks. However, we intend this to be a simple graph;
the two lines still signify a single edge.
Now here’s the key insight: suppose that we could color each vertex either red
or blue so that adjacent vertices are colored differently. Then all constraints are
satisfied if we send the red packets through the upper network and the blue packets
through the lower network. Such a 2-coloring of the graph corresponds to a solu-
tion to the routing problem. The only remaining question is whether the constraint
graph is 2-colorable, which is easy to verify:
Lemma 11.3.3. Prove that if the edges of a graph can be grouped into two sets such
that every vertex has at most 1 edge from each set incident to it, then the graph is
2-colorable.

Proof. It is not hard to show that a graph is 2-colorable iff every cycle in it has even
length (see Theorem 12.8.3). We’ll take this for granted here.
So all we have to do is show that every cycle has even length. Since the two sets
of edges may overlap, let’s call an edge that is in both sets a doubled edge.
There are two cases:
Case 1: [The cycle contains a doubled edge.] No other edge can be incident
to either of the endpoints of a doubled edge, since that endpoint would then be
incident to two edges from the same set. So a cycle traversing a doubled edge has
nowhere to go but back and forth along the edge an even number of times.
Case 2: [No edge on the cycle is doubled.] Since each vertex is incident to
at most one edge from each set, any path with no doubled edges must traverse
successive edges that alternate from one set to the other. In particular, a cycle must
traverse a path of alternating edges that begins and ends with edges from different
sets. This means the cycle has to be of even length.

For example, here is a 2-coloring of the constraint graph:
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11.3. Network Designs 445

blue red
1 5
red 0 2 red

blue 4 6 blue
7 3
blue red

The solution to this graph-coloring problem provides a start on the packet routing
problem:
We can complete the routing in the two smaller Beneš networks by induction!
Back to the proof. End of Digression.
Let be an arbitrary permutation of Œ0::N 1. Let G be the graph whose
vertices are packet numbers 0; 1; : : : ; N 1 and whose edges come from the union
of these two sets:

E1 WWDfhu—vi j ju vj D N=2g; and
E2 WWDfhu—wi j j.u/ .w/j D N=2g:

Now any vertex u is incident to at most two edges: a unique edge hu—vi 2 E1
and a unique edge hu—wi 2 E2 . So according to Lemma 11.3.3, there is a 2-
coloring for the vertices of G. Now route packets of one color through the upper
subnetwork and packets of the other color through the lower subnetwork. Since
for each edge in E1 , one vertex goes to the upper subnetwork and the other to the
lower subnetwork, there will not be any conflicts in the first level. Since for each
edge in E2 , one vertex comes from the upper subnetwork and the other from the
lower subnetwork, there will not be any conflicts in the last level. We can complete
the routing within each subnetwork by the induction hypothesis P .n/.

Problems for Section 11.2
Exam Problems
Problem 11.1.
Consider the following communication network:
(a) What is the max congestion?

(b) Give an input/output permutation 0 that forces maximum congestion.

446 Chapter 11 Communication Networks

in0 in1 in2

out0 out1 out2

(d) What is the latency for the permutation 1 ? (If you could not find 1 , just
choose a permutation and find its latency.)

Problems for Section 11.3
Class Problems
Problem 11.2.
The Beneš network has a max congestion of one—every permutation can be routed
in such a way that a single packet passes through each switch. Let’s work through
an example. A diagram of the Beneš network B3 of size N D 8 appears in Fig-
ure 11.7. The two subnetworks of size N D 4 are marked. We’ll refer to these as
the upper and lower subnetworks.
(a) Now consider the following permutation routing problem:

.0/ D 3 .4/ D 2
.1/ D 1 .5/ D 0
.2/ D 6 .6/ D 7
.3/ D 5 .7/ D 4

Each packet must be routed through either the upper subnetwork or the lower sub-
network. Construct a graph with vertices numbered by integers 0 to 7 and draw a
dashed edge between each pair of packets that cannot go through the same subnet-
work because a collision would occur in the second column of switches.

(b) Add a solid edge in your graph between each pair of packets that cannot go
through the same subnetwork because a collision would occur in the next-to-last
column of switches.
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11.3. Network Designs 447

(c) Assign colors red and blue to the vertices of your graph so that vertices that
are adjacent by either a dashed or a solid edge get different colors. Why must this
be possible, regardless of the permutation ?

(d) Suppose that red vertices correspond to packets routed through the upper sub-
network and blue vertices correspond to packets routed through the lower subnet-
work. Referring to the Beneš network shown in Figure 11.6, indicate the first and
last edge traversed by each packet.

(e) All that remains is to route packets through the upper and lower subnetworks.
One way to do this is by applying the procedure described above recursively on
each subnetwork. However, since the remaining problems are small, see if you can
complete all the paths on your own.

Problem 11.3.
A multiple binary-tree network has N inputs and N outputs, where N is a power
of 2. Each input is connected to the root of a binary tree with N=2 leaves and with
edges pointing away from the root. Likewise, each output is connected to the root
of a binary tree with N=2 leaves and with edges pointing toward the root.
Two edges point from each leaf of an input tree, and each of these edges points
to a leaf of an output tree. The matching of leaf edges is arranged so that for every
input and output tree, there is an edge from a leaf of the input tree to a leaf of the
output tree, and every output tree leaf has exactly two edges pointing to it.
(a) Draw such a multiple binary-tree net for N D 4.

(b) Fill in the table, and explain your entries.

# switches switch size diameter max congestion

Problem 11.4.
The n-input 2-D array network was shown to have congestion 2. An n-input 2-
layer array consisting of two n-input 2-D Arrays connected as pictured below for
n D 4.
In general, an n-input 2-layer array has two layers of switches, with each layer
connected like an n-input 2-D array. There is also an edge from each switch in the
first layer to the corresponding switch in the second layer. The inputs of the 2-layer
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448 Chapter 11 Communication Networks

in0

in1

in2

in3

out0 out1 out2 out3

array enter the left side of the first layer, and the n outputs leave from the bottom
row of either layer.
(a) For any given input-output permutation, there is a way to route packets that
achieves congestion 1. Describe how to route the packets in this way.

(b) What is the latency of a routing designed to minimize latency?

(c) Explain why the congestion of any minimum latency (CML) routing of packets
through this network is greater than the network’s congestion.

Problem 11.5.
A 5-path communication network is shown below. From this, it’s easy to see what
an n-path network would be. Fill in the table of properties below, and be prepared
to justify your answers.

network # switches switch size diameter max congestion
5-path
n-path

Problem 11.6.
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11.3. Network Designs 449

in0 in1 in2 in3 in4

out0 out1 out2 out3 out4

Figure 11.8 5-Path

Tired of being a TA, Megumi has decided to become famous by coming up with a
new, better communication network design. Her network has the following specifi-
cations: every input node will be sent to a butterfly network, a Beneš network and
a 2-d array network. At the end, the outputs of all three networks will converge on
the new output.
In the Megumi-net a minimum latency routing does not have minimum conges-
tion. The latency for min-congestion (LMC) of a net is the best bound on latency
achievable using routings that minimize congestion. Likewise, the congestion for
min-latency (CML) is the best bound on congestion achievable using routings that
minimize latency.

in1 out1
in2 Butterfly out2
in3 out3
. .
. Beneš .
. .

inN 2-d Array outN

Fill in the following chart for Megumi’s new net and explain your answers.
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450 Chapter 11 Communication Networks

network diameter # switches congestion LMC CML
Megumi’s net

Homework Problems
Problem 11.7.
Louis Reasoner figures that, wonderful as the Beneš network may be, the butterfly
network has a few advantages, namely: fewer switches, smaller diameter, and an
easy way to route packets through it. So Louis designs an N -input/output network
he modestly calls a Reasoner-net with the aim of combining the best features of
both the butterfly and Beneš nets:

The i th input switch in a Reasoner-net connects to two switches, ai and
bi , and likewise, the j th output switch has two switches, yj and zj ,
connected to it. Then the Reasoner-net has an N -input Beneš network
connected using the ai switches as input switches and the yj switches
as its output switches. The Reasoner-net also has an N -input butterfly
net connected using the bi switches as inputs and¡ the zj switches as
outputs.

In the Reasoner-net a minimum latency routing does not have minimum conges-
tion. The latency for min-congestion (LMC) of a net is the best bound on latency
achievable using routings that minimize congestion. Likewise, the congestion for
min-latency (CML) is the best bound on congestion achievable using routings that
minimize latency.
Fill in the following chart for the Reasoner-net and briefly explain your answers.

diameter switch size(s) # switches congestion LMC CML

Problem 11.8. p
Show that the congestion of the butterfly net, Fn , is exactly N when n is even.
Hint:

There is a unique path from each input to each output, so the congestion is
the maximum number of messages passing through a vertex for any routing
problem.
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11.3. Network Designs 451

If v is a vertex in column i of the butterfly network, there is a path from ex-
actly 2i input vertices to v and a path from v to exactly 2n i output vertices.

At which column of the butterfly network must the congestion be worst?
What is the congestion of the topmost switch in that column of the network?
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12 Simple Graphs
Simple graphs model relationships that are symmetric, meaning that the relationship
is mutual. Examples of such mutual relationships are being married, speaking the
same language, not speaking the same language, occurring during overlapping time
intervals, or being connected by a conducting wire. They come up in all sorts of
applications, including scheduling, constraint satisfaction, computer graphics, and
communications, but we’ll start with an application designed to get your attention:
we are going to make a professional inquiry into sexual behavior. Specifically,
we’ll look at some data about who, on average, has more opposite-gender partners:
men or women.
Sexual demographics have been the subject of many studies. In one of the largest,
researchers from the University of Chicago interviewed a random sample of 2500
people over several years to try to get an answer to this question. Their study,
published in 1994 and entitled The Social Organization of Sexuality, found that
men have on average 74% more opposite-gender partners than women.
Other studies have found that the disparity is even larger. In particular, ABC
News claimed that the average man has 20 partners over his lifetime, and the av-
erage woman has 6, for a percentage disparity of 233%. The ABC News study,
aired on Primetime Live in 2004, purported to be one of the most scientific ever
done, with only a 2.5% margin of error. It was called “American Sex Survey: A
peek between the sheets”—raising some questions about the seriousness of their
reporting.
Yet again in August, 2007, the New York Times reported on a study by the
National Center for Health Statistics of the U.S. government showing that men had
seven partners while women had four. So, whose numbers do you think are more
accurate: the University of Chicago, ABC News, or the National Center?
Don’t answer—this is a trick question designed to trip you up. Using a little
graph theory, we’ll explain why none of these findings can be anywhere near the
truth.

12.1 Vertex Adjacency and Degrees
Simple graphs are defined in almost the same way as digraphs, except that edges are
undirected—they connect two vertices without pointing in either direction between
the vertices. So instead of a directed edge hv ! wi which starts at vertex v and
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454 Chapter 12 Simple Graphs

ends at vertex w, a simple graph only has an undirected edge hv—wi that connects
v and w.
Definition 12.1.1. A simple graph G consists of a nonempty set, V .G/, called the
vertices of G, and a set E.G/ called the edges of G. An element of V .G/ is called
a vertex. A vertex is also called a node; the words “vertex” and “node” are used
interchangeably. An element of E.G/ is an undirected edge or simply an “edge.”
An undirected edge has two vertices u ¤ v called its endpoints. Such an edge
can be represented by the two element set fu; vg. The notation hu—vi denotes this
edge.
Both hu—vi and hv—ui define the same undirected edge, whose endpoints are
u and v.
b h

a f d

g i
c e

Figure 12.1 An example of a graph with 9 nodes and 8 edges.

For example, let H be the graph pictured in Figure 12.1. The vertices of H
correspond to the nine dots in Figure 12.1, that is,

V .H / D fa; b; c; d; e; f; g; h; i g :

The edges correspond to the eight lines, that is,

E.H / D f ha—bi ; ha—ci ; hb—d i ; hc—d i ; hc—ei ; he—f i ; he—gi ; hh—i i g:

Mathematically, that’s all there is to the graph H .
Definition 12.1.2. Two vertices in a simple graph are said to be adjacent iff they
are the endpoints of the same edge, and an edge is said to be incident to each of its
endpoints. The number of edges incident to a vertex v is called the degree of the
vertex and is denoted by deg.v/. Equivalently, the degree of a vertex is the number
of vertices adjacent to it.
For example, for the graph H of Figure 12.1, vertex a is adjacent to vertex b, and
b is adjacent to d . The edge ha—ci is incident to its endpoints a and c. Vertex h
has degree 1, d has degree 2, and deg.e/ D 3. It is possible for a vertex to have
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12.2. Sexual Demographics in America 455

degree 0, in which case it is not adjacent to any other vertices. A simple graph G
does not need to have any edges at all. jE.G/j could be zero, implying that the
degree of every vertex would also be zero. But a simple graph must have at least
one vertex—jV .G/j is required to be at least one.
An edge whose endpoints are the same is called a self-loop. Self-loops aren’t
allowed in simple graphs.1 In a more general class of graphs called multigraphs,
there can be more than one edge with the same two endpoints, but this doesn’t
happen in simple graphs, because every edge is uniquely determined by its two
endpoints. Sometimes graphs with no vertices, with self-loops, or with more than
one edge between the same two vertices are convenient to have, but we don’t need
them, and sticking with simple graphs is simpler.
For the rest of this chapter we’ll use “graphs” as an abbreviation for “simple
graphs.”
A synonym for “vertices” is “nodes,” and we’ll use these words interchangeably.
Simple graphs are sometimes called networks, edges are sometimes called arcs.
We mention this as a “heads up” in case you look at other graph theory literature;
we won’t use these words.

12.2 Sexual Demographics in America
Let’s model the question of heterosexual partners in graph theoretic terms. To do
this, we’ll let G be the graph whose vertices V are all the people in America. Then
we split V into two separate subsets: M which contains all the males, and F which
contains all the females.2 We’ll put an edge between a male and a female iff they
have been sexual partners. This graph is pictured in Figure 12.2 with males on the
left and females on the right.
Actually, this is a pretty hard graph to figure out, let alone draw. The graph is
enormous: the US population is about 300 million, so jV j 300M . Of these,
approximately 50.8% are female and 49.2% are male, so jM j 147:6M , and
jF j 152:4M . And we don’t even have trustworthy estimates of how many edges
there are, let alone exactly which couples are adjacent. But it turns out that we
don’t need to know any of this—we just need to figure out the relationship between
the average number of partners per male and partners per female. To do this, we
note that every edge has exactly one endpoint at an M vertex (remember, we’re
only considering male-female relationships); so the sum of the degrees of the M
1 You might try to represent a self-loop going between a vertex v and itself as fv; vg, but this

equals fvg. It wouldn’t be an edge, which is defined to be a set of two vertices.
2 For simplicity, we’ll ignore the possibility of someone being both a man and a woman, or neither.
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456 Chapter 12 Simple Graphs

M F

Figure 12.2 The sex partners graph.

vertices equals the number of edges. For the same reason, the sum of the degrees
of the F vertices equals the number of edges. So these sums are equal:
X X
deg.x/ D deg.y/:
x2M y2F

Now suppose we divide both sides of this equation by the product of the sizes of
the two sets, jM j jF j:
P !
y2F deg.y/
P
x2M deg.x/ 1 1
D
jM j jF j jF j jM j

The terms above in parentheses are the average degree of an M vertex and the
average degree of an F vertex. So we know:
jF j
Avg. deg in M D Avg. deg in F (12.1)
jM j
In other words, we’ve proved that the average number of female partners of
males in the population compared to the average number of males per female is
determined solely by the relative number of males and females in the population.
Now the Census Bureau reports that there are slightly more females than males
in America; in particular jF j=jM j is about 1.035. So we know that males have
on average 3.5% more opposite-gender partners than females, and that this tells us
nothing about any sex’s promiscuity or selectivity. Rather, it just has to do with the
relative number of males and females. Collectively, males and females have the
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12.3. Some Common Graphs 457

same number of opposite gender partners, since it takes one of each set for every
partnership, but there are fewer males, so they have a higher ratio. This means that
the University of Chicago, ABC, and the Federal government studies are way off.
After a huge effort, they gave a totally wrong answer.
There’s no definite explanation for why such surveys are consistently wrong.
One hypothesis is that males exaggerate their number of partners—or maybe fe-
males downplay theirs—but these explanations are speculative. Interestingly, the
principal author of the National Center for Health Statistics study reported that she
knew the results had to be wrong, but that was the data collected, and her job was
to report it.
The same underlying issue has led to serious misinterpretations of other survey
data. For example, a couple of years ago, the Boston Globe ran a story on a survey
of the study habits of students on Boston area campuses. Their survey showed that
on average, minority students tended to study with non-minority students more than
the other way around. They went on at great length to explain why this “remarkable
phenomenon” might be true. But it’s not remarkable at all. Using our graph theory
formulation, we can see that all it says is that there are fewer students in a minority
than students not in that minority, which is, of course, what “minority” means.

12.2.1 Handshaking Lemma
The previous argument hinged on the connection between a sum of degrees and the
number of edges. There is a simple connection between these in any graph:

Lemma 12.2.1. The sum of the degrees of the vertices in a graph equals twice the
number of edges.

Proof. Every edge contributes two to the sum of the degrees, one for each of its
endpoints.

We refer to Lemma 12.2.1 as the Handshaking Lemma: if we total up the number
of people each person at a party shakes hands with, the total will be twice the
number of handshakes that occurred.

12.3 Some Common Graphs
Some graphs come up so frequently that they have names. A complete graph Kn
has n vertices and an edge between every two vertices, for a total of n.n 1/=2
edges. For example, K5 is shown in Figure 12.3.
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458 Chapter 12 Simple Graphs

Figure 12.3 K5 : the complete graph on 5 nodes.

Figure 12.4 An empty graph with 5 nodes.

The empty graph has no edges at all. For example, the empty graph with 5 nodes
is shown in Figure 12.4.
An n-node graph containing n 1 edges in sequence is known as a line graph Ln .
More formally, Ln has
V .Ln / D fv1 ; v2 ; : : : ; vn g
and
E.Ln / D f hv1 —v2 i ; hv2 —v3 i ; : : : ; hvn 1 —vn i g

For example, L5 is pictured in Figure 12.5.
There is also a one-way infinite line graph L1 which can be defined by letting
the nonnegative integers N be the vertices with edges hk—.k C 1/i for all k 2 N.

Figure 12.5 L5 : a 5-node line graph.
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12.4. Isomorphism 459

Figure 12.6 C5 : a 5-node cycle graph.

a b 1 2

d c 4 3
(a) (b)

Figure 12.7 Two Isomorphic graphs.

If we add the edge hvn —v1 i to the line graph Ln , we get a graph called a length-
n cycle Cn . Figure 12.6 shows a picture of length-5 cycle.

12.4 Isomorphism
Two graphs that look different might actually be the same in a formal sense. For
example, the two graphs in Figure 12.7 are both 4-vertex, 5-edge graphs and you
get graph (b) by a 90o clockwise rotation of graph (a).
Strictly speaking, these graphs are different mathematical objects, but this dif-
ference doesn’t reflect the fact that the two graphs can be described by the same
picture—except for the labels on the vertices. This idea of having the same picture
“up to relabeling” can be captured neatly by adapting Definition 10.7.1 of isomor-
phism of digraphs to handle simple graphs. An isomorphism between two graphs
is an edge-preserving bijection between their sets of vertices:

Definition 12.4.1. An isomorphism between graphs G and H is a bijection f W
V .G/ ! V .H / such that

hu—vi 2 E.G/ iff hf .u/—f .v/i 2 E.H /
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460 Chapter 12 Simple Graphs

Figure 12.8 Isomorphic C5 graphs.

for all u; v 2 V .G/. Two graphs are isomorphic when there is an isomorphism
between them.
Here is an isomorphism f between the two graphs in Figure 12.7:

f .a/ WWD 2 f .b/ WWD 3
f .c/ WWD 4 f .d / WWD 1:

You can check that there is an edge between two vertices in the graph on the left if
and only if there is an edge between the two corresponding vertices in the graph on
the right.
Two isomorphic graphs may be drawn very differently. For example, Figure 12.8
shows two different ways of drawing C5 .
Notice that if f is an isomorphism between G and H , then f 1 is an isomor-
phism between H and G. Isomorphism is also transitive because the composition
of isomorphisms is an isomorphism. In fact, isomorphism is an equivalence rela-
tion.
Isomorphism preserves the connection properties of a graph, abstracting out what
the vertices are called, what they are made out of, or where they appear in a drawing
of the graph. More precisely, a property of a graph is said to be preserved under
isomorphism if whenever G has that property, every graph isomorphic to G also
has that property. For example, since an isomorphism is a bijection between sets of
vertices, isomorphic graphs must have the same number of vertices. What’s more,
if f is a graph isomorphism that maps a vertex v of one graph to the vertex f .v/
of an isomorphic graph, then by definition of isomorphism, every vertex adjacent
to v in the first graph will be mapped by f to a vertex adjacent to f .v/ in the
isomorphic graph. Thus, v and f .v/ will have the same degree. If one graph has
a vertex of degree 4 and another does not, then they can’t be isomorphic. In fact,
they can’t be isomorphic if the number of degree 4 vertices in each of the graphs is
not the same.
Looking for preserved properties can make it easy to determine that two graphs
are not isomorphic, or to guide the search for an isomorphism when there is one.
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12.5. Bipartite Graphs & Matchings 461

It’s generally easy in practice to decide whether two graphs are isomorphic. How-
ever, no one has yet found a procedure for determining whether two graphs are
isomorphic that is guaranteed to run in polynomial time on all pairs of graphs.3
Having such a procedure would be useful. For example, it would make it easy
to search for a particular molecule in a database given the molecular bonds. On
the other hand, knowing there is no such efficient procedure would also be valu-
able: secure protocols for encryption and remote authentication can be built on the
hypothesis that graph isomorphism is computationally exhausting.
The definitions of bijection and isomorphism apply to infinite graphs as well as
finite graphs, as do most of the results in the rest of this chapter. But graph theory
focuses mostly on finite graphs, and we will too. In the rest of this chapter we’ll
assume graphs are finite.
We’ve actually been taking isomorphism for granted ever since we wrote “Kn
has n vertices. . . ” at the beginning of Section 12.3.
Graph theory is all about properties preserved by isomorphism.

12.5 Bipartite Graphs & Matchings
There were two kinds of vertices in the “Sex in America” graph, males and females,
and edges only went between the two kinds. Graphs like this come up so frequently
that they have earned a special name: bipartite graphs.

Definition 12.5.1. A bipartite graph is a graph whose vertices can be divided into
two sets, L.G/ and R.G/, such that every edge has one endpoint in L.G/ and the
other endpoint in R.G/.

So every bipartite graph looks something like the graph in Figure 12.2.

12.5.1 The Bipartite Matching Problem
The bipartite matching problem is related to the sex-in-America problem that we
just studied; only now, the goal is to get everyone happily married. As you might
imagine, this is not possible for a variety of reasons, not the least of which is the
fact that there are more women in America than men. So, it is simply not possible
to marry every woman to a man so that every man is married at most once.
But what about getting a mate for every man so that every woman is married at
most once? Is it possible to do this so that each man is paired with a woman that
3A procedure runs in polynomial time when it needs an amount of time of at most p.n/, where n
is the total number of vertices and p./ is a fixed polynomial.
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462 Chapter 12 Simple Graphs

Alice
Chuck
Martha
Tom
Sara
Michael
Jane
John
Mergatroid

Figure 12.9 A graph where an edge between a man and woman denotes that the
man likes the woman.

he likes? The answer, of course, depends on the bipartite graph that represents who
likes who, but the good news is that it is possible to find natural properties of the
who-likes-who graph that completely determine the answer to this question.
In general, suppose that we have a set of men and an equal-sized or larger set of
women, and there is a graph with an edge between a man and a woman if the man
likes the woman. In this scenario, the “likes” relationship need not be symmetric,
since for the time being, we will only worry about finding a mate for each man
that he likes.4 Later, we will consider the “likes” relationship from the female
perspective as well. For example, we might obtain the graph in Figure 12.9.
A matching is defined to be an assignment of a woman to each man so that
different men are assigned to different women, and a man is always assigned a
woman that he likes. For example, one possible matching for the men is shown in
Figure 12.10.

12.5.2 The Matching Condition
A famous result known as Hall’s Matching Theorem gives necessary and sufficient
conditions for the existence of a matching in a bipartite graph. It turns out to be a
remarkably useful mathematical tool.
We’ll state and prove Hall’s Theorem using man-likes-woman terminology. De-
fine the set of women liked by a given set of men to consist of all women liked by
4 By the way, we do not mean to imply that marriage should or should not be heterosexual. Nor

do we mean to imply that men should get their choice instead of women. It’s just that there are fewer
men than women in America, making it impossible to match up all the women with different men.
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12.5. Bipartite Graphs & Matchings 463

Alice
Chuck
Martha
Tom
Sara
Michael
Jane
John
Mergatroid

Figure 12.10 One possible matching for the men is shown with bold edges. For
example, John is matched with Mergatroid.

at least one of those men. For example, the set of women liked by Tom and John in
Figure 12.9 consists of Martha, Sara, and Mergatroid. For us to have any chance at
all of matching up the men, the following matching condition must hold:
The Matching Condition: every subset of men likes at least as large a set of women.
For example, we cannot find a matching if some set of 4 men like only 3 women.
Hall’s Theorem says that this necessary condition is actually sufficient; if the match-
ing condition holds, then a matching exists.

Theorem 12.5.2. A matching for a set M of men with a set W of women can be
found if and only if the matching condition holds.

Proof. First, let’s suppose that a matching exists and show that the matching condi-
tion holds. For any subset of men, each man likes at least the woman he is matched
with and a woman is matched with at most one man. Therefore, every subset of
men likes at least as large a set of women. Thus, the matching condition holds.
Next, let’s suppose that the matching condition holds and show that a matching
exists. We use strong induction on jM j, the number of men, on the predicate:

P .m/ WWD if the matching condition holds for a set, M ,
of m men, then there is a matching for M .

Base case (jM j D 1): If jM j D 1, then the matching condition implies that the
lone man likes at least one woman, and so a matching exists.
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464 Chapter 12 Simple Graphs

Inductive Step: Suppose that jM j D m C 1 2. To find a matching for M , there
are two cases.

Case 1: Every nonempty subset of at most m men likes a strictly larger set of
women. In this case, we have some latitude: we pair an arbitrary man with
a woman he likes and send them both away. This leaves m men and one
fewer women, and the matching condition will still hold. So the induction
hypothesis P .m/ implies we can match the remaining m men.

Case 2: Some nonempty subset X of at most m men likes an equal-size set Y of
women. The matching condition must hold within X, so the strong induction
hypothesis implies we can match the men in X with the women in Y . This
leaves the problem of matching the set M X of men to the set W Y of
women.
But the problem of matching M X against W Y also satisfies the Match-
ing condition, because any subset of men in M X who liked fewer women
in W Y would imply there was a set of men who liked fewer women in the
whole set W . Namely, if a subset M0 M X liked only a strictly smaller
subset of women W0 W Y , then the set M0 [ X of men would like only
women in the strictly smaller set W0 [ Y . So again the strong induction hy-
pothesis implies we can match the men in M X with the women in W Y ,
which completes a matching for M .

So in both cases, there is a matching for the men, which completes the proof of
the Inductive step. The theorem follows by induction.

The proof of Theorem 12.5.2 gives an algorithm for finding a matching in a bi-
partite graph, albeit not a very efficient one. However, efficient algorithms for find-
ing a matching in a bipartite graph do exist. Thus, if a problem can be reduced to
finding a matching, instances of the problem can be solved in a reasonably efficient
way.

A Formal Statement
Let’s restate Theorem 12.5.2 in abstract terms so that you’ll not always be con-
demned to saying, “Now this group of men likes at least as many women. . . ”

Definition 12.5.3. A matching in a graph G is a set M of edges of G such that no
vertex is an endpoint of more than one edge in M . A matching is said to cover a
set S of vertices iff each vertex in S is an endpoint of an edge of the matching. A
matching is said to be perfect if it covers V .G/. In any graph G the set N.S / of
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12.5. Bipartite Graphs & Matchings 465

neighbors of some set S of vertices is the image of S under the edge-relation, that
is,
N.S/ WWD f r j hs—ri 2 E.G/ for some s 2 S g:
S is called a bottleneck if
jS j > j N.S /j:
Theorem 12.5.4 (Hall’s Theorem). Let G be a bipartite graph. There is a matching
in G that covers L.G/ iff no subset of L.G/ is a bottleneck.

An Easy Matching Condition
The bipartite matching condition requires that every subset of men has a certain
property. In general, verifying that every subset has some property, even if it’s easy
to check any particular subset for the property, quickly becomes overwhelming
because the number of subsets of even relatively small sets is enormous—over a
billion subsets for a set of size 30. However, there is a simple property of vertex
degrees in a bipartite graph that guarantees the existence of a matching. Call a
bipartite graph degree-constrained if vertex degrees on the left are at least as large
as those on the right. More precisely,
Definition 12.5.5. A bipartite graph G is degree-constrained when deg.l/ deg.r/
for every l 2 L.G/ and r 2 R.G/.
For example, the graph in Figure 12.9 is degree-constrained since every node on
the left is adjacent to at least two nodes on the right while every node on the right
is adjacent to at most two nodes on the left.
Theorem 12.5.6. If G is a degree-constrained bipartite graph, then there is a
matching that covers L.G/.
Proof. We will show that G satisfies Hall’s condition, namely, if S is an arbitrary
subset of L.G/, then
j N.S /j jS j: (12.2)
Since G is degree-constrained, there is a d > 0 such that deg.l/ d deg.r/
for every l 2 L and r 2 R. Since every edge with an endpoint in S has its other
endpoint in N.S / by definition, and every node in N.S / is incident to at most d
edges, we know that

d j N.S /j #edges with an endpoint in S :

Also, since every node in S is the endpoint of at least d edges,

#edges incident to a vertex in S d jS j:
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466 Chapter 12 Simple Graphs

It follows that d j N.S /j d jSj. Cancelling d completes the derivation of equa-
tion (12.2).

Regular graphs are a large class of degree-constrained graphs that often arise in
practice. Hence, we can use Theorem 12.5.6 to prove that every regular bipartite
graph has a perfect matching. This turns out to be a surprisingly useful result in
computer science.

Definition 12.5.7. A graph is said to be regular if every node has the same degree.

Theorem 12.5.8. Every regular bipartite graph has a perfect matching.

Proof. Let G be a regular bipartite graph. Since regular graphs are degree-constrained,
we know by Theorem 12.5.6 that there must be a matching in G that covers L.G/.
Such a matching is only possible when jL.G/j jR.G/j. But G is also degree-
constrained if the roles of L.G/ and R.G/ are switched, which implies that jR.G/j
jL.G/j also. That is, L.G/ and R.G/ are the same size, and any matching covering
L.G/ will also cover R.G/. So every node in G is an endpoint of an edge in the
matching, and thus G has a perfect matching.

12.6 Coloring
In Section 12.2, we used edges to indicate an affinity between a pair of nodes.
But there are lots of situations in which edges will correspond to conflicts between
nodes. Exam scheduling is a typical example.

12.6.1 An Exam Scheduling Problem
Each term, the MIT Schedules Office must assign a time slot for each final exam.
This is not easy, because some students are taking several classes with finals, and
(even at MIT) a student can take only one test during a particular time slot. The
Schedules Office wants to avoid all conflicts. Of course, you can make such a
schedule by having every exam in a different slot, but then you would need hun-
dreds of slots for the hundreds of courses, and the exam period would run all year!
So, the Schedules Office would also like to keep exam period short.
The Schedules Office’s problem is easy to describe as a graph. There will be a
vertex for each course with a final exam, and two vertices will be adjacent exactly
when some student is taking both courses. For example, suppose we need to sched-
ule exams for 6.041, 6.042, 6.002, 6.003 and 6.170. The scheduling graph might
appear as in Figure 12.11.
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12.6. Coloring 467

6:170

6:002 6:003

6:041 6:042

Figure 12.11 A scheduling graph for five exams. Exams connected by an edge
cannot be given at the same time.

blue

red green

green blue

Figure 12.12 A 3-coloring of the exam graph from Figure 12.11.

6.002 and 6.042 cannot have an exam at the same time since there are students in
both courses, so there is an edge between their nodes. On the other hand, 6.042 and
6.170 can have an exam at the same time if they’re taught at the same time (which
they sometimes are), since no student can be enrolled in both (that is, no student
should be enrolled in both when they have a timing conflict).
We next identify each time slot with a color. For example, Monday morning
is red, Monday afternoon is blue, Tuesday morning is green, etc. Assigning an
exam to a time slot is then equivalent to coloring the corresponding vertex. The
main constraint is that adjacent vertices must get different colors—otherwise, some
student has two exams at the same time. Furthermore, in order to keep the exam
period short, we should try to color all the vertices using as few different colors as
possible. As shown in Figure 12.12, three colors suffice for our example.
The coloring in Figure 12.12 corresponds to giving one final on Monday morning
(red), two Monday afternoon (blue), and two Tuesday morning (green). Can we use
fewer than three colors? No! We can’t use only two colors since there is a triangle
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468 Chapter 12 Simple Graphs

in the graph, and three vertices in a triangle must all have different colors.
This is an example of a graph coloring problem: given a graph G, assign colors
to each node such that adjacent nodes have different colors. A color assignment
with this property is called a valid coloring of the graph—a “coloring,” for short.
A graph G is k-colorable if it has a coloring that uses at most k colors.

Definition 12.6.1. The minimum value of k for which a graph G has a valid color-
ing is called its chromatic number, .G/.

So G is k-colorable iff .G/ k.
In general, trying to figure out if you can color a graph with a fixed number of
colors can take a long time. It’s a classic example of a problem for which no fast
algorithms are known. In fact, it is easy to check if a coloring works, but it seems
really hard to find it. (If you figure out how, then you can get a $1 million Clay
prize.)

12.6.2 Some Coloring Bounds
There are some simple properties of graphs that give useful bounds on colorability.
The simplest property is being a cycle: an even-length closed cycle is 2-colorable.
Cycles in simple graphs by convention have positive length and so are not 1-
colorable. So
.Ceven / D 2:
On the other hand, an odd-length cycle requires 3 colors, that is,

.Codd / D 3: (12.3)

You should take a moment to think about why this equality holds.
Another simple example is a complete graph Kn :

.Kn / D n

since no two vertices can have the same color.
Being bipartite is another property closely related to colorability. If a graph is
bipartite, then you can color it with 2 colors using one color for the nodes on the
“left” and a second color for the nodes on the “right.” Conversely, graphs with
chromatic number 2 are all bipartite with all the vertices of one color on the “left”
and those with the other color on the right. Since only graphs with no edges—the
empty graphs—have chromatic number 1, we have:

Lemma 12.6.2. A graph G with at least one edge is bipartite iff .G/ D 2.
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12.6. Coloring 469

The chromatic number of a graph can also be shown to be small if the vertex
degrees of the graph are small. In particular, if we have an upper bound on the
degrees of all the vertices in a graph, then we can easily find a coloring with only
one more color than the degree bound.

Theorem 12.6.3. A graph with maximum degree at most k is .k C 1/-colorable.

Since k is the only nonnegative integer valued variable mentioned in the the-
orem, you might be tempted to try to prove this theorem using induction on k.
Unfortunately, this approach leads to disaster—we don’t know of any reasonable
way to do this and expect it would ruin your week if you tried it on a problem set.
When you encounter such a disaster using induction on graphs, it is usually best to
change what you are inducting on. In graphs, typical good choices for the induction
parameter are n, the number of nodes, or e, the number of edges.

Proof of Theorem 12.6.3. We use induction on the number of vertices in the graph,
which we denote by n. Let P .n/ be the proposition that an n-vertex graph with
maximum degree at most k is .k C 1/-colorable.
Base case (n D 1): A 1-vertex graph has maximum degree 0 and is 1-colorable, so
P .1/ is true.
Inductive step: Now assume that P .n/ is true, and let G be an .nC1/-vertex graph
with maximum degree at most k. Remove a vertex v (and all edges incident to it),
leaving an n-vertex subgraph H . The maximum degree of H is at most k, and so
H is .k C 1/-colorable by our assumption P .n/. Now add back vertex v. We can
assign v a color (from the set of k C 1 colors) that is different from all its adjacent
vertices, since there are at most k vertices adjacent to v and so at least one of the
k C 1 colors is still available. Therefore, G is .k C 1/-colorable. This completes
the inductive step, and the theorem follows by induction.

Sometimes k C 1 colors is the best you can do. For example, .Kn / D n
and every node in Kn has degree k D n 1 and so this is an example where
Theorem 12.6.3 gives the best possible bound. By a similar argument, we can
show that Theorem 12.6.3 gives the best possible bound for any graph with degree
bounded by k that has KkC1 as a subgraph.
But sometimes k C 1 colors is far from the best that you can do. For example,
the n-node star graph shown in Figure 12.13 has maximum degree n 1 but can
be colored using just 2 colors.
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470 Chapter 12 Simple Graphs

Figure 12.13 A 7-node star graph.

12.6.3 Why coloring?
One reason coloring problems frequently arise in practice is because scheduling
conflicts are so common. For example, at the internet company Akamai, cofounded
by Tom Leighton, a new version of software is deployed over each of its servers
(200,000 servers in 2016) every few days. It would take more than twenty years to
update all these the servers one at a time, so the deployment must be carried out
for many servers simultaneouly. On the other hand, certain pairs of servers with
common critical functions cannot be updated simultaneouly, since a server needs
to be taken offline while being updated.
This problem gets solved by making a 200,000-node conflict graph and coloring
it with with a dozen or so colors—so only a dozen or so waves of installs are
needed!
Another example comes from the need to assign frequencies to radio stations. If
two stations have an overlap in their broadcast area, they can’t be given the same
frequency. Frequencies are precious and expensive, it is important to minimize the
number handed out. This amounts to finding the minimum coloring for a graph
whose vertices are the stations and whose edges connect stations with overlapping
areas.
Coloring also comes up in allocating registers for program variables. While a
variable is in use, its value needs to be saved in a register. Registers can be reused
for different variables, but two variables need different registers if they are refer-
enced during overlapping intervals of program execution. So register allocation is
the coloring problem for a graph whose vertices are the variables: vertices are ad-
jacent if their intervals overlap, and the colors are registers. Once again, the goal is
to minimize the number of colors needed to color the graph.
Finally, there’s the famous map coloring problem stated in Proposition 1.1.4. The
question is how many colors are needed to color a map so that adjacent territories
get different colors? This is the same as the number of colors needed to color a
graph that can be drawn in the plane without edges crossing. A proof that four
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12.7. Simple Walks 471

colors are enough for planar graphs was acclaimed when it was discovered about
forty years ago. Implicit in that proof was a 4-coloring procedure that takes time
proportional to the number of vertices in the graph (countries in the map).
Surprisingly, it’s another of those million dollar prize questions to find an effi-
cient procedure to tell if any particular planar graph really needs four colors, or if
three will actually do the job. A proof that testing 3-colorability of graphs is as hard
as the million dollar SAT problem is given in Problem 12.29; this turns out to be
true even for planar graphs. (It is easy to tell if a graph is 2-colorable, as explained
in Section 12.8.2.) In Chapter 13, we’ll develop enough planar graph theory to
present an easy proof that all planar graphs are 5-colorable.

12.7 Simple Walks
12.7.1 Walks, Paths, Cycles in Simple Graphs
Walks and paths in simple graphs are esentially the same as in digraphs. We just
modify the digraph definitions using undirected edges instead of directed ones. For
example, the formal definition of a walk in a simple graph is a virtually the same
as the Definition 10.2.1 of a walk in a digraph:

Definition 12.7.1. A walk in a simple graph G is an alternating sequence of vertices
and edges that begins with a vertex, ends with a vertex, and such that for every edge
hu—vi in the walk, one of the endpoints u, v is the element just before the edge,
and the other endpoint is the next element after the edge. The length of a walk is
the total number of occurrences of edges in it.
So a walk v is a sequence of the form

v WWD v0 hv0 —v1 i v1 hv1 —v2 i v2 : : : hvk 1 —vk i vk

where hvi —vi C1 i 2 E.G/ for i 2 Œ0::k/. The walk is said to start at v0 , to end
at vk , and the length, jvj, of the walk is k. The walk is a path iff all the vi ’s are
different, that is, if i ¤ j , then vi ¤ vj .
A walk that begins and ends at the same vertex is a closed walk. A single vertex
counts as a length zero closed walk as well as a length zero path.
A cycle can be represented by a closed walk of length three or more whose
vertices are distinct except for the beginning and end vertices.

Note that in contrast to digraphs, we don’t count length two closed walks as
cycles in simple graphs. That’s because a walk going back and forth on the same
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472 Chapter 12 Simple Graphs

b d e

c
a g h
f

Figure 12.14 A graph with 3 cycles: bhecb, cdec, bcdehb.

edge is always possible in a simple graph, and it has no importance. Also, there are
no closed walks of length one, since simple graphs don’t have self loops.
As in digraphs, the length of a walk is one less than the number of occurrences of
vertices in it. For example, the graph in Figure 12.14 has a length 6 path through the
seven successive vertices abcdefg. This is the longest path in the graph. The graph
in Figure 12.14 also has three cycles through successive vertices bhecb, cdec and
bcdehb.

12.7.2 Cycles as Subgraphs
We don’t want think of a cycle as having a beginning or an end, so any of the paths
that go around it can represent the cycle. For example, in the graph in Figure 12.14,
the cycle starting at b and going through vertices bcdehb can also be described as
starting at d and going through dehbcd . Furthermore, cycles in simple graphs
don’t have a direction: dcbhed describes the same cycle as though it started and
ended at d but went in the opposite direction.
A precise way to explain which closed walks represent the same cycle is to define
cycle as a subgraph. Specifically, we could define a cycle in G to be a subgraph of
G that looks like a length-n cycle for n 3.
Definition 12.7.2. A graph G is said to be a subgraph of a graph H if V .G/
V .H / and E.G/ E.H /.
For example, the one-edge graph G where

V .G/ D fg; h; i g and E.G/ D f hh—i i g

is a subgraph of the graph H in Figure 12.1. On the other hand, any graph con-
taining an edge hg—hi will not be a subgraph of H because this edge is not in
E.H /. Another example is an empty graph on n nodes, which will be a subgraph
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12.8. Connectivity 473

of an Ln with the same set of nodes; similarly, Ln is a subgraph of Cn , and Cn is
a subgraph of Kn .

Definition 12.7.3. For n 3, let Cn be the graph with vertices 1; : : : ; n and edges

h1—2i ; h2—3i ; : : : ; h.n 1/—ni ; hn—1i :

A cycle of a graph G is a subgraph of G that is isomorphic to Cn for some n 3.

This definition formally captures the idea that cycles don’t have direction or be-
ginnings or ends.

12.8 Connectivity
Definition 12.8.1. Two vertices are connected in a graph when there is a path that
begins at one and ends at the other. By convention, every vertex is connected to
itself by a path of length zero. A graph is connected when every pair of vertices
are connected.

12.8.1 Connected Components
Being connected is usually a good property for a graph to have. For example, it
could mean that it is possible to get from any node to any other node, or that it is
possible to communicate between any pair of nodes, depending on the application.
But not all graphs are connected. For example, the graph where nodes represent
cities and edges represent highways might be connected for North American cities,
but would surely not be connected if you also included cities in Australia. The
same is true for communication networks like the internet—in order to be protected
from viruses that spread on the internet, some government networks are completely
isolated from the internet.

Figure 12.15 One graph with 3 connected components.
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474 Chapter 12 Simple Graphs

Another example is shown in Figure 12.15, which looks like a picture of three
graphs, but is intended to be a picture of one graph. This graph consists of three
pieces. Each piece is a subgraph that by itself is connected, but there are no paths
between vertices in different pieces. These connected pieces of a graph are called
its connected components.

Definition 12.8.2. A connected component of a graph is a subgraph consisting of
some vertex and every node and edge that is connected to that vertex.

So, a graph is connected iff it has exactly one connected component. At the
other extreme, the empty graph on n vertices has n connected components, each
consisting of a single vertex.

12.8.2 Odd Cycles and 2-Colorability
We have already seen that determining the chromatic number of a graph is a chal-
lenging problem. There is one special case where this problem is very easy, namely,
when the graph is 2-colorable.

Theorem 12.8.3. The following graph properties are equivalent:

1. The graph contains an odd length cycle.

2. The graph is not 2-colorable.

3. The graph contains an odd length closed walk.

In other words, if a graph has any one of the three properties above, then it has
all of the properties.
We will show the following implications among these properties:

1. IMPLIES 2. IMPLIES 3. IMPLIES 1:

So each of these properties implies the other two, which means they all are equiva-
lent.

1 IMPLIES 2 Proof. This follows from equation 12.3.

2 IMPLIES 3 If we prove this implication for connected graphs, then it will hold
for an arbitrary graph because it will hold for each connected component. So
we can assume that G is connected.
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12.8. Connectivity 475

Proof. Pick an arbitrary vertex r of G. Since G is connected, for every node
u 2 V .G/, there will be a walk wu starting at u and ending at r. Assign
colors to vertices of G as follows:
(
black; if jwu j is even;
color.u/ D
white; otherwise:

Now since G is not colorable, this can’t be a valid coloring. So there must
be an edge between two nodes u and v with the same color. But in that case

wu breverse.wv /b hv—ui

is a closed walk starting and ending at u, and its length is

jwu j C jwv j C 1

which is odd.

3 IMPLIES 1 Proof. Since there is an odd length closed walk, the WOP implies
there is an odd length closed walk w of minimum length. We claim w must
be a cycle. To show this, assume to the contrary that w is not a cycle, so
there is a repeat vertex occurrence besides the start and end. There are then
two cases to consider depending on whether the additional repeat is different
from, or the same as, the start vertex.
In the first case, the start vertex has an extra occurrence. That is,

w D fb
xr

for some positive length walks f and r that begin and end at x. Since

jwj D jfj C jrj

is odd, exactly one of f and r must have odd length, and that one will be an
odd length closed walk shorter than w, a contradiction.
In the second case,
w D fb
y gb
yr
where f is a walk from x to y for some y ¤ x, and r is a walk from y to
x, and jgj > 0. Now g cannot have odd length or it would be an odd-length
y r must
closed walk shorter than w. So g has even length. That implies that fb
be an odd-length closed walk shorter than w, again a contradiction.
This completes the proof of Theorem 12.8.3.
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476 Chapter 12 Simple Graphs

Theorem 12.8.3 turns out to be useful, since bipartite graphs come up fairly often
in practice.5

12.8.3 k-connected Graphs
If we think of a graph as modeling cables in a telephone network, or oil pipelines,
or electrical power lines, then we not only want connectivity, but we want connec-
tivity that survives component failure. So more generally, we want to define how
strongly two vertices are connected. One measure of connection strength is how
many links must fail before connectedness fails. In particular, two vertices are k-
edge connected when it takes at least k “edge-failures” to disconnect them. More
precisely:

Definition 12.8.4. Two vertices in a graph are k-edge connected when they remain
connected in every subgraph obtained by deleting up to k 1 edges. A graph is k-
edge connected when it has more than one vertex, and every pair of distinct vertices
in the graph are k-edge connected.

From now on we’ll drop the “edge” modifier and just say “k-connected.”6
Notice that according to Definition 12.8.4, if a graph is k-connected, it is also
j -connected for j k. This convenient convention implies that two vertices are
connected according to definition 12.8.1 iff they are 1-connected according to Def-
inition 12.8.4.
For example, in the graph in figure 12.14, vertices c and e are 3-connected, b
and e are 2-connected, g and e are 1 connected, and no vertices are 4-connected.
The graph as a whole is only 1-connected. A complete graph Kn is .n 1/-
connected. Every cycle is 2-connected.
The idea of a cut edge is a useful way to explain 2-connectivity.

Definition 12.8.5. If two vertices are connected in a graph G, but not connected
when an edge e is removed, then e is called a cut edge of G.

So a graph with more than one vertex is 2-connected iff it is connected and
has no cut edges. The following Lemma is another immediate consequence of the
definition:

Lemma 12.8.6. An edge is a cut edge iff it is not on a cycle.
5 One example concerning routing networks already came up in Lemma 11.3.3. Corollary 13.5.4
reveals the importance of another example in planar graph theory.
6 There is a corresponding definition of k-vertex connectedness based on deleting vertices rather

than edges. Graph theory texts usually use “k-connected” as shorthand for “k-vertex connected.” But
edge-connectedness will be enough for us.
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12.8. Connectivity 477

More generally, if two vertices are connected by k edge-disjoint paths—that is,
no edge occurs in two paths—then they must be k-connected, since at least one
edge will have to be removed from each of the paths before they could disconnect.
A fundamental fact, whose ingenious proof we omit, is Menger’s theorem which
confirms that the converse is also true: if two vertices are k-connected, then there
are k edge-disjoint paths connecting them. It takes some ingenuity to prove this
just for the case k D 2.

12.8.4 The Minimum Number of Edges in a Connected Graph
The following theorem says that a graph with few edges must have many connected
components.

Theorem 12.8.7. Every graph G has at least jV .G/j jE.G/j connected compo-
nents.

Of course for Theorem 12.8.7 to be of any use, there must be fewer edges than
vertices.

Proof. We use induction on the number k of edges. Let P .k/ be the proposition
that

every graph G with k edges has at least jV .G/j k connected compo-
nents.

Base case (k D 0): In a graph with 0 edges, each vertex is itself a connected
component, and so there are exactly jV .G/j D jV .G/j 0 connected components.
So P .0/ holds.
Inductive step:
Let Ge be the graph that results from removing an edge, e 2 E.G/. So Ge
has k edges, and by the induction hypothesis P .k/, we may assume that Ge has
at least .jV .G/j k/ connected components. Now add back the edge e to obtain
the original graph G. If the endpoints of e were in the same connected component
of Ge , then G has the same sets of connected vertices as Ge , so G has at least
.jV .G/j k/ > .jV .G/j .k C 1// components. Alternatively, if the endpoints
of e were in different connected components of Ge , then these two components are
merged into one component in G, while all other components remain unchanged,
so that G has one fewer connected component than Ge . That is, G has at least
.jV .G/j k/ 1 D .jV .G/j .k C 1// connected components. So in either case,
G has at least jV .G/j .k C 1/ components, as claimed.
This completes the inductive step and hence the entire proof by induction.
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478 Chapter 12 Simple Graphs

Figure 12.16 A 6-node forest consisting of 2 component trees.

Corollary 12.8.8. Every connected graph with n vertices has at least n 1 edges.

A couple of points about the proof of Theorem 12.8.7 are worth noticing. First,
we used induction on the number of edges in the graph. This is very common in
proofs involving graphs, as is induction on the number of vertices. When you’re
presented with a graph problem, these two approaches should be among the first
you consider.
The second point is more subtle. Notice that in the inductive step, we took an
arbitrary .k C1/-edge graph, threw out an edge so that we could apply the induction
assumption, and then put the edge back. You’ll see this shrink-down, grow-back
process very often in the inductive steps of proofs related to graphs. This might
seem like needless effort: why not start with an k-edge graph and add one more to
get an .k C 1/-edge graph? That would work fine in this case, but opens the door
to a nasty logical error called buildup error, illustrated in Problem 12.40.

12.9 Forests & Trees
We’ve already made good use of digraphs without cycles, but simple graphs without
cycles are arguably the most important graphs in computer science.

12.9.1 Leaves, Parents & Children
Definition 12.9.1. An acyclic graph is called a forest. A connected acyclic graph
is called a tree.

The graph shown in Figure 12.16 is a forest. Each of its connected components
is by definition a tree.
One of the first things you will notice about trees is that they tend to have a lot
of nodes with degree one. Such nodes are called leaves.

Definition 12.9.2. A degree 1 node in a forest is called a leaf.

The forest in Figure 12.16 has 4 leaves. The tree in Figure 12.17 has 5 leaves.
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12.9. Forests & Trees 479

a e h
c

g i
b d
f

Figure 12.17 A 9-node tree with 5 leaves.
e

d g

b c f h i

Figure 12.18 The tree from Figure 12.17 redrawn with node e as the root and the
other nodes arranged in levels.

Trees are a fundamental data structure in computer science. For example, in-
formation is often stored in tree-like data structures, and the execution of many
recursive programs can be modeled as the traversal of a tree. In such cases, it is
often useful to arrange the nodes in levels, where the node at the top level is iden-
tified as the root and where every edge joins a parent to a child one level below.
Figure 12.18 shows the tree of Figure 12.17 redrawn in this way. Node d is a child
of node e and the parent of nodes b and c.

12.9.2 Properties
Trees have many unique properties. We have listed some of them in the following
theorem.

Theorem 12.9.3. Every tree has the following properties:

1. Every connected subgraph is a tree.

2. There is a unique path between every pair of vertices.
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480 Chapter 12 Simple Graphs

3. Adding an edge between nonadjacent nodes in a tree creates a graph with a
cycle.

4. Removing any edge disconnects the graph. That is, every edge is a cut edge.

5. If the tree has at least two vertices, then it has at least two leaves.

6. The number of vertices in a tree is one larger than the number of edges.

Proof. 1. A cycle in a subgraph is also a cycle in the whole graph, so any sub-
graph of an acyclic graph must also be acyclic. If the subgraph is also con-
nected, then by definition, it is a tree.

2. Since a tree is connected, there is at least one path between every pair of ver-
tices. Suppose for the purposes of contradiction, that there are two different
paths between some pair of vertices. Then there are two distinct paths p ¤ q
between the same two vertices with minimum total length jpj C jqj. If these
paths shared a vertex w other than at the start and end of the paths, then the
parts of p and q from start to w, or the parts of p and q from w to the end,
must be distinct paths between the same vertices with total length less than
jpj C jqj, contradicting the minimality of this sum. Therefore, p and q have
no vertices in common besides their endpoints, and so p breverse.q/ is a
cycle.

3. An additional edge hu—vi together with the unique path between u and v
forms a cycle.

4. Suppose that we remove edge hu—vi. Since the tree contained a unique path
between u and v, that path must have been hu—vi. Therefore, when that
edge is removed, no path remains, and so the graph is not connected.

5. Since the tree has at least two vertices, the longest path in the tree will have
different endpoints u and v. We claim u is a leaf. This follows because, by
definition of endpoint, u is incident to at most one edge on the path. Also, if u
was incident to an edge not on the path, then the path could be lengthened by
adding that edge, contradicting the fact that the path was as long as possible.
It follows that u is incident only to a single edge, that is u is a leaf. The same
hold for v.

6. We use induction on the proposition

P .n/ WWD there are n 1 edges in any n-vertex tree:
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12.9. Forests & Trees 481

Figure 12.19 A graph where the edges of a spanning tree have been thickened.

Base case (n D 1): P .1/ is true since a tree with 1 node has 0 edges and
1 1 D 0.

Inductive step: Now suppose that P .n/ is true and consider an .nC1/-vertex
tree T . Let v be a leaf of the tree. You can verify that deleting a vertex of
degree 1 (and its incident edge) from any connected graph leaves a connected
subgraph. So by Theorem 12.9.3.1, deleting v and its incident edge gives a
smaller tree, and this smaller tree has n 1 edges by induction. If we re-
attach the vertex v and its incident edge, we find that T has n D .n C 1/ 1
edges. Hence, P .n C 1/ is true, and the induction proof is complete.

Various subsets of properties in Theorem 12.9.3 provide alternative characteri-
zations of trees. For example,

Lemma 12.9.4. A graph G is a tree iff G is a forest and jV .G/j D jE.G/j C 1.

The proof is an easy consequence of Theorem 12.9.3.6 (Problem 12.47).

12.9.3 Spanning Trees
Trees are everywhere. In fact, every connected graph contains a subgraph that is a
tree with the same vertices as the graph. This is called a spanning tree for the graph.
For example, Figure 12.19 is a connected graph with a spanning tree highlighted.

Definition 12.9.5. Define a spanning subgraph of a graph G to be a subgraph
containing all the vertices of G.

Theorem 12.9.6. Every connected graph contains a spanning tree.

Proof. Suppose G is a connected graph, so the graph G itself is a connected, span-
ning subgraph. So by WOP, G must have a minimum-edge connected, spanning
subgraph T . We claim T is a spanning tree. Since T is a connected, spanning
subgraph by definition, all we have to show is that T is acyclic.
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482 Chapter 12 Simple Graphs

1
2 2 3
3 3
2 2
3 3
1 1 4 1 1

1
7 7
(a) (b)

Figure 12.20 A spanning tree (a) with weight 19 for a graph (b).

But suppose to the contrary that T contained a cycle C . By Lemma 12.8.6,
an edge e of C will not be a cut edge, so removing it would leave a connected,
spanning subgraph that was smaller than T , contradicting the minimality to T .

12.9.4 Minimum Weight Spanning Trees
Spanning trees are interesting because they connect all the nodes of a graph using
the smallest possible number of edges. For example the spanning tree for the 6-
node graph shown in Figure 12.19 has 5 edges.
In many applications, there are numerical costs or weights associated with the
edges of the graph. For example, suppose the nodes of a graph represent buildings
and edges represent connections between them. The cost of a connection may vary
a lot from one pair of buildings or towns to another. Another example is where the
nodes represent cities and the weight of an edge is the distance between them: the
weight of the Los Angeles/New York City edge is much higher than the weight of
the NYC/Boston edge. The weight of a graph is simply defined to be the sum of
the weights of its edges. For example, the weight of the spanning tree shown in
Figure 12.20 is 19.

Definition 12.9.7. A minimum weight spanning tree (MST) of an edge-weighted
graph G is a spanning tree of G with the smallest possible sum of edge weights.

Is the spanning tree shown in Figure 12.20(a) an MST of the weighted graph
shown in Figure 12.20(b)? It actually isn’t, since the tree shown in Figure 12.21 is
also a spanning tree of the graph shown in Figure 12.20(b), and this spanning tree
has weight 17.
What about the tree shown in Figure 12.21? It seems to be an MST, but how do
we prove it? In general, how do we find an MST for a connected graph G? We
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12.9. Forests & Trees 483

1
2

2
3
1 1

Figure 12.21 An MST with weight 17 for the graph in Figure 12.20(b).

could try enumerating all subtrees of G, but that approach would be hopeless for
large graphs.
There actually are many good ways to find MST’s based on a property of some
subgraphs of G called pre-MST’s.

Definition 12.9.8. A pre-MST for a graph G is a spanning subgraph of G that is
also a subgraph of some MST of G.

So a pre-MST will necessarily be a forest.
For example, the empty graph with the same vertices as G is guaranteed to be a
pre-MST of G, and so is any actual MST of G.
If e is an edge of G and S is a spanning subgraph, we’ll write S C e for the
spanning subgraph with edges E.S / [ feg.

Definition 12.9.9. If F is a pre-MST and e is a new edge, that is e 2 E.G/ E.F /,
then e extends F when F C e is also a pre-MST.

So being a pre-MST is contrived to be an invariant under addition of extending
edges, by the definition of extension.
The standard methods for finding MST’s all start with the empty spanning forest
and build up to an MST by adding one extending edge after another. Since the
empty spanning forest is a pre-MST, and being a pre-MST is, by definition, in-
variant under extensions, every forest built in this way will be a pre-MST. But no
spanning tree can be a subgraph of a different spanning tree. So when the pre-MST
finally grows enough to become a tree, it will be an MST. By Lemma 12.9.4, this
happens after exactly jV .G/j 1 edge extensions.
So the problem of finding MST’s reduces to the question of how to tell if an edge
is an extending edge. Here’s how:
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484 Chapter 12 Simple Graphs

Definition 12.9.10. Let F be a pre-MST, and color the vertices in each connected
component of F either all black or all white. At least one component of each color
is required. Call this a solid coloring of F . A gray edge of a solid coloring is an
edge of G with different colored endpoints.

Any path in G from a white vertex to a black vertex obviously must include a
gray edge, so for any solid coloring, there is guaranteed to be at least one gray edge.
In fact, there will have to be at least as many gray edges as there are components
with the same color. Here’s the punchline:

Lemma 12.9.11. An edge extends a pre-MST F if it is a minimum weight gray edge
in some solid coloring of F .

So to extend a pre-MST, choose any solid coloring, find the gray edges, and
among them choose one with minimum weight. Each of these steps is easy to do,
so it is easy to keep extending and arrive at an MST. For example, here are three
known algorithms that are explained by Lemma 12.9.11:

Algorithm 1. [Prim] Grow a tree one edge at a time by adding a minimum weight
edge among the edges that have exactly one endpoint in the tree.

This is the algorithm that comes from coloring the growing tree white and all the
vertices not in the tree black. Then the gray edges are the ones with exactly one
endpoint in the tree.

Algorithm 2. [Kruskal] Grow a forest one edge at a time by adding a minimum
weight edge among the edges with endpoints in different connected components.

An edge does not create a cycle iff it connects different components. The edge
chosen by Kruskal’s algorithm will be the minimum weight gray edge when the
components it connects are assigned different colors.
For example, in the weighted graph we have been considering, we might run
Algorithm 1 as follows. Start by choosing one of the weight 1 edges, since this
is the smallest weight in the graph. Suppose we chose the weight 1 edge on the
bottom of the triangle of weight 1 edges in our graph. This edge is incident to the
same vertex as two weight 1 edges, a weight 4 edge, a weight 7 edge, and a weight 3
edge. We would then choose the incident edge of minimum weight. In this case,
one of the two weight 1 edges. At this point, we cannot choose the third weight 1
edge: it won’t be gray because its endpoints are both in the tree, and so are both
colored white. But we can continue by choosing a weight 2 edge. We might end
up with the spanning tree shown in Figure 12.22, which has weight 17, the smallest
we’ve seen so far.
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12.9. Forests & Trees 485

1
2 3

2
1

1
7

Figure 12.22 A spanning tree found by Algorithm 1.

Now suppose we instead ran Algorithm 2 on our graph. We might again choose
the weight 1 edge on the bottom of the triangle of weight 1 edges in our graph.
Now, instead of choosing one of the weight 1 edges it touches, we might choose
the weight 1 edge on the top of the graph. This edge still has minimum weight, and
will be gray if we simply color its endpoints differently, so Algorithm 2 can choose
it. We would then choose one of the remaining weight 1 edges. Note that neither
causes us to form a cycle. Continuing the algorithm, we could end up with the same
spanning tree in Figure 12.22, though this will depend on the tie breaking rules used
to choose among gray edges with the same minimum weight. For example, if the
weight of every edge in G is one, then all spanning trees are MST’s with weight
jV .G/j 1, and both of these algorithms can arrive at each of these spanning trees
by suitable tie-breaking.
The coloring that explains Algorithm 1 also justifies a more flexible algorithm
which has Algorithm 1 as a special case:
Algorithm 3. Grow a forest one edge at a time by picking any component and
adding a minimum weight edge among the edges leaving that component.
This algorithm allows components that are not too close to grow in parallel and
independently, which is great for “distributed” computation where separate proces-
sors share the work with limited communication between processors.7
These are examples of greedy approaches to optimization. Sometimes greediness
works and sometimes it doesn’t. The good news is that it does work to find the
MST. Therefore, we can be sure that the MST for our example graph has weight 17,
since it was produced by Algorithm 2. Furthermore we have a fast algorithm for
finding a minimum weight spanning tree for any graph.
7 The idea of growing trees seems first to have been developed in by Boru̇vka (1926), ref TBA.

Efficient MST algorithms running in parallel time O.log jV j/ are described in Karger, Klein, and
Tarjan (1995), ref TBA.
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486 Chapter 12 Simple Graphs

Ok, to wrap up this story, all that’s left is the proof that minimal gray edges are
extending edges. This might sound like a chore, but it just uses the same reasoning
we used to be sure there would be a gray edge when you need it.

Proof. (of Lemma 12.9.11)
Let F be a pre-MST that is a subgraph of some MST M of G, and suppose e is a
minimum weight gray edge under some solid coloring of F . We want to show that
F C e is also a pre-MST.
If e happens to be an edge of M , then F C e remains a subgraph of M , and so
is a pre-MST.
The other case is when e is not an edge of M . In that case, M C e will be a
connected, spanning subgraph. Also M has a path p between the different colored
endpoints of e, so M C e has a cycle consisting of e together with p. Now p has
both a black endpoint and a white one, so it must contain some gray edge g ¤ e.
The trick is to remove g from M C e to obtain a subgraph M C e g. Since gray
edges by definition are not edges of F , the graph M C e g contains F C e. We
claim that M C e g is an MST, which proves the claim that e extends F .
To prove this claim, note that M C e is a connected, spanning subgraph, and
g is on a cycle of M C e, so by Lemma 12.8.6, removing g won’t disconnect
anything. Therefore, M C e g is still a connected, spanning subgraph. Moreover,
M C e g has the same number of edges as M , so Lemma 12.9.4 implies that it
must be a spanning tree. Finally, since e is minimum weight among gray edges,

w.M C e g/ D w.M / C w.e/ w.g/ w.M /:

This means that M C e g is a spanning tree whose weight is at most that of an
MST, which implies that M C e g is also an MST.

Another interesting fact falls out of the proof of Lemma 12.9.11:

Corollary 12.9.12. If all edges in a weighted graph have distinct weights, then the
graph has a unique MST.

The proof of Corollary 12.9.12 is left to Problem 12.63.

12.10 References
[8], [13], [22], [25], [27]
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12.10. References 487

Problems for Section 12.2
Practice Problems
Problem 12.1.
The average degree of the vertices in an n-vertex graph is twice the average number
of edges of per vertex. Explain why.

Problem 12.2.
Among connected simple graphs whose sum of vertex degrees is 20:
(a) what is the largest possible number of vertices?

(b) what is the smallest possible number of vertices?

Class Problems
Problem 12.3. (a) Prove that in every simple graph, there are an even number of
vertices of odd degree.

(b) Conclude that at a party where some people shake hands, the number of people
who shake hands an odd number of times is an even number.

(c) Call a sequence of people at the party a handshake sequence if each person in
the sequence has shaken hands with the next person, if any, in the sequence.
Suppose George was at the party and has shaken hands with an odd number of
people. Explain why, starting with George, there must be a handshake sequence
ending with a different person who has shaken an odd number of hands.

Exam Problems
Problem 12.4.
A researcher analyzing data on heterosexual sexual behavior in a group of m males
and f females found that within the group, the male average number of female
partners was 10% larger that the female average number of male partners.
(a) Comment on the following claim. “Since we’re assuming that each encounter
involves one man and one woman, the average numbers should be the same, so the
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488 Chapter 12 Simple Graphs

males must be exaggerating.”

(b) For what constant c is m D c f ?

(c) The data shows that approximately 20% of the females were virgins, while
only 5% of the males were. The researcher wonders how excluding virgins from
the population would change the averages. If he knew graph theory, the researcher
would realize that the nonvirgin male average number of partners will be x.f =m/
times the nonvirgin female average number of partners. What is x?

(d) For purposes of further research, it would be helpful to pair each female in the
group with a unique male in the group. Explain why this is not possible.

Problems for Section 12.4
Practice Problems
Problem 12.5.
Which of the items below are simple-graph properties preserved under isomor-
phism?
(a) There is a cycle that includes all the vertices.

(b) The vertices are numbered 1 through 7.

(d) There are two degree 8 vertices.

(e) Two edges are of equal length.

(f) No matter which edge is removed, there is a path between any two vertices.

(g) There are two cycles that do not share any vertices.

(h) The vertices are sets.

(i) The graph can be drawn in a way that all the edges have the same length.

(j) No two edges cross.

(k) The OR of two properties that are preserved under isomorphism.

(l) The negation of a property that is preserved under isomorphism.
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12.10. References 489

Class Problems
Problem 12.6.
For each of the following pairs of simple graphs, either define an isomorphism
between them, or prove that there is none. (We write ab as shorthand for ha—bi.)
(a)

G1 with V1 D f1; 2; 3; 4; 5; 6g; E1 D f12; 23; 34; 14; 15; 35; 45g
G2 with V2 D f1; 2; 3; 4; 5; 6g; E2 D f12; 23; 34; 45; 51; 24; 25g

(b)

G3 with V3 D f1; 2; 3; 4; 5; 6g; E3 D f12; 23; 34; 14; 45; 56; 26g
G4 with V4 D fa; b; c; d; e; f g; E4 D fab; bc; cd; de; ae; ef; cf g

Problem 12.7.
List all the isomorphisms between the two graphs given in Figure 12.23. Explain
why there are no others.

1 a

3 2 4 c b d

5 6 e f

Figure 12.23 Graphs with several isomorphisms

Homework Problems
Problem 12.8.
Determine which among the four graphs pictured in Figure 12.24 are isomorphic.
For each pair of isomorphic graphs, describe an isomorphism between them. For
each pair of graphs that are not isomorphic, give a property that is preserved under
isomorphism such that one graph has the property, but the other does not. For
at least one of the properties you choose, prove that it is indeed preserved under
isomorphism (you only need prove one of them).
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490 Chapter 12 Simple Graphs

1 1

6 6
5 8 9 2 5 9 7 2

10 7 10 8

4 3 4 3
(a) G1 (b) G2

1
1
9 2
6
5 9 7 2 8 3
10

10 8 7 4

4 3 6 5
(c) G3 (d) G4

Figure 12.24 Which graphs are isomorphic?
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12.10. References 491

Problem 12.9. (a) For any vertex v in a graph, let N.v/ be the set of neighbors of
v, namely, the vertices adjacent to v:

N.v/ WWD fu j hu—vi is an edge of the graphg:

Suppose f is an isomorphism from graph G to graph H . Prove that f .N.v// D
N.f .v//.
Your proof should follow by simple reasoning using the definitions of isomorphism
and neighbors—no pictures or handwaving.
Hint: Prove by a chain of iff’s that

h 2 N.f .v// iff h 2 f .N.v//

for every h 2 VH . Use the fact that h D f .u/ for some u 2 VG .

(b) Conclude that if G and H are isomorphic graphs, then for each k 2 N, they
have the same number of degree k vertices.

Problem 12.10.
Let’s say that a graph has “two ends” if it has exactly two vertices of degree 1 and
all its other vertices have degree 2. For example, here is one such graph:

(a) A line graph is a graph whose vertices can be listed in a sequence with edges
between consecutive vertices only. So the two-ended graph above is also a line
graph of length 4.
Prove that the following theorem is false by drawing a counterexample.
False Theorem. Every two-ended graph is a line graph.

(b) Point out the first erroneous statement in the following bogus proof of the false
theorem and describe the error.

Bogus proof. We use induction. The induction hypothesis is that every two-ended
graph with n edges is a line graph.
Base case (n D 1): The only two-ended graph with a single edge consists of two
vertices joined by an edge:
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492 Chapter 12 Simple Graphs

Sure enough, this is a line graph.
Inductive case: We assume that the induction hypothesis holds for some n 1
and prove that it holds for n C 1. Let Gn be any two-ended graph with n edges.
By the induction assumption, Gn is a line graph. Now suppose that we create a
two-ended graph GnC1 by adding one more edge to Gn . This can be done in only
one way: the new edge must join one of the two endpoints of Gn to a new vertex;
otherwise, GnC1 would not be two-ended.
gn

↑
new edge

Clearly, GnC1 is also a line graph. Therefore, the induction hypothesis holds for
all graphs with n C 1 edges, which completes the proof by induction.

Problems for Section 12.5
Practice Problems
Problem 12.11.
Let B be a bipartite graph with vertex sets L.B/; R.B/. Explain why the sum of
the degrees of the vertices in L.B/ equals the sum of the degrees of the vertices in
R.B/.

Class Problems
Problem 12.12.
A certain Institute of Technology has a lot of student clubs; these are loosely over-
seen by the Student Association. Each eligible club would like to delegate one of its
members to appeal to the Dean for funding, but the Dean will not allow a student to
be the delegate of more than one club. Fortunately, the Association VP took Math
for Computer Science and recognizes a matching problem when she sees one.
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12.10. References 493

(a) Explain how to model the delegate selection problem as a bipartite matching
problem. (This is a modeling problem; we aren’t looking for a description of an
algorithm to solve the problem.)

(b) The VP’s records show that no student is a member of more than 9 clubs. The
VP also knows that to be eligible for support from the Dean’s office, a club must
have at least 13 members. That’s enough for her to guarantee there is a proper
delegate selection. Explain. (If only the VP had taken an Algorithms class, she
could even have found a delegate selection without much effort.)

Problem 12.13.
A simple graph is called regular when every vertex has the same degree. Call
a graph balanced when it is regular and is also a bipartite graph with the same
number of left and right vertices.
Prove that if G is a balanced graph, then the edges of G can be partitioned into
blocks such that each block is a perfect matching.
For example, if G is a balanced graph with 2k vertices each of degree j , then the
edges of G can be partitioned into j blocks, where each block consists of k edges,
each of which is a perfect matching.

Exam Problems
Problem 12.14.
Overworked and over-caffeinated, the Teaching Assistant’s (TA’s) decide to oust
the lecturer and teach their own recitations. They will run a recitation session at 4
different times in the same room. There are exactly 20 chairs to which a student can
be assigned in each recitation. Each student has provided the TA’s with a list of the
recitation sessions her schedule allows and each student’s schedule conflicts with
at most two sessions. The TA’s must assign each student to a chair during recitation
at a time she can attend, if such an assignment is possible.
(a) Describe how to model this situation as a matching problem. Be sure to spec-
ify what the vertices/edges should be and briefly describe how a matching would
determine seat assignments for each student in a recitation that does not conflict
with his schedule. (This is a modeling problem; we aren’t looking for a description
of an algorithm to solve the problem.)

(b) Suppose there are 41 students. Given the information provided above, is a
matching guaranteed? Briefly explain.
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494 Chapter 12 Simple Graphs

Problem 12.15.
Because of the incredible popularity of his class Math for Computer Science, TA
Mike decides to give up on regular office hours. Instead, he arranges for each
student to join some study groups. Each group must choose a representative to talk
to the staff, but there is a staff rule that a student can only represent one group. The
problem is to find a representative from each group while obeying the staff rule.
(a) Explain how to model the delegate selection problem as a bipartite matching
problem. (This is a modeling problem; we aren’t looking for a description of an
algorithm to solve the problem.)

(b) The staff’s records show that each student is a member of at most 4 groups,
and all the groups have 4 or more members. Is that enough to guarantee there is a
proper delegate selection? Explain.

Problem 12.16.
Let Rb be the “implies” binary relation on propositional formulas defined by the rule
that
F R
b G iff Œ.F IMPLIES G/ is a valid formula: (12.4)
For example, .P AND Q/ R b P , because the formula .P AND Q/ IMPLIES P is
valid. Also, it is not true that .P OR Q/ Rb P since .P OR Q/ IMPLIES P is not
valid.
(a) Let A and B be the sets of formulas listed below. Explain why R
b is not a weak
partial order on the set A [ B.

(b) Fill in the R
b arrows from A to B.
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12.10. References 495

A arrows B
Q

P XOR Q

P OR Q

P AND Q

P OR Q OR .P AND Q/

NOT .P AND Q/

(c) The diagram in part (b) defines a bipartite graph G with L.G/ D A, R.G/ D
B and an edge between F and G iff F R b G. Exhibit a subset S of A such that both
S and A S are nonempty, and the set N.S / of neighbors of S is the same size as
S , that is, jN.S /j D jS j.

(d) Let G be an arbitrary, finite, bipartite graph. For any subset S L.G/, let
S WWD L.G/ S , and likewise for any M R.G/, let M WWD R.G/ M . Suppose
S is a subset of L.G/ such that jN.S /j D jS j, and both S and S are nonempty.
Circle the formula that correctly completes the following statement:
There is a matching from L.G/ to R.G/ if and only if there is both a matching
from S to its neighbors, N.S /, and also a matching from S to
1 1
N.S / N.S / N .N.S // N .N.S // N.S / N.S / N.S / N.S /

Hint: The proof of Hall’s Bottleneck Theorem.

Problem 12.17. (a) Show that there is no matching for the bipartite graph G in
Figure 12.25 that covers L.G/.

(b) The bipartite graph H in Figure 12.26 has an easily verified property that
implies it has a matching that covers L.H /. What is the property?
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496 Chapter 12 Simple Graphs

a v

b w

c x

d y

e z
L(G) R(G)

Figure 12.25 Bipartite graph G.

Homework Problems
Problem 12.18.
A Latin square is n n array whose entries are the number 1; : : : ; n. These en-
tries satisfy two constraints: every row contains all n integers in some order, and
also every column contains all n integers in some order. Latin squares come up
frequently in the design of scientific experiments for reasons illustrated by a little
story in a footnote8
8 At Guinness brewery in the eary 1900’s, W. S. Gosset (a chemist) and E. S. Beavan (a “maltster”)

were trying to improve the barley used to make the brew. The brewery used different varieties of
barley according to price and availability, and their agricultural consultants suggested a different
fertilizer mix and best planting month for each variety.
Somewhat sceptical about paying high prices for customized fertilizer, Gosset and Beavan planned
a season long test of the influence of fertilizer and planting month on barley yields. For as many
months as there were varieties of barley, they would plant one sample of each variety using a different
one of the fertilizers. So every month, they would have all the barley varieties planted and all the
fertilizers used, which would give them a way to judge the overall quality of that planting month.
But they also wanted to judge the fertilizers, so they wanted each fertilizer to be used on each variety
during the course of the season. Now they had a little mathematical problem, which we can abstract
as follows.
Suppose there are n barley varieties and an equal number of recommended fertilizers. Form an
n n array with a column for each fertilizer and a row for each planting month. We want to fill in
the entries of this array with the integers 1,. . . ,n numbering the barley varieties, so that every row
contains all n integers in some order (so every month each variety is planted and each fertilizer is
used), and also every column contains all n integers (so each fertilizer is used on all the varieties over
the course of the growing season).
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12.10. References 497

v
a
w
b
x
c
y
d
z
L(H)
R(H)

Figure 12.26 Bipartite Graph H .

For example, here is a 4 4 Latin square:

1 2 3 4
3 4 2 1
2 1 4 3
4 3 1 2

(a) Here are three rows of what could be part of a 5 5 Latin square:

2 4 5 3 1
4 1 3 2 5
3 2 1 5 4

Fill in the last two rows to extend this “Latin rectangle” to a complete Latin square.

(b) Show that filling in the next row of an n n Latin rectangle is equivalent to
finding a matching in some 2n-vertex bipartite graph.

(c) Prove that a matching must exist in this bipartite graph and, consequently, a
Latin rectangle can always be extended to a Latin square.
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498 Chapter 12 Simple Graphs

Problem 12.19.
Take a regular deck of 52 cards. Each card has a suit and a value. The suit is one of
four possibilities: heart, diamond, club, spade. The value is one of 13 possibilities,
A; 2; 3; : : : ; 10; J; Q; K. There is exactly one card for each of the 4 13 possible
combinations of suit and value.
Ask your friend to lay the cards out into a grid with 4 rows and 13 columns.
They can fill the cards in any way they’d like. In this problem you will show that
you can always pick out 13 cards, one from each column of the grid, so that you
wind up with cards of all 13 possible values.
(a) Explain how to model this trick as a bipartite matching problem between the
13 column vertices and the 13 value vertices. Is the graph necessarily degree-
constrained?

(b) Show that any n columns must contain at least n different values and prove
that a matching must exist.

Problem 12.20.
Scholars through the ages have identified twenty fundamental human virtues: hon-
esty, generosity, loyalty, prudence, completing the weekly course reading-response,
etc. At the beginning of the term, every student in Math for Computer Science pos-
sessed exactly eight of these virtues. Furthermore, every student was unique; that
is, no two students possessed exactly the same set of virtues. The Math for Com-
puter Science course staff must select one additional virtue to impart to each student
by the end of the term. Prove that there is a way to select an additional virtue for
each student so that every student is unique at the end of the term as well.
Suggestion: Use Hall’s theorem. Try various interpretations for the vertices on
the left and right sides of your bipartite graph.

Problems for Section 12.6
Class Problems
Problem 12.21.
Let G be the graph below.9 Carefully explain why .G/ D 4.

9 From [30], Exercise 13.3.1
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Problem 12.22.
A portion of a computer program consists of a sequence of calculations where the
results are stored in variables, like this:

Inputs: a; b
Step 1: c D aCb
2: d D ac
3: e D cC3
4: f D c e
5: g D aCf
6: h D f C1
Outputs: d; g; h

A computer can perform such calculations most quickly if the value of each variable
is stored in a register, a chunk of very fast memory inside the microprocessor.
Programming language compilers face the problem of assigning each variable in a
program to a register. Computers usually have few registers, however, so they must
be used wisely and reused often. This is called the register allocation problem.
In the example above, variables a and b must be assigned different registers,
because they hold distinct input values. Furthermore, c and d must be assigned
different registers; if they used the same one, then the value of c would be over-
written in the second step and we’d get the wrong answer in the third step. On the
other hand, variables b and d may use the same register; after the first step, we no
longer need b and can overwrite the register that holds its value. Also, f and h may
use the same register; once f C 1 is evaluated in the last step, the register holding
the value of f can be overwritten.
(a) Recast the register allocation problem as a question about graph coloring.
What do the vertices correspond to? Under what conditions should there be an edge
between two vertices? Construct the graph corresponding to the example above.

(b) Color your graph using as few colors as you can. Call the computer’s registers
R1, R2 etc. Describe the assignment of variables to registers implied by your
coloring. How many registers do you need?
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500 Chapter 12 Simple Graphs

:::
t Dr Cs
uDt 3
t Dm k
v Dt Cu
:::

How might you cope with this complication?

Problem 12.23.
Suppose an n-vertex bipartite graph has exactly k connected components, each of
which has two or more vertices. How many ways are there color it using a given
set of two colors?

Homework Problems
Problem 12.24.
6.042 is often taught using recitations. Suppose it happened that 8 recitations were
needed, with two or three staff members running each recitation. The assignment
of staff to recitation sections, using their secret codenames, is as follows:

R1: Maverick, Goose, Iceman

R2: Maverick, Stinger, Viper

R3: Goose, Merlin

R4: Slider, Stinger, Cougar

R5: Slider, Jester, Viper

R6: Jester, Merlin

R7: Jester, Stinger

R8: Goose, Merlin, Viper
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12.10. References 501

Two recitations can not be held in the same 90-minute time slot if some staff
member is assigned to both recitations. The problem is to determine the minimum
number of time slots required to complete all the recitations.
(a) Recast this problem as a question about coloring the vertices of a particular
graph. Draw the graph and explain what the vertices, edges, and colors represent.

(b) Show a coloring of this graph using the fewest possible colors. What schedule
of recitations does this imply?

Problem 12.25.
This problem generalizes the result proved Theorem 12.6.3 that any graph with
maximum degree at most w is .w C 1/-colorable.
A simple graph G is said to have width w iff its vertices can be arranged in a
sequence such that each vertex is adjacent to at most w vertices that precede it in
the sequence. If the degree of every vertex is at most w, then the graph obviously
has width at most w—just list the vertices in any order.
(a) Prove that every graph with width at most w is .w C 1/-colorable.

(b) Describe a 2-colorable graph with minimum width n.

(d) Describe an example of a graph with 100 vertices, width 3, but average degree
more than 5.

Problem 12.26.
A sequence of vertices of a graph has width w iff each vertex is adjacent to at most
w vertices that precede it in the sequence. A simple graph G has width w if there
is a width-w sequence of all its vertices.
(a) Explain why the width of a graph must be at least the minimum degree of its
vertices.

(b) Prove that if a finite graph has width w, then there is a width-w sequence of
all its vertices that ends with a minimum degree vertex.

502 Chapter 12 Simple Graphs

Problem 12.27.
Let G be a simple graph whose vertex degrees are all k. Prove by induction on
number of vertices that if every connected component of G has a vertex of degree
strictly less than k, then G is k-colorable.

Problem 12.28.
A basic example of a simple graph with chromatic number n is the complete graph
on n vertices, that is .Kn / D n. This implies that any graph with Kn as a subgraph
must have chromatic number at least n. It’s a common misconception to think that,
conversely, graphs with high chromatic number must contain a large complete sub-
graph. In this problem we exhibit a simple example countering this misconception,
namely a graph with chromatic number four that contains no triangle—length three
cycle—and hence no subgraph isomorphic to Kn for n 3. Namely, let G be the
11-vertex graph of Figure 12.27. The reader can verify that G is triangle-free.

Figure 12.27 Graph G with no triangles and .G/ D 4.

(a) Show that G is 4-colorable.

(b) Prove that G can’t be colored with 3 colors.

Problem 12.29.
This problem will show that 3-coloring a graph is just as difficult as finding a sat-
isfying truth assignment for a propositional formula. The graphs considered will
all be taken to have three designated color-vertices connected in a triangle to force
them to have different colors in any coloring of the graph. The colors assigned to
the color-vertices will be called T; F and N .
Suppose f is an n-argument truth function. That is,
f W fT; F gn ! fT; F g:
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12.10. References 503

NOT(P)

P N

T
F

[h]

Figure 12.28 A 3-color NOT-gate

A graph G is called a 3-color-f-gate iff G has n designated input vertices and a
designated output vertex, such that

G can be 3-colored only if its input vertices are colored with T ’s and F ’s.

For every sequence b1 ; b2 ; : : : ; bn 2 fT; F g, there is a 3-coloring of G in
which the input vertices v1 ; v2 ; : : : ; vn 2 V .G/ have the colors b1 ; b2 ; : : : ; bn 2
fT; F g.

In any 3-coloring of G where the input vertices v1 ; v2 ; : : : ; vn 2 V .G/ have
colors b1 ; b2 ; : : : ; bn 2 fT; F g, the output vertex has color f .b1 ; b2 ; : : : ; bn /.

For example, a 3-color-NOT-gate consists simply of two adjacent vertices. One
vertex is designated to be the input vertex P and the other is designated to be the
output vertex. Both vertices have to be constrained so they can only be colored with
T ’s or F ’s in any proper 3-coloring. This constraint can be imposed by making
them adjacent to the color-vertex N , as shown in Figure 12.28.
(a) Verify that the graph in Figure 12.29 is a 3-color-OR-gate. (The dotted lines
indicate edges to color-vertex N ; these edges constrain the P , Q and P OR Q
vertices to be colored T or F in any proper 3-coloring.)
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504 Chapter 12 Simple Graphs

P OR Q

F P Q

[h]

Figure 12.29 A 3-color OR-gate

(b) Let E be an n-variable propositional formula, and suppose E defines a truth
function f W fT; F gn ! fT; F g. Explain a simple way to construct a graph that is
a 3-color-f -gate.

(c) Explain why an efficient procedure for determining if a graph was 3-colorable
would lead to an efficient procedure to solve the satisfiability problem, SAT.

Problem 12.30.
The 3-coloring problem for planar graphs turns out to be no easier than the 3-
coloring problem for arbitrary graphs. This claim follows very simply from the
existence of a “3-color cross-over gadget.” Such a gadget is a planar graph whose
outer face is a cycle with four designated vertices u; v; w; x occurring in clockwise
order such that

1. Any assignment of colors to vertices u and v can be completed into a 3-
coloring of the gadget.

2. In every 3-coloring of the gadget, the colors of u and w are the same, and the
colors of v and x are the also same.
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12.10. References 505

u
v

x
w
[h]

Figure 12.30 A 3-color cross-over gadget.

Figure 12.30 shows such a 3-color cross-over gadget.10
So to find a 3-coloring for any simple graph, simply draw it in the plane with
edges crossing as needed, and then replace each occurrence of an edge crossing by
a copy of the gadget as shown in Figure 12.31. This yields a planar graph which
has a 3-coloring iff the original graph had one.
(a) Prove that the graph in Figure 12.30 satisfies condition (1) by exhibiting the
claimed 3-colorings.
10 Thisgadget and reduction of 3-colorability to planar 3-colorability are due to Larry Stock-
meyer [43].

[h]

Figure 12.31 Replacing an edge-crossing with a planar gadget.
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506 Chapter 12 Simple Graphs

Hint: Only two colorings are needed, one where u and v are the same color and
another where they are not the same color.

(b) Prove that the graph in Figure 12.30 satisfies condition (2).
Hint: The colorings for part (a) are almost completely forced by the coloring of u
and v.

Exam Problems
Problem 12.31.

False Claim. Let G be a graph whose vertex degrees are all k. If G has a vertex
of degree strictly less than k, then G is k-colorable.
(a) Give a counterexample to the False Claim when k D 2.

(b) Underline the exact sentence or part of a sentence that is the first unjustified
step in the following bogus proof of the False Claim.

Bogus proof. Proof by induction on the number n of vertices:
The induction hypothesis P .n/ is:
Let G be an n-vertex graph whose vertex degrees are all k. If G
also has a vertex of degree strictly less than k, then G is k-colorable.

Base case: (n D 1) G has one vertex, the degree of which is 0. Since G is
1-colorable, P .1/ holds.
Inductive step: We may assume P .n/. To prove P .n C 1/, let GnC1 be a
graph with nC1 vertices whose vertex degrees are all k or less. Also, suppose
GnC1 has a vertex v of degree strictly less than k. Now we only need to prove
that GnC1 is k-colorable.
To do this, first remove the vertex v to produce a graph Gn with n vertices.
Let u be a vertex that is adjacent to v in GnC1 . Removing v reduces the
degree of u by 1. So in Gn , vertex u has degree strictly less than k. Since no
edges were added, the vertex degrees of Gn remain k. So Gn satisfies the
conditions of the induction hypothesis P .n/, and so we conclude that Gn is
k-colorable.
Now a k-coloring of Gn gives a coloring of all the vertices of GnC1 , except for
v. Since v has degree less than k, there will be fewer than k colors assigned
to the nodes adjacent to v. So among the k possible colors, there will be a
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12.10. References 507

color not used to color these adjacent nodes, and this color can be assigned to
v to form a k-coloring of GnC1 .

(c) With a slightly strengthened condition, the preceding proof of the False Claim
could be revised into a sound proof of the following Claim:
Claim. Let G be a graph whose vertex degrees are all k.
If hstatement inserted from belowi has a vertex of degree strictly less than k, then
G is k-colorable.
Indicate each of the statements below that could be inserted to make the proof
correct.

G is connected and
G has no vertex of degree zero and
G does not contain a complete graph on k vertices and
every connected component of G
some connected component of G

Problem 12.32.
In the graph shown in Figure 12.32, the vertices connected in the triangle on the left
are called color-vertices; since they form a triangle, they are forced to have different
colors in any coloring of the graph. The colors assigned to the color-vertices will
be called T; F and N. The dotted lines indicate edges to the color-vertex N.
(a) Explain why for any assignment of different truth-colors to P and Q, there is
a unique 3-coloring of the graph.

(b) Prove that in any 3-coloring of the whole graph, the vertex labeled P XOR Q
is colored with the XOR of the colors of vertices P and Q.

Problems for Section 12.7
Exam Problems
Problem 12.33.
Since you can go back and forth on an edge in a simple graph, every vertex is on
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508 Chapter 12 Simple Graphs

XOR P ⊕ Q

T b P e c F

a d
Q

[h]

Figure 12.32 A 3-color XOR-gate

an even length closed walk. So even length closed walks don’t tell you much about
even length cycles. The situation with odd-length closed walks is more interesting.
(a) Give an example of a simple graph in which every vertex is on a unique odd-
length cycle and a unique even-length cycle.
Hint: Four vertices.

(b) Give an example of a simple graph in which every vertex is on a unique odd-
length cycle and no vertex is on an even-length cycle.

(c) Prove that in a digraph, a smallest size odd-length closed walk must be a cycle.
Note that there will always be lots of even-length closed walks that are shorter than
the smallest odd-length one.
Hint: Let e be an odd-length closed walk of minimum size, and suppose it begins
and ends at vertex a. If it is not a cycle, then it must include a repeated vertex
b ¤ a. That is, e starts with a walk f from a to b, followed by a walk g from b to
b, followed by a walk h from b to a.11
11 In the notation of the text
e D a fb
b gb
b h a:
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12.10. References 509

Homework Problems
Problem 12.34. (a) Give an example of a simple graph that has two vertices u ¤ v
and two distinct paths between u and v, but neither u nor v is on a cycle.

(b) Prove that if there are different paths between two vertices in a simple graph,
then the graph has a cycle.

Problem 12.35.
The entire field of graph theory began when Euler asked whether there was a walk
through his home city of Königsberg in which all seven of its famous bridges were
each crossed exactly once. Abstractly, we can represent the parts of the city sep-
arated by rivers as vertices and the bridges as edges between the vertices. Then
Euler’s question asks whether there is a closed walk through the graph that in-
cludes every edge in a graph exactly once. In his honor, such a walk is called an
Euler tour.
So how do you tell in general whether a graph has an Euler tour? At first glance
this may seem like a daunting problem. The similar sounding problem of finding
a cycle that touches every vertex exactly once is one of those Millenium Prize NP-
complete problems known as the Hamiltonian Cycle Problem). But it turns out to
be easy to characterize which graphs have Euler tours.

Theorem. A connected graph has an Euler tour if and only if every vertex has even
degree.

(a) Show that if a graph has an Euler tour, then the degree of each of its vertices
is even.
In the remaining parts, we’ll work out the converse: if the degree of every vertex
of a connected finite graph is even, then it has an Euler tour. To do this, let’s define
an Euler walk to be a walk that includes each edge at most once.
(b) Suppose that an Euler walk in a connected graph does not include every edge.
Explain why there must be an unincluded edge that is incident to a vertex on the
walk.
In the remaining parts, let w be the longest Euler walk in some finite, connected
graph.
(c) Show that if w is a closed walk, then it must be an Euler tour.
Hint: part (b)

(d) Explain why all the edges incident to the end of w must already be in w.
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510 Chapter 12 Simple Graphs

(e) Show that if the end of w was not equal to the start of w, then the degree of
the end would be odd.
Hint: part (d)

(f) Conclude that if every vertex of a finite, connected graph has even degree, then
it has an Euler tour.

Problems for Section 12.8
Class Problems
Problem 12.36.
A simple graph G is 2-removable iff it contains two vertices v ¤ w such that G v
is connected, and G w is also connected. Prove that every connected graph with
at least two vertices is 2-removable.
Hint: Consider a maximum length path.

Problem 12.37.
The n-dimensional hypercube Hn is a graph whose vertices are the binary strings
of length n. Two vertices are adjacent if and only if they differ in exactly 1 bit. For
example, in H3 , vertices 111 and 011 are adjacent because they differ only in the
first bit, while vertices 101 and 011 are not adjacent because they differ at both
the first and second bits.
(a) Prove that it is impossible to find two spanning trees of H3 that do not share
some edge.

(b) Verify that for any two vertices x ¤ y of H3 , there are 3 paths from x to y in
H3 , such that, besides x and y, no two of those paths have a vertex in common.

(d) Try extending your reasoning to H4 . (In fact, the connectivity of Hn is n for
all n 1. A proof appears in the problem solution.)

Problem 12.38.
A set M of vertices of a graph is a maximal connected set if every pair of vertices
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12.10. References 511

in the set are connected, and any set of vertices properly containing M will contain
two vertices that are not connected.
(a) What are the maximal connected subsets of the following (unconnected) graph?

(b) Explain the connection between maximal connected sets and connected com-
ponents. Prove it.

Problem 12.39. (a) Prove that Kn is .n 1/-edge connected for n > 1.
Let Mn be a graph defined as follows: begin by taking n graphs with non-
overlapping sets of vertices, where each of the n graphs is .n 1/-edge connected
(they could be disjoint copies of Kn , for example). These will be subgraphs of Mn .
Then pick n vertices, one from each subgraph, and add enough edges between pairs
of picked vertices that the subgraph of the n picked vertices is also .n 1/-edge
connected.
(b) Draw a picture of M3 .: : : M4 /.

Problem 12.40.

False Claim. If every vertex in a graph has positive degree, then the graph is
connected.
(a) Prove that this Claim is indeed false by providing a counterexample.
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512 Chapter 12 Simple Graphs

(b) Since the Claim is false, there must be a logical mistake in the following bogus
proof. Pinpoint the first logical mistake (unjustified step) in the proof.

Bogus proof. We prove the Claim above by induction. Let P .n/ be the proposition
that if every vertex in an n-vertex graph has positive degree, then the graph is
connected.
Base cases: (n 2). In a graph with 1 vertex, that vertex cannot have positive
degree, so P .1/ holds vacuously.
P .2/ holds because there is only one graph with two vertices of positive degree,
namely, the graph with an edge between the vertices, and this graph is connected.
Inductive step: We must show that P .n/ implies P .n C 1/ for all n 2. Consider
an n-vertex graph in which every vertex has positive degree. By the assumption
P .n/, this graph is connected; that is, there is a path between every pair of vertices.
Now we add one more vertex x to obtain an .n C 1/-vertex graph:

n-node
x
connected
graph
y

All that remains is to check that there is a path from x to every other vertex z. Since
x has positive degree, there is an edge from x to some other vertex y. Thus, we can
obtain a path from x to z by going from x to y and then following the path from y
to z. This proves P .n C 1/.
By the principle of induction, P .n/ is true for all n 0, which proves the Claim.

Homework Problems
Problem 12.41.
An edge is said to leave a set of vertices if one end of the edge is in the set and the
other end is not.
(a) An n-node graph is said to be mangled if there is an edge leaving every set of
bn=2c or fewer vertices. Prove the following:
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12.10. References 513

Claim. Every mangled graph is connected.

An n-node graph is said to be tangled if there is an edge leaving every set of
dn=3e or fewer vertices.
(b) Draw a tangled graph that is not connected.

Bogus proof. The proof is by strong induction on the number of vertices in the
graph. Let P .n/ be the proposition that if an n-node graph is tangled, then it is
connected. In the base case, P .1/ is true because the graph consisting of a single
node is trivially connected.
For the inductive case, assume n 1 and P .1/; : : : ; P .n/ hold. We must prove
P .n C 1/, namely, that if an .n C 1/-node graph is tangled, then it is connected.
So let G be a tangled, .n C 1/-node graph. Choose dn=3e of the vertices and let G1
be the tangled subgraph of G with these vertices and G2 be the tangled subgraph
with the rest of the vertices. Note that since n 1, the graph G has a least two
vertices, and so both G1 and G2 contain at least one vertex. Since G1 and G2 are
tangled, we may assume by strong induction that both are connected. Also, since
G is tangled, there is an edge leaving the vertices of G1 which necessarily connects
to a vertex of G2 . This means there is a path between any two vertices of G: a path
within one subgraph if both vertices are in the same subgraph, and a path traversing
the connecting edge if the vertices are in separate subgraphs. Therefore, the entire
graph G is connected. This completes the proof of the inductive case, and the Claim
follows by strong induction.

Problem 12.42.
In the cycle C2n of length 2n, we’ll call two vertices opposite if they are on opposite
sides of the cycle, that is that are distance n apart in C2n . Let G be the graph formed
from C2n by adding an edge, which we’ll call a crossing edge, between each pair
of opposite vertices. So G has n crossing edges.
(a) Give a simple description of the shortest path between any two vertices of G.
Hint: Argue that a shortest path between two vertices in G uses at most one crossing
edge.

(b) What is the diameter of G, that is, the largest distance between two vertices?
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514 Chapter 12 Simple Graphs

(d) Prove that the graph is 3-connected.

Exam Problems
Problem 12.43.
We apply the following operation to a simple graph G: pick two vertices u ¤ v
such that either
1. there is an edge of G between u and v, and there is also a path from u to v
which does not include this edge; in this case, delete the edge hu—vi.

2. there is no path from u to v; in this case, add the edge hu—vi.
Keep repeating these operations until it is no longer possible to find two vertices
u ¤ v to which an operation applies.
Assume the vertices of G are the integers 1; 2; : : : ; n for some n 2. This
procedure can be modelled as a state machine whose states are all possible simple
graphs with vertices 1; 2; : : : ; n. G is the start state, and the final states are the
graphs on which no operation is possible.
(a) Let G be the graph with vertices f1; 2; 3; 4g and edges

ff1; 2g; f3; 4gg

How many possible final states are reachable from start state G? 1in

(b) On the line next to each of the derived state variables below, indicate the
strongest property from the list below that the variable is guaranteed to satisfy,
no matter what the starting graph G is. The properties are:
constant increasing decreasing
nonincreasing nondecreasing none of these
For any state, let e be the number of edges in it, and let c be the number of con-
nected components it has. Since e may increase or decrease in a transition, it does
not have any of the first four properties. The derived variables are:

0) e none of these
i) c
ii) c C e
iii) 2c C e
e
iv) c C eC1
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12.10. References 515

(c) Explain why, starting from any state G, the procedure terminates. If your ex-
planation depends on answers you gave to part (b), you must justify those answers.

(d) Prove that any final state must be an unordered tree on the set of vertices, that
is, a spanning tree.

Problem 12.44.
If a simple graph has e edges, v vertices, and k connected components, then it has
at least e v C k cycles.
Prove this by induction on the number of edges e.

Problems for Section 12.9
Practice Problems
Problem 12.45. (a) Prove that the average degree of a tree is less than 2.

(b) Suppose every vertex in a graph has degree at least k. Explain why the graph
has a path of length k.
Hint: Consider a longest path.

c
b d

f g

Figure 12.33 The graph G.
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516 Chapter 12 Simple Graphs

Problem 12.46. (a) How many spanning trees are there for the graph G in Fig-
ure 12.33?

(b) For G e, the graph G with vertex e deleted, describe two spanning trees that
have no edges in common.

(c) For G e with edge ha—d i deleted, explain why there cannot be two edge-
disjoint spanning trees.
Hint: : Count vertices and edges.

Problem 12.47.
Prove that if G is a forest and

jV .G/j D jE.G/j C 1; (12.5)

then G is a tree.

Problem 12.48.
Let H3 be the graph shown in Figure 12.34. Explain why it is impossible to find
two spanning trees of H3 that have no edges in common.
000 010

100 110

101 111

001 011

Figure 12.34 H3 .

Exam Problems
Problem 12.49. (a) Let T be a tree and e a new edge between two vertices of T .
Explain why T C e must contain a cycle.

(b) Conclude that T C e must have another spanning tree besides T .
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12.10. References 517

Problem 12.50.
The diameter of a connected graph is the largest distance between any two vertices.
(a) What is the largest possible diameter in any connected graph with n vertices?
Describe a graph with this maximum diameter.

(b) What is the smallest possible diameter of an n-vertex tree for n > 2? Describe
an n-vertex tree with this minimum diameter.

Problem 12.51.

(a) Indicate all the properties below that are preserved under graph isomorphism.

There is a cycle that includes all the vertices.
Two edges are of equal length.
The graph remains connected if any two edges are removed.
There exists an edge that is an edge of every spanning tree.
The negation of a property that is preserved under isomorphism.

(b) For the following statements about finite trees, indicate whether they are true
or false, and provide counterexamples for those that are false.

Any connected subgraph is a tree. true false
Adding an edge between two nonadjacent vertices creates a cycle. true
false
The number of vertices is one less than twice the number of leaves. true
false
The number of vertices is one less than the number of edges. true false
For every finite graph (not necessarily a tree), there is one (a finite tree) that
spans it. true false

Problem 12.52.

Circle true or false for the following statements about finite simple
graphs G.
(a) G has a spanning tree. true false
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518 Chapter 12 Simple Graphs

(b) jV .G/j D O.jE.G/j/ for connected G. true false

(d) jV .G/j D O..G//. true false

Problem 12.53.
A simple graph G is said to have width 1 iff there is a way to list all its vertices so
that each vertex is adjacent to at most one vertex that appears earlier in the list. All
the graphs mentioned below are assumed to be finite.
(a) Prove that every graph with width one is a forest.
Hint: By induction, removing the last vertex.

(b) Prove that every finite tree has width one. Conclude that a graph is a forest iff
it has width one.

Problem 12.54.
Prove by induction that, using a fixed set of n > 1 colors, there are exactly n .n
1/m 1 different colorings of any tree with m vertices.

Problem 12.55.
Let G be a connected weighted simple graph and let v be a vertex of G. Suppose
e WWD hv—wi is an edge of G that is strictly smaller than the weight of every other
edge incident to v. Let T be a minimum weight spanning tree of G. Prove that e is
an edge of T . Hint: By contradiction.

Problem 12.56.
Let G be a connected simple graph, T be a spanning tree of G, and e be an edge of
G.
(a) Prove that if e is not on a cycle in G, then e is an edge of T .

(b) Prove that if e is on a cycle in G, and e is in T , then there is an edge f ¤ e
such that T e C f is also a spanning tree.

12 .G/ is the chromatic number of G.
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12.10. References 519

(c) Suppose G is edge-weighted, the weight of e is larger than the weights of all
the other edges, e is on a cycle in G, and e is an edge of T . Conclude that T is not
a minimum weight spanning tree of G.

Class Problems
Problem 12.57.
Procedure Mark starts with a connected, simple graph with all edges unmarked and
then marks some edges. At any point in the procedure a path that includes only
marked edges is called a fully marked path, and an edge that has no fully marked
path between its endpoints is called eligible.
Procedure Mark simply keeps marking eligible edges, and terminates when there
are none.
Prove that Mark terminates, and that when it does, the set of marked edges forms
a spanning tree of the original graph.

Problem 12.58.
A procedure for connecting up a (possibly disconnected) simple graph and creating
a spanning tree can be modelled as a state machine whose states are finite simple
graphs. A state is final when no further transitions are possible. The transitions are
determined by the following rules:

Procedure create-spanning-tree
1. If there is an edge hu—vi on a cycle, then delete hu—vi.

2. If vertices u and v are not connected, then add the edge hu—vi.

(a) Draw all the possible final states reachable starting with the graph with vertices
f1; 2; 3; 4g and edges
fh1—2i ; h3—4ig:

(b) Prove that if the machine reaches a final state, then the final state will be a tree
on the vertices of the agraph on which it started.

(c) For any graph G 0 , let e be the number of edges in G 0 , c be the number of con-
nected components it has, and s be the number of cycles. For each of the quantities
below, indicate the strongest of the properties that it is guaranteed to satisfy, no
matter what the starting graph is.
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520 Chapter 12 Simple Graphs

The choices for properties are: constant, strictly increasing, strictly decreasing,
weakly increasing, weakly decreasing, none of these.

(i) e
(ii) c
(iii) s
(iv) e s
(v) c C e
(vi) 3c C 2e
(vii) c C s

(d) Prove that one of the quantities from part (c) strictly decreases at each transi-
tion. Conclude that for every starting state, the machine will reach a final state.

Problem 12.59.
Let G be a weighted graph and suppose there is a unique edge e 2 E.G/ with
smallest weight, that is, w.e/ < w.f / for all edges f 2 E.G/ feg. Prove that
any minimum weight spanning tree (MST) of G must include e.

Problem 12.60.
Let G be the 4 4 grid with vertical and horizontal edges between neighboring
vertices and edge weights as shown in Figure 12.35.
In this problem you will practice some of the ways to build minimum weight
spanning trees. For each part, list the edge weights in the order in which the edges
with those weights were chosen by the given rules.
(a) Construct a minimum weight spanning tree (MST) for G by initially selecting
the minimum weight edge, and then successively selecting the minimum weight
edge that does not create a cycle with the previously selected edges. Stop when the
selected edges form a spanning tree of G. (This is Kruskal’s MST algorithm.)
For any step in Kruskal’s procedure, describe a black-white coloring of the graph
components so that the edge Kruskal chooses is the minimum weight “gray edge”
according to Lemma 12.9.11.

(b) Grow an MST for G by starting with the tree consisting of the single vertex u
and then successively adding the minimum weight edge with exactly one endpoint
in the tree. Stop when the tree spans G. (This is Prim’s MST algorithm.)
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12.10. References 521

0 0.04 0.08

1 1.04 1.08 1.12

0.01 u 0.05 0.09
w
1.01 1.05 1.09 1.13

0.02 0.06 0.10

1.02 1.06 1.10 1.14

0.03 v 0.07 0.11

Figure 12.35 The 4x4 array graph G

For any step in Prim’s procedure, describe a black-white coloring of the graph
components so that the edge Prim chooses is the minimum weight “gray edge”
according to Lemma 12.9.11.

(c) The 6.042 “parallel” MST algorithm can grow an MST for G by starting with
the upper left corner vertex along with the vertices labelled v and w. Regard each
of the three vertices as one-vertex trees. Successively add, for each tree in parallel,
the minimum weight edge among the edges with exactly one endpoint in the tree.
Stop working on a tree when it is within distance two of another tree. Continue
until there are no more eligible trees—that is, each tree is within distance two of
some other tree—then go back to applying the general gray-edge method until the
parallel trees merge to form a spanning tree of G.

(d) Verify that you got the same MST each time. Problem 12.63 explains why
there is a unique MST for any finite, connected, weighted graph where no two
edges have the same weight.
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522 Chapter 12 Simple Graphs

Problem 12.61.
In this problem you will prove:

Theorem. A graph G is 2-colorable iff it contains no odd length closed walk.

As usual with “iff” assertions, the proof splits into two proofs: part (a) asks you
to prove that the left side of the “iff” implies the right side. The other problem parts
prove that the right side implies the left.
(a) Assume the left side and prove the right side. Three to five sentences should
suffice.

(b) Now assume the right side. As a first step toward proving the left side, explain
why we can focus on a single connected component H within G.

(d) Choose any 2-coloring of a spanning tree T of H . Prove that H is 2-colorable
by showing that any edge not in T must also connect different-colored vertices.

Homework Problems
Problem 12.62.
Suppose D D .d1 ; d2 ; : : : ; dn / is a list of the vertex degrees of some n-vertex tree
T for n 2. That is, we assume the vertices of T are numbered, and di > 0 is the
degree of the i th vertex of T .
(a) Explain why
n
X
di D 2.n 1/: (12.6)
i D1

(b) Prove conversely that if D is a sequence of positive integers satisfying equa-
tion (12.6), then D is a list of the degrees of the vertices of some n-vertex tree.
Hint: Induction.

(c) Assume that D satisfies equation (12.6). Show that it is possible to partition
D into two sets S1 ; S2 such that the sum of the elements in each set is the same.
Hint: Trees are bipartite.

Problem 12.63.
Prove Corollary 12.9.12: If all edges in a finite weighted graph have distinct weights,
then the graph has a unique MST.
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12.10. References 523

Hint: Suppose M and N were different MST’s of the same graph. Let e be the
smallest edge in one and not the other, say e 2 M N , and observe that N C e
must have a cycle.
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13 Planar Graphs

13.1 Drawing Graphs in the Plane
Suppose there are three dog houses and three human houses, as shown in Fig-
ure 13.1. Can you find a route from each dog house to each human house such that
no route crosses any other route?
A similar question comes up about a little-known animal called a quadrapus that
looks like an octopus with four stretchy arms instead of eight. If five quadrapi are
resting on the sea floor, as shown in Figure 13.2, can each quadrapus simultane-
ously shake hands with every other in such a way that no arms cross?
Both these puzzles can be understood as asking about drawing graphs in the
plane. Replacing dogs and houses by nodes, the dog house puzzle can be rephrased
as asking whether there is a planar drawing of the graph with six nodes and edges
between each of the first three nodes and each of the second three nodes. This
graph is called the complete bipartite graph K3;3 and is shown in Figure 13.3.(a).
The quadrapi puzzle asks whether there is a planar drawing of the complete graph
K5 shown in Figure 13.3.(b).
In each case, the answer is, “No—but almost!” In fact, if you remove an edge
from either of these graphs, then the resulting graph can be redrawn in the plane so
that no edges cross, as shown in Figure 13.4.
Planar drawings have applications in circuit layout and are helpful in displaying
graphical data such as program flow charts, organizational charts and scheduling
conflicts. For these applications, the goal is to draw the graph in the plane with as
few edge crossings as possible. (See the box on the following page for one such
example.)

13.2 Definitions of Planar Graphs
We took the idea of a planar drawing for granted in the previous section, but if
we’re going to prove things about planar graphs, we better have precise definitions.

Definition 13.2.1. A drawing of a graph assigns to each node a distinct point in
the plane and assigns to each edge a smooth curve in the plane whose endpoints
correspond to the nodes incident to the edge. The drawing is planar if none of the
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526 Chapter 13 Planar Graphs

Figure 13.1 Three dog houses and and three human houses. Is there a route from
each dog house to each human house so that no pair of routes cross each other?
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13.2. Definitions of Planar Graphs 527

Figure 13.2 Five quadrapi (4-armed creatures).

(a) (b)

Figure 13.3 K3;3 (a) and K5 (b). Can you redraw these graphs so that no pairs
of edges cross?
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528 Chapter 13 Planar Graphs

u
u

v v

(a) (b)

Figure 13.4 Planar drawings of (a) K3;3 without hu—vi, and (b) K5 without
hu—vi.

Steve Wozniak and a Planar Circuit Design
When wires are arranged on a surface, like a circuit board or microchip, cross-
ings require troublesome three-dimensional structures. When Steve Wozniak
designed the disk drive for the early Apple II computer, he struggled might-
ily to achieve a nearly planar design according to the following excerpt from
apple2history.org which in turn quotes Fire in the Valley by Freiberger
and Swaine:

For two weeks, he worked late each night to make a satisfactory de-
sign. When he was finished, he found that if he moved a connector
he could cut down on feedthroughs, making the board more reliable.
To make that move, however, he had to start over in his design. This
time it only took twenty hours. He then saw another feedthrough
that could be eliminated, and again started over on his design. “The
final design was generally recognized by computer engineers as bril-
liant and was by engineering aesthetics beautiful. Woz later said, ’It’s
something you can only do if you’re the engineer and the PC board
layout person yourself. That was an artistic layout. The board has
virtually no feedthroughs.’
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13.2. Definitions of Planar Graphs 529

curves cross themselves or other curves, namely, the only points that appear more
than once on any of the curves are the node points. A graph is planar when it has a
planar drawing.

Definition 13.2.1 is precise but depends on further concepts: “smooth planar
curves” and “points appearing more than once” on them. We haven’t defined these
concepts—we just showed the simple picture in Figure 13.4 and hoped you would
get the idea.
Pictures can be a great way to get a new idea across, but it is generally not a good
idea to use a picture to replace precise mathematics. Relying solely on pictures can
sometimes lead to disaster—or to bogus proofs, anyway. There is a long history of
bogus proofs about planar graphs based on misleading pictures.
The bad news is that to prove things about planar graphs using the planar draw-
ings of Definition 13.2.1, we’d have to take a chapter-long excursion into contin-
uous mathematics just to develop the needed concepts from plane geometry and
point-set topology. The good news is that there is another way to define planar
graphs that uses only discrete mathematics. In particular, we can define planar
graphs as a recursive data type. In order to understand how it works, we first need
to understand the concept of a face in a planar drawing.

13.2.1 Faces
The curves in a planar drawing divide up the plane into connected regions called
the continuous faces1 of the drawing. For example, the drawing in Figure 13.5 has
four continuous faces. Face IV, which extends off to infinity in all directions, is
called the outside face.
The vertices along the boundary of each continuous face in Figure 13.5 form a
cycle. For example, labeling the vertices as in Figure 13.6, the cycles for each of
the face boundaries can be described by the vertex sequences

abca abda bcdb acda: (13.1)

These four cycles correspond nicely to the four continuous faces in Figure 13.6—
so nicely, in fact, that we can identify each of the faces in Figure 13.6 by its cycle.
For example, the cycle abca identifies face III. The cycles in list 13.1 are called the
discrete faces of the graph in Figure 13.6. We use the term “discrete” since cycles
in a graph are a discrete data type—as opposed to a region in the plane, which is a
continuous data type.
1 Most texts drop the adjective continuous from the definition of a face as a connected region. We
need the adjective to distinguish continuous faces from the discrete faces we’re about to define.
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530 Chapter 13 Planar Graphs

III

II I

Figure 13.5 A planar drawing with four continuous faces.

b
III

a II I
c

IV
d

Figure 13.6 The drawing with labeled vertices.
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13.2. Definitions of Planar Graphs 531

f
b

a c e

g
d

Figure 13.7 A planar drawing with a bridge.

Unfortunately, continuous faces in planar drawings are not always bounded by
cycles in the graph—things can get a little more complicated. For example, the
planar drawing in Figure 13.7 has what we will call a bridge, namely, a cut edge
hc—ei. The sequence of vertices along the boundary of the outer region of the
drawing is
abcefgecda:
This sequence defines a closed walk, but does not define a cycle since the walk has
two occurrences of the bridge hc—ei and each of its endpoints.
The planar drawing in Figure 13.8 illustrates another complication. This drawing
has what we will call a dongle, namely, the nodes v, x, y and w, and the edges
incident to them. The sequence of vertices along the boundary of the inner region
is
rst vxyxvwvt ur:
This sequence defines a closed walk, but once again does not define a cycle be-
cause it has two occurrences of every edge of the dongle—once “coming” and once
“going.”
It turns out that bridges and dongles are the only complications, at least for con-
nected graphs. In particular, every continuous face in a planar drawing corresponds
to a closed walk in the graph. These closed walks will be called the discrete faces
of the drawing, and we’ll define them next.

13.2.2 A Recursive Definition for Planar Embeddings
The association between the continuous faces of a planar drawing and closed walks
provides the discrete data type we can use instead of continuous drawings. We’ll
define a planar embedding of connected graph to be the set of closed walks that are
its face boundaries. Since all we care about in a graph are the connections between
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532 Chapter 13 Planar Graphs

y
x
v
r t

Figure 13.8 A planar drawing with a dongle.

vertices—not what a drawing of the graph actually looks like—planar embeddings
are exactly what we need.
The question is how to define planar embeddings without appealing to continu-
ous drawings. There is a simple way to do this based on the idea that any continuous
drawing can drawn step by step:

either draw a new point somewhere in the plane to represent a vertex,

or draw a curve between two vertex points that have already been laid down,
making sure the new curve doesn’t cross any of the previously drawn curves.

A new curve won’t cross any other curves precisely when it stays within one
of the continuous faces. Alternatively, a new curve won’t have to cross any other
curves if it can go between the outer faces of two different drawings. So to be sure
it’s ok to draw a new curve, we just need to check that its endpoints are on the
boundary of the same face, or that its endpoints are on the outer faces of different
drawings. Of course drawing the new curve changes the faces slightly, so the face
boundaries will have to be updated once the new curve is drawn. This is the idea
behind the following recursive definition.

Definition 13.2.2. A planar embedding of a connected graph consists of a nonempty
set of closed walks of the graph called the discrete faces of the embedding. Planar
embeddings are defined recursively as follows:
Base case: If G is a graph consisting of a single vertex v then a planar embedding
of G has one discrete face, namely, the length zero closed walk v.
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13.2. Definitions of Planar Graphs 533

w
a

y b

Figure 13.9 The “split a face” case: awxbyza splits into awxba and abyza.

Constructor case (split a face): Suppose G is a connected graph with a planar
embedding, and suppose a and b are distinct, nonadjacent vertices of G that occur
in some discrete face of the planar embedding. That is, is a closed walk of the
form
D ˛bˇ
where ˛ is a walk from a to b and ˇ is a walk from b to a. Then the graph obtained
by adding the edge ha—bi to the edges of G has a planar embedding with the same
discrete faces as G, except that face is replaced by the two discrete faces2

˛b hb—ai and ha—bi bˇ (13.2)

as illustrated in Figure 13.9.3
Constructor case (add a bridge): Suppose G and H are connected graphs with
planar embeddings and disjoint sets of vertices. Let be a discrete face of the
embedding of G and suppose that begins and ends at vertex a.
Similarly, let ı be a discrete face of the embedding of H that begins and ends at
vertex b.
2 There is a minor exception to this definition of embedding in the special case when G is a line

graph beginning with a and ending with b. In this case the cycles into which splits are actually
the same. That’s because adding edge ha—bi creates a cycle that divides the plane into “inner” and
“outer” continuous faces that are both bordered by this cycle. In order to maintain the correspondence
between continuous faces and discrete faces in this case, we define the two discrete faces of the
embedding to be two “copies” of this same cycle.
3 Formally, merge is an operation on walks, not a walk and an edge, so in (13.2), we should have

used a walk .a ha—bi b/ instead of an edge ha—bi and written

˛b.b hb—ai a/ and .a ha—bi b/bˇ
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534 Chapter 13 Planar Graphs

t
z

u
a b
y
w
v
x

Figure 13.10 The “add a bridge” case.

Then the graph obtained by connecting G and H with a new edge ha—bi has a
planar embedding whose discrete faces are the union of the discrete faces of G and
H , except that faces and ı are replaced by one new face

b ha—bi bıb hb—ai :

This is illustrated in Figure 13.10, where the vertex sequences of the faces of G
and H are:

G W faxyza; axya; ayzag H W fbt uvwb; bt vwb; t uvtg;

and after adding the bridge ha—bi, there is a single connected graph whose faces
have the vertex sequences

faxyzabt uvwba; axya; ayza; bt vwb; t uvtg:

A bridge is simply a cut edge, but in the context of planar embeddings, the
bridges are precisely the edges that occur twice on the same discrete face—as op-
posed to once on each of two faces. Dongles are trees made of bridges; we only
use dongles in illustrations, so there’s no need to define them more precisely.

13.2.3 Does It Work?
Yes! In general, a graph is planar because it has a planar drawing according to
Definition 13.2.1 if and only if each of its connected components has a planar em-
bedding as specified in Definition 13.2.2. Of course we can’t prove this without an
excursion into exactly the kind of continuous math that we’re trying to avoid. But
now that the recursive definition of planar graphs is in place, we won’t ever need to
fall back on the continuous stuff. That’s the good news.
The bad news is that Definition 13.2.2 is a lot more technical than the intuitively
simple notion of a drawing whose edges don’t cross. In many cases it’s easier to
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13.2. Definitions of Planar Graphs 535

r r

u s s u

t t

Figure 13.11 Two illustrations of the same embedding.

stick to the idea of planar drawings and give proofs in those terms. For example,
erasing edges from a planar drawing will surely leave a planar drawing. On the
other hand, it’s not so obvious, though of course it is true, that you can delete an
edge from a planar embedding and still get a planar embedding (see Problem 13.9).
In the hands of experts, and perhaps in your hands too with a little more expe-
rience, proofs about planar graphs by appeal to drawings can be convincing and
reliable. But given the long history of mistakes in such proofs, it’s safer to work
from the precise definition of planar embedding. More generally, it’s also important
to see how the abstract properties of curved drawings in the plane can be modelled
successfully using a discrete data type.

13.2.4 Where Did the Outer Face Go?
Every planar drawing has an immediately-recognizable outer face —it’s the one
that goes to infinity in all directions. But where is the outer face in a planar embed-
ding?
There isn’t one! That’s because there really isn’t any need to distinguish one face
from another. In fact, a planar embedding could be drawn with any given face on
the outside. An intuitive explanation of this is to think of drawing the embedding
on a sphere instead of the plane. Then any face can be made the outside face by
“puncturing” that face of the sphere, stretching the puncture hole to a circle around
the rest of the faces, and flattening the circular drawing onto the plane.
So pictures that show different “outside” boundaries may actually be illustra-
tions of the same planar embedding. For example, the two embeddings shown in
Figure 13.11 are really the same—check it: they have the same boundary cycles.
This is what justifies the “add bridge” case in Definition 13.2.2: whatever face
is chosen in the embeddings of each of the disjoint planar graphs, we can draw
a bridge between them without needing to cross any other edges in the drawing,
because we can assume the bridge connects two “outer” faces.
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536 Chapter 13 Planar Graphs

13.3 Euler’s Formula
The value of the recursive definition is that it provides a powerful technique for
proving properties of planar graphs, namely, structural induction. For example,
we will now use Definition 13.2.2 and structural induction to establish one of the
most basic properties of a connected planar graph, namely, that the number of ver-
tices and edges completely determines the number of faces in every possible planar
embedding of the graph.

Theorem 13.3.1 (Euler’s Formula). If a connected graph has a planar embedding,
then
v eCf D2
where v is the number of vertices, e is the number of edges and f is the number of
faces.

For example, in Figure 13.5, v D 4, e D 6 and f D 4. Sure enough, 4 6 C 4 D
2, as Euler’s Formula claims.

Proof. The proof is by structural induction on the definition of planar embeddings.
Let P .E/ be the proposition that v e C f D 2 for an embedding E.
Base case (E is the one-vertex planar embedding): By definition, v D 1, e D 0 and
f D 1, and 1 0 C 1 D 2, so P .E/ indeed holds.
Constructor case (split a face): Suppose G is a connected graph with a planar
embedding, and suppose a and b are distinct, nonadjacent vertices of G that appear
on some discrete face D a : : : b a of the planar embedding.
Then the graph obtained by adding the edge ha—bi to the edges of G has a
planar embedding with one more face and one more edge than G. So the quantity
v e C f will remain the same for both graphs, and since by structural induction
this quantity is 2 for G’s embedding, it’s also 2 for the embedding of G with the
added edge. So P holds for the constructed embedding.
Constructor case (add bridge): Suppose G and H are connected graphs with pla-
nar embeddings and disjoint sets of vertices. Then connecting these two graphs
with a bridge merges the two bridged faces into a single face, and leaves all other
faces unchanged. So the bridge operation yields a planar embedding of a connected
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13.4. Bounding the Number of Edges in a Planar Graph 537

graph with vG C vH vertices, eG C eH C 1 edges, and fG C fH 1 faces. Since

.vG C vH / .eG C eH C 1/ C .fG C fH 1/
D .vG eG C fG / C .vH eH C fH / 2
D .2/ C .2/ 2 (by structural induction hypothesis)
D 2;

v e C f remains equal to 2 for the constructed embedding. That is, P .E/ also
holds in this case.
This completes the proof of the constructor cases, and the theorem follows by
structural induction.

13.4 Bounding the Number of Edges in a Planar Graph
Like Euler’s formula, the following lemmas follow by structural induction directly
from Definition 13.2.2.

Lemma 13.4.1. In a planar embedding of a connected graph, each edge occurs
once in each of two different faces, or occurs exactly twice in one face.

Lemma 13.4.2. In a planar embedding of a connected graph with at least three
vertices, each face is of length at least three.

Combining Lemmas 13.4.1 and 13.4.2 with Euler’s Formula, we can now prove
that planar graphs have a limited number of edges:

Theorem 13.4.3. Suppose a connected planar graph has v 3 vertices and e
edges. Then
e 3v 6: (13.3)

Proof. By definition, a connected graph is planar iff it has a planar embedding. So
suppose a connected graph with v vertices and e edges has a planar embedding
with f faces. By Lemma 13.4.1, every edge has exactly two occurrences in the
face boundaries. So the sum of the lengths of the face boundaries is exactly 2e.
Also by Lemma 13.4.2, when v 3, each face boundary is of length at least three,
so this sum is at least 3f . This implies that

3f 2e: (13.4)
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538 Chapter 13 Planar Graphs

But f D e v C 2 by Euler’s formula, and substituting into (13.4) gives

3.e v C 2/ 2e
e 3v C 6 0
e 3v 6

13.5 Returning to K5 and K3;3
Finally we have a simple way to answer the quadrapi question at the beginning of
this chapter: the five quadrapi can’t all shake hands without crossing. The reason
is that we know the quadrupi question is the same as asking whether a complete
graph K5 is planar, and Theorem 13.4.3 has the immediate:

Corollary 13.5.1. K5 is not planar.

Proof. K5 is connected and has 5 vertices and 10 edges. But since 10 > 3 5 6,
K5 does not satisfy the inequality (13.3) that holds in all planar graphs.

We can also use Euler’s Formula to show that K3;3 is not planar. The proof is
similar to that of Theorem 13.3 except that we use the additional fact that K3;3 is a
bipartite graph.

Lemma 13.5.2. In a planar embedding of a connected bipartite graph with at least
3 vertices, each face has length at least 4.

Proof. By Lemma 13.4.2, every face of a planar embedding of the graph has length
at least 3. But by Lemma 12.6.2 and Theorem 12.8.3.3, a bipartite graph can’t have
odd length closed walks. Since the faces of a planar embedding are closed walks,
there can’t be any faces of length 3 in a bipartite embedding. So every face must
have length at least 4.

Theorem 13.5.3. Suppose a connected bipartite graph with v 3 vertices and e
edges is planar. Then
e 2v 4: (13.5)

Proof. Lemma 13.5.2 implies that all the faces of an embedding of the graph have
length at least 4. Now arguing as in the proof of Theorem 13.4.3, we find that the
sum of the lengths of the face boundaries is exactly 2e and at least 4f . Hence,

4f 2e (13.6)
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13.6. Coloring Planar Graphs 539

for any embedding of a planar bipartite graph. By Euler’s theorem, f D 2 v C e.
Substituting 2 v C e for f in (13.6), we have

4.2 v C e/ 2e;

which simplies to (13.5).

Corollary 13.5.4. K3;3 is not planar.

Proof. K3;3 is connected, bipartite and has 6 vertices and 9 edges. But since 9 >
2 6 4, K3;3 does not satisfy the inequality (13.3) that holds in all bipartite planar
graphs.

13.6 Coloring Planar Graphs
We’ve covered a lot of ground with planar graphs, but not nearly enough to prove
the famous 4-color theorem. But we can get awfully close. Indeed, we have done
almost enough work to prove that every planar graph can be colored using only 5
colors.
There are two familiar facts about planarity that we will need.

Lemma 13.6.1. Any subgraph of a planar graph is planar.

Lemma 13.6.2. Merging two adjacent vertices of a planar graph leaves another
planar graph.

Merging two adjacent vertices, n1 and n2 of a graph means deleting the two
vertices and then replacing them by a new “merged” vertex m adjacent to all the
vertices that were adjacent to either of n1 or n2 , as illustrated in Figure 13.12.
Many authors take Lemmas 13.6.1 and 13.6.2 for granted for continuous draw-
ings of planar graphs described by Definition 13.2.1. With the recursive Defini-
tion 13.2.2 both Lemmas can actually be proved using structural induction (see
Problem 13.9).
We need only one more lemma:

Lemma 13.6.3. Every planar graph has a vertex of degree at most five.

Proof. Assuming to the contrary that every vertex of some planar graph had degree
at least 6, then the sum of the vertex degrees is at least 6v. But the sum of the
vertex degrees equals 2e by the Handshake Lemma 12.2.1, so we have e 3v
contradicting the fact that e 3v 6 < 3v by Theorem 13.4.3.
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540 Chapter 13 Planar Graphs

n1
n1
! n2
! m

Figure 13.12 Merging adjacent vertices n1 and n2 into new vertex m.

Theorem 13.6.4. Every planar graph is five-colorable.

Proof. The proof will be by strong induction on the number v of vertices, with
induction hypothesis:

Every planar graph with v vertices is five-colorable.

Base cases (v 5): immediate.
Inductive case: Suppose G is a planar graph with v C 1 vertices. We will describe
a five-coloring of G.
First, choose a vertex g of G with degree at most 5; Lemma 13.6.3 guarantees
there will be such a vertex.

Case 1: (deg.g/ < 5): Deleting g from G leaves a graph H that is planar by
Lemma 13.6.1, and since H has v vertices, it is five-colorable by induction
hypothesis. Now define a five coloring of G as follows: use the five-coloring
of H for all the vertices besides g, and assign one of the five colors to g that
is not the same as the color assigned to any of its neighbors. Since there are
fewer than 5 neighbors, there will always be such a color available for g.

Case 2: (deg.g/ D 5): If the five neighbors of g in G were all adjacent to each
other, then these five vertices would form a nonplanar subgraph isomorphic
to K5 , contradicting Lemma 13.6.1 (since K5 is not planar). So there must
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13.7. Classifying Polyhedra 541

be two neighbors, n1 and n2 , of g that are not adjacent. Now merge n1 and g
into a new vertex, m. In this new graph, n2 is adjacent to m, and the graph is
planar by Lemma 13.6.2. So we can then merge m and n2 into a another new
vertex m0 , resulting in a new graph G 0 which by Lemma 13.6.2 is also planar.
Since G 0 has v 1 vertices, it is five-colorable by the induction hypothesis.
Now define a five coloring of G as follows: use the five-coloring of G 0 for
all the vertices besides g, n1 and n2 . Next assign the color of m0 in G 0 to
be the color of the neighbors n1 and n2 . Since n1 and n2 are not adjacent
in G, this defines a proper five-coloring of G except for vertex g. But since
these two neighbors of g have the same color, the neighbors of g have been
colored using fewer than five colors altogether. So complete the five-coloring
of G by assigning one of the five colors to g that is not the same as any of
the colors assigned to its neighbors.

13.7 Classifying Polyhedra
p
The Pythagoreans had two great mathematical secrets, the irrationality of 2 and
a geometric construct that we’re about to rediscover!
A polyhedron is a convex, three-dimensional region bounded by a finite number
of polygonal faces. If the faces are identical regular polygons and an equal number
of polygons meet at each corner, then the polyhedron is regular. Three examples
of regular polyhedra are shown in Figure 13.13: the tetrahedron, the cube, and the
octahedron.
We can determine how many more regular polyhedra there are by thinking about
planarity. Suppose we took any polyhedron and placed a sphere inside it. Then we
could project the polyhedron face boundaries onto the sphere, which would give
an image that was a planar graph embedded on the sphere, with the images of the
corners of the polyhedron corresponding to vertices of the graph. We’ve already
observed that embeddings on a sphere are the same as embeddings on the plane, so
Euler’s formula for planar graphs can help guide our search for regular polyhedra.
For example, planar embeddings of the three polyhedra in Figure 13.1 are shown
in Figure 13.14.
Let m be the number of faces that meet at each corner of a polyhedron, and let n
be the number of edges on each face. In the corresponding planar graph, there are
m edges incident to each of the v vertices. By the Handshake Lemma 12.2.1, we
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542 Chapter 13 Planar Graphs

(a) (b) (c)

Figure 13.13 The tetrahedron (a), cube (b), and octahedron (c).
v

(a) (b) (c)

Figure 13.14 Planar embeddings of the tetrahedron (a), cube (b), and octahe-
dron (c).
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13.7. Classifying Polyhedra 543

n m v e f polyhedron
3 3 4 6 4 tetrahedron
4 3 8 12 6 cube
3 4 6 12 8 octahedron
3 5 12 30 20 icosahedron
5 3 20 30 12 dodecahedron

Figure 13.15 The only possible regular polyhedra.

know:
mv D 2e:
Also, each face is bounded by n edges. Since each edge is on the boundary of two
faces, we have:
nf D 2e
Solving for v and f in these equations and then substituting into Euler’s formula
gives:
2e 2e
eC D2
m n
which simplifies to
1 1 1 1
C D C (13.7)
m n e 2
Equation 13.7 places strong restrictions on the structure of a polyhedron. Every
nondegenerate polygon has at least 3 sides, so n 3. And at least 3 polygons
must meet to form a corner, so m 3. On the other hand, if either n or m were
6 or more, then the left side of the equation could be at most 1=3 C 1=6 D 1=2,
which is less than the right side. Checking the finitely-many cases that remain turns
up only five solutions, as shown in Figure 13.15. For each valid combination of n
and m, we can compute the associated number of vertices v, edges e, and faces f .
And polyhedra with these properties do actually exist. The largest polyhedron, the
dodecahedron, was the other great mathematical secret of the Pythagorean sect.
The 5 polyhedra in Figure 13.15 are the only possible regular polyhedra. So if
you want to put more than 20 geocentric satellites in orbit so that they uniformly
blanket the globe—tough luck!
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544 Chapter 13 Planar Graphs

13.8 Another Characterization for Planar Graphs
We did not pick K5 and K3;3 as examples because of their application to dog
houses or quadrapi shaking hands. We really picked them because they provide
another, famous, discrete characterizarion of planar graphs:

Theorem 13.8.1 (Kuratowski). A graph is not planar if and only if it contains K5
or K3;3 as a minor.

Definition 13.8.2. A minor of a graph G is a graph that can be obtained by repeat-
edly4 deleting vertices, deleting edges, and merging adjacent vertices of G.

For example, Figure 13.16 illustrates why C3 is a minor of the graph in Fig-
ure 13.16(a). In fact C3 is a minor of a connected graph G if and only if G is not a
tree.
The known proofs of Kuratowski’s Theorem 13.8.1 are a little too long to include
in an introductory text, so we won’t give one.

Problems for Section 13.2
Practice Problems
Problem 13.1.
What are the discrete faces of the following two graphs?
Write each cycle as a sequence of letters without spaces, starting with the alpha-
betically earliest letter in the clockwise direction, for example “adbfa.” Separate
the sequences with spaces.
(a)
f
b

a c e

g
d

(b)
4 The three operations can each be performed any number of times in any order.
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13.8. Another Characterization for Planar Graphs 545

v2
e1

(a) (b) (c)

v3
e2

(d) (e) (f)

Figure 13.16 One method by which the graph in (a) can be reduced to C3 (f),
thereby showing that C3 is a minor of the graph. The steps are: merging the nodes
incident to e1 (b), deleting v1 and all edges incident to it (c), deleting v2 (d), delet-
ing e2 , and deleting v3 (f).
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546 Chapter 13 Planar Graphs

y
x
v
r t

Problems for Section 13.8
Exam Problems
Problem 13.2.

h
l
c
d
j
g
b
e i n o
k m
a
f
g1 g2 g3

(a) Describe an isomorphism between graphs G1 and G2 , and another isomor-
phism between G2 and G3 .

(b) Why does part .a/ imply that there is an isomorphism between graphs G1 and
G3 ?
Let G and H be planar graphs. An embedding EG of G is isomorphic to an em-
bedding EH of H iff there is an isomorphism from G to H that also maps each
face of EG to a face of EH .

(d) Explain why all embeddings of two isomorphic planar graphs must have the
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13.8. Another Characterization for Planar Graphs 547

same number of faces.

Problem 13.3. (a) Give an example of a planar graph with two planar embeddings,
where the first embedding has a face whose length is not equal to the length of any
face in the secoind embedding. Draw the two embeddings to demonstrate this.

(b) Define the length of a planar embedding E to be the sum of the lengths of
the faces of E. Prove that all embeddings of the same planar graph have the same
length.

Problem 13.4.
Definition 13.2.2 of planar graph embeddings applied only to connected planar
graphs. The definition can be extended to planar graphs that are not necessarily
connected by adding the following additional constructor case to the definition:

Constructor Case: (collect disjoint graphs) Suppose E1 and E2 are planar
embeddings with no vertices in common. Then E1 [ E2 is a planar embed-
ding.

Euler’s Planar Graph Theorem now generalizes to unconnected graphs as fol-
lows: if a planar embedding E has v vertices, e edges, f faces and c connected
components, then
v e C f 2c D 0: (13.8)
This can be proved by structural induction on the definition of planar embedding.
(a) State and prove the base case of the structural induction.

(b) Let vi ; ei ; fi ; and ci be the number of vertices, edges, faces, and connected
components in embedding Ei and let v; e; f; c be the numbers for the embedding
from the (collect disjoint graphs) constructor case. Express v; e; f; c in terms of
vi ; ei ; fi ; ci .

Problem 13.5. (a) A simple graph has 8 vertices and 24 edges. What is the average
degree per vertex?

(b) A connected planar simple graph has 5 more edges than it has vertices. How
many faces does it have?
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548 Chapter 13 Planar Graphs

(d) How many faces does a planar graph from part c have?

(e) How many distinct isomorphisms are there between the graph given in Fig-
ure 13.17 and itself? (Include the identity isomorphism.)
a

d e b f

Figure 13.17

Class Problems
x

Problem 13.6.
Figure 13.18 shows four different pictures of planar graphs.
(a) For each picture, describe its discrete faces (closed walks that define the region
borders).

(b) Which of the pictured graphs are isomorphic? Which pictures represent the
same planar embedding?—that is, they have the same discrete faces.

(c) Describe a way to construct the embedding in Figure 4 according to the recur-
sive Definition 13.2.2 of planar embedding. For each application of a constructor
rule, be sure to indicate the faces (cycles) to which the rule was applied and the
cycles which result from the application.

Problem 13.7.
Prove the following assertions by structural induction on the definition of planar
embedding.
(a) In a planar embedding of a graph, each edge occurs exactly twice in the faces
of the embedding.
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13.8. Another Characterization for Planar Graphs 549

b c b

c
a d a d

figure 1 figure 2

b c b

c
a d a d

e e
figure 3 figure 4

Figure 13.18

(b) In a planar embedding of a connected graph with at least three vertices, each
face is of length at least three.

Homework Problems
Problem 13.8.
A simple graph is triangle-free when it has no cycle of length three.
(a) Prove for any connected triangle-free planar graph with v > 2 vertices and e
edges,
e 2v 4: (13.9)

(b) Show that any connected triangle-free planar graph has at least one vertex of
degree three or less.
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550 Chapter 13 Planar Graphs

Problem 13.9. (a) Prove
Lemma (Switch Edges). Suppose that, starting from some embeddings of planar
graphs with disjoint sets of vertices, it is possible by two successive applications of
constructor operations to add edges e and then f to obtain a planar embedding F.
Then starting from the same embeddings, it is also possible to obtain F by adding
f and then e with two successive applications of constructor operations.
Hint: There are four cases to analyze, depending on which two constructor opera-
tions are applied to add e and then f . Structural induction is not needed.

(b) Prove
Corollary (Permute Edges). Suppose that, starting from some embeddings of pla-
nar graphs with disjoint sets of vertices, it is possible to add a sequence of edges
e0 ; e1 ; : : : ; en by successive applications of constructor operations to obtain a pla-
nar embedding F. Then starting from the same embeddings, it is also possible
to obtain F by applications of constructor operations that successively add any
permutation5 of the edges e0 ; e1 ; : : : ; en .
Hint: By induction on the number of switches of adjacent elements needed to con-
vert the sequence 0,1,. . . ,n into a permutation .0/; .1/; : : : ; .n/.

(d) Conclude that any subgraph of a planar graph is planar.

5 If W f0; 1; : : : ; ng ! f0; 1; : : : ; ng is a bijection, then the sequence e.0/ ; e.1/ ; : : : ; e.n/ is
called a permutation of the sequence e0 ; e1 ; : : : ; en .
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III Counting
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Introduction
Counting seems easy enough: 1, 2, 3, 4, etc. This direct approach works well
for counting simple things—like your toes—and may be the only approach for ex-
tremely complicated things with no identifiable structure. However, subtler meth-
ods can help you count many things in the vast middle ground, such as:
The number of different ways to select a dozen doughnuts when there are
five varieties available.

The number of 16-bit numbers with exactly 4 ones.
Perhaps surprisingly, but certainly not coincidentally, these two numbers are the
same: 1820.
Counting is useful in computer science for several reasons:
Determining the time and storage required to solve a computational problem—
a central objective in computer science—often comes down to solving a
counting problem.

Password and encryption security counts on having a very large set of possi-
ble passwords and encryption keys.

Counting is the basis of probability theory, which plays a central role in all
sciences, including computer science.
We begin our study of counting in Chapter 14 with a collection of rules and
methods for finding
P closed-formQexpressions for commonly-occurring sums and
products such as niD0 x i and niD1 i . We also introduce asymptotic notations
such as , O and ‚ that are commonly used in computer science to express how a
quantity such as the running time of a program grows with the size of the input.
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554 Part III Counting

Chapter 15 describes the most basic rules for determining the cardinality of a set.
These rules are actually theorems, but our focus here will be less on their proofs
than on teaching their use in simple counting as a practical skill, like integration.
But counting can be tricky, and people make counting mistakes all the time,
so a crucial part of counting skill is being able to verify a counting argument.
Sometimes this can be done simply by finding an alternative way to count and
then comparing answers—they better agree. But most elementary counting argu-
ments reduce to finding a bijection between objects to be counted and easy-to-count
sequences. The chapter shows how explicitly defining these bijections—and veri-
fying that they are bijections—is another useful way to verify counting arguments.
The material in Chapter 15 is simple yet powerful, and it provides a great tool set
for use in your future career.
Finally, Chapter 16 introduces generating functions which allow many counting
problems to be solved by simple algebraic formula simplification.
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14 Sums and Asymptotics
Sums and products arise regularly in the analysis of algorithms, financial appli-
cations, physical problems, and probabilistic systems. For example, according to
Theorem 2.2.1,
n.n C 1/
1 C 2 C 3 C C n D : (14.1)
2
Of course, the left-hand sum could be expressed concisely as a subscripted summa-
tion
Xn
i
i D1

but the right-hand expression n.n C 1/=2 is not only concise but also easier to
evaluate. Furthermore, it more clearly reveals properties such as the growth rate
of the sum. Expressions like n.n C 1/=2 that do not make use of subscripted
summations or products—or those handy but sometimes troublesome sequences
of three dots—are called closed forms.
Another example is the closed form for a geometric sum

1 x nC1
1 C x C x2 C x3 C C xn D (14.2)
1 x
given in Problem 5.4. The sum as described on the left-hand side of (14.2) involves
n additions and 1 C 2 C C .n 1/ D .n 1/n=2 multiplications, but its closed
form on the right-hand side can be evaluated using fast exponentiation with at most
2 log n multiplications, a division, and a couple of subtractions. Also, the closed
form makes the growth and limiting behavior of the sum much more apparent.
Equations (14.1) and (14.2) were easy to verify by induction, but, as is often the
case, the proofs by induction gave no hint about how these formulas were found in
the first place. Finding them is part math and part art, which we’ll start examining
in this chapter.
Our first motivating example will be the value of a financial instrument known as
an annuity. This value will be a large and nasty-looking sum. We will then describe
several methods for finding closed forms for several sorts of sums, including those
for annuities. In some cases, a closed form for a sum may not exist, and so we
will provide a general method for finding closed forms for good upper and lower
bounds on the sum.
The methods we develop for sums will also work for products, since any product
can be converted into a sum by taking its logarithm. For instance, later in the
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556 Chapter 14 Sums and Asymptotics

chapter we will use this approach to find a good closed-form approximation to the
factorial function
nŠ WWD 1 2 3 n:
We conclude the chapter with a discussion of asymptotic notation, especially
“Big Oh” notation. Asymptotic notation is often used to bound the error terms
when there is no exact closed form expression for a sum or product. It also provides
a convenient way to express the growth rate or order of magnitude of a sum or
product.

14.1 The Value of an Annuity
Would you prefer a million dollars today or $50,000 a year for the rest of your life?
On the one hand, instant gratification is nice. On the other hand, the total dollars
received at $50K per year is much larger if you live long enough.
Formally, this is a question about the value of an annuity. An annuity is a finan-
cial instrument that pays out a fixed amount of money at the beginning of every year
for some specified number of years. In particular, an n-year, m-payment annuity
pays m dollars at the start of each year for n years. In some cases, n is finite, but
not always. Examples include lottery payouts, student loans, and home mortgages.
There are even firms on Wall Street that specialize in trading annuities.1
A key question is, “What is an annuity worth?” For example, lotteries often pay
out jackpots over many years. Intuitively, $50,000 a year for 20 years ought to be
worth less than a million dollars right now. If you had all the cash right away, you
could invest it and begin collecting interest. But what if the choice were between
$50,000 a year for 20 years and a half million dollars today? Suddenly, it’s not
clear which option is better.

14.1.1 The Future Value of Money
In order to answer such questions, we need to know what a dollar paid out in the
future is worth today. To model this, let’s assume that money can be invested at a
fixed annual interest rate p. We’ll assume an 8% rate2 for the rest of the discussion,
so p D 0:08.
1 Such trading ultimately led to the subprime mortgage disaster in 2008–2009. We’ll talk more
about that in a later chapter.
2 U.S. interest rates have dropped steadily for several years, and ordinary bank deposits now earn

around 1.0%. But just a few years ago the rate was 8%; this rate makes some of our examples a little
more dramatic. The rate has been as high as 17% in the past thirty years.
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14.1. The Value of an Annuity 557

Here is why the interest rate p matters. Ten dollars invested today at interest rate
p will become .1 C p/ 10 D 10:80 dollars in a year, .1 C p/2 10 11:66 dollars
in two years, and so forth. Looked at another way, ten dollars paid out a year from
now is only really worth 1=.1 C p/ 10 9:26 dollars today, because if we had the
$9.26 today, we could invest it and would have $10.00 in a year anyway. Therefore,
p determines the value of money paid out in the future.
So for an n-year, m-payment annuity, the first payment of m dollars is truly worth
m dollars. But the second payment a year later is worth only m=.1 C p/ dollars.
Similarly, the third payment is worth m=.1 C p/2 , and the n-th payment is worth
only m=.1 C p/n 1 . The total value V of the annuity is equal to the sum of the
payment values. This gives:
n
X m
V D
.1 C p/i 1
i D1
n
X1 1 j
Dm (substitute j D i 1)
1Cp
j D0
n
X1
Dm xj (substitute x D 1=.1 C p/): (14.3)
j D0

The goal of the preceding substitutions was to get the summation into the form
of a simple geometric sum. This leads us to an explanation of a way you could have
discovered the closed form (14.2) in the first place using the Perturbation Method.

14.1.2 The Perturbation Method
Given a sum that has a nice structure, it is often useful to “perturb” the sum so that
we can somehow combine the sum with the perturbation to get something much
simpler. For example, suppose

S D 1 C x C x2 C C xn:

An example of a perturbation would be

xS D x C x 2 C C x nC1 :

The difference between S and xS is not so great, and so if we were to subtract xS
from S , there would be massive cancellation:

S D 1 C x C x2 C x3 C C xn
xS D x x2 x3 xn x nC1 :
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558 Chapter 14 Sums and Asymptotics

The result of the subtraction is

S xS D 1 x nC1 :

Solving for S gives the desired closed-form expression in equation 14.2, namely,

1 x nC1
SD :
1 x
We’ll see more examples of this method when we introduce generating functions
in Chapter 16.

14.1.3 A Closed Form for the Annuity Value
Using equation 14.2, we can derive a simple formula for V , the value of an annuity
that pays m dollars at the start of each year for n years.

1 xn

V Dm (by equations 14.3 and 14.2) (14.4)
1 x
!
1 C p .1=.1 C p//n 1
Dm (substituting x D 1=.1 C p/): (14.5)
p

Equation 14.5 is much easier to use than a summation with dozens of terms. For
example, what is the real value of a winning lottery ticket that pays $50,000 per
year for 20 years? Plugging in m D $50,000, n D 20 and p D 0:08 gives
V $530,180. So because payments are deferred, the million dollar lottery is
really only worth about a half million dollars! This is a good trick for the lottery
advertisers.

14.1.4 Infinite Geometric Series
We began this chapter by asking whether you would prefer a million dollars today
or $50,000 a year for the rest of your life. Of course, this depends on how long
you live, so optimistically assume that the second option is to receive $50,000 a
year forever. This sounds like infinite money! But we can compute the value of an
annuity with an infinite number of payments by taking the limit of our geometric
sum in equation 14.2 as n tends to infinity.

Theorem 14.1.1. If jxj < 1, then
1
X 1
xi D :
1 x
i D0
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14.1. The Value of an Annuity 559

Proof.
1
X n
X
x i WWD lim xi
n!1
i D0 iD0
1 x nC1
D lim (by equation 14.2)
n!1 1 x
1
D :
1 x
The final line follows from the fact that limn!1 x nC1 D 0 when jxj < 1.

In our annuity problem x D 1=.1 C p/ < 1, so Theorem 14.1.1 applies, and we
get
1
X
V Dm xj (by equation 14.3)
j D0
1
Dm (by Theorem 14.1.1)
1 x
1Cp
Dm .x D 1=.1 C p//:
p

Plugging in m D $50,000 and p D 0:08, we see that the value V is only $675,000.
It seems amazing that a million dollars today is worth much more than $50,000
paid every year for eternity! But on closer inspection, if we had a million dollars
today in the bank earning 8% interest, we could take out and spend $80,000 a year,
forever. So as it turns out, this answer really isn’t so amazing after all.

14.1.5 Examples
Equation 14.2 and Theorem 14.1.1 are incredibly useful in computer science.
Here are some other common sums that can be put into closed form using equa-
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560 Chapter 14 Sums and Asymptotics

tion 14.2 and Theorem 14.1.1:
1 i
X 11
1 C 1=2 C 1=4 C D D2 D (14.6)
2 1 .1=2/
i D0
1
! !
1 i 1 10
X
0:99999 D 0:9 D 0:9 D 0:9 D1 (14.7)
10 1 1=10 9
i D0
1
1 i 1

X 2
1 1=2 C 1=4 D D D (14.8)
2 1 . 1=2/ 3
i D0
n
X1 1 2n
1 C 2 C 4 C C 2n 1
D 2i D D 2n 1 (14.9)
1 2
i D0
n
X1 1 3n 3n 1
1 C 3 C 9 C C 3n 1
D 3i D D (14.10)
1 3 2
i D0
If the terms in a geometric sum grow smaller, as in equation 14.6, then the sum is
said to be geometrically decreasing. If the terms in a geometric sum grow progres-
sively larger, as in equations 14.9 and 14.10, then the sum is said to be geometrically
increasing. In either case, the sum is usually approximately equal to the term in the
sum with the greatest absolute value. For example, in equations 14.6 and 14.8, the
largest term is equal to 1 and the sums are 2 and 2/3, both relatively close to 1. In
equation 14.9, the sum is about twice the largest term. In equation 14.10, the largest
term is 3n 1 and the sum is .3n 1/=2, which is only about a factor of 1:5 greater.
You can see why this rule of thumb works by looking carefully at equation 14.2
and Theorem 14.1.1.

14.1.6 Variations of Geometric Sums
We now know all about geometric sums—if you have one, life is easy. But in
practice one often encounters
P i sums that cannot be transformed by simple variable
substitutions to the form x .
A non-obvious but useful way to obtain new summation formulas from old ones
is by differentiating or integrating with respect to x. As an example, consider the
following sum:
n
X1
ix i D x C 2x 2 C 3x 3 C C .n 1/x n 1

i D1
This is not a geometric sum. The ratio between successive terms is not fixed, and
so our formula for the sum of a geometric sum cannot be directly applied. But
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14.1. The Value of an Annuity 561

differentiating equation 14.2 leads to:
n 1
!
d 1 xn

d X i
x D : (14.11)
dx dx 1 x
i D0

The left-hand side of equation 14.11 is simply
n
X1 n
X1
d i
.x / D ix i 1
:
dx
i D0 i D0

The right-hand side of equation 14.11 is

nx n 1 .1 x/ . 1/.1 xn/ nx n 1
C nx n C 1 x n
D
.1 x/2 .1 x/2
1 nx 1 C .n 1/x n
n
D :
.1 x/2
Hence, equation 14.11 means that
n
X1 1 nx n 1 C .n 1/x n
ix i 1
D :
.1 x/2
i D0

Incidentally, Problem 14.2 shows how the perturbation method could also be ap-
plied to derive this formula.
Often, differentiating or integrating messes up the exponent
P of x in every term.
i 1
P a formula for a sum of the form ix , but we want a
In this case, we now have
formula for the series ix i . The solution is simple: multiply by x. This gives:
n
X1 x nx n C .n 1/x nC1
ix i D (14.12)
.1 x/2
i D1

and we have the desired closed-form expression for our sum. It seems a little com-
plicated, but it’s easier to work with than the sum.
Notice that if jxj < 1, then this series converges to a finite value even if there
are infinitely many terms. Taking the limit of equation 14.12 as n tends to infinity
gives the following theorem:
Theorem 14.1.2. If jxj < 1, then
1
X x
ix i D : (14.13)
.1 x/2
i D1
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562 Chapter 14 Sums and Asymptotics

As a consequence, suppose that there is an annuity that pays i m dollars at the
end of each year i , forever. For example, if m D $50,000, then the payouts are
$50,000 and then $100,000 and then $150,000 and so on. It is hard to believe that
the value of this annuity is finite! But we can use Theorem 14.1.2 to compute the
value:
1
X im
V D
.1 C p/i
i D1
1=.1 C p/
Dm 1 2
.1 1Cp /
1Cp
Dm :
p2
The second line follows by an application of Theorem 14.1.2. The third line is
obtained by multiplying the numerator and denominator by .1 C p/2 .
For example, if m D $50,000, and p D 0:08 as usual, then the value of the
annuity is V D $8,437,500. Even though the payments increase every year, the in-
crease is only additive with time; by contrast, dollars paid out in the future decrease
in value exponentially with time. The geometric decrease swamps out the additive
increase. Payments in the distant future are almost worthless, so the value of the
annuity is finite.
The important thing to remember is the trick of taking the derivative (or integral)
of a summation formula. Of course, this technique requires one to compute nasty
derivatives correctly, but this is at least theoretically possible!

14.2 Sums of Powers
In Chapter 5, we verified the formula (14.1), but the source of this formula is still
a mystery. Sure, we can prove that it’s true by using well ordering or induction,
but where did the expression on the right come from in the first place? Even more
inexplicable is the closed form expression for the sum of consecutive squares:
n
X .2n C 1/.n C 1/n
i2 D : (14.14)
6
i D1

It turns out that there is a way to derive these expressions, but before we explain
it, we thought it would be fun—OK, our definition of “fun” may be different than
“mcs” — 2017/3/10 — 22:22 — page 563 — #571

14.2. Sums of Powers 563

yours—to show you how Gauss is supposed to have proved equation 14.1 when he
was a young boy.
Gauss’s idea is related to the perturbation method we used in Section 14.1.2. Let
n
X
SD i:
i D1

Then we can write the sum in two orders:
S D 1 C 2 C : : : C .n 1/ C n;
S D n C .n 1/ C : : : C 2 C 1:
Adding these two equations gives

2S D .n C 1/ C .n C 1/ C C .n C 1/ C .n C 1/
D n.n C 1/:

Hence,
n.n C 1/
SD :
2
Not bad for a young child—Gauss showed some potential. . . .
Unfortunately, the same trick does not work for summing consecutive squares.
However, we can observe that the result might be a third-degree polynomial in n,
since the sum contains n terms that average out to a value that grows quadratically
in n. So we might guess that
n
X
i 2 D an3 C bn2 C cn C d:
i D1

If our guess is correct, then we can determine the parameters a, b, c and d by
plugging in a few values for n. Each such value gives a linear equation in a, b,
c and d . If we plug in enough values, we may get a linear system with a unique
solution. Applying this method to our example gives:

nD0 implies 0 D d
nD1 implies 1 D a C b C c C d
nD2 implies 5 D 8a C 4b C 2c C d
nD3 implies 14 D 27a C 9b C 3c C d:

Solving this system gives the solution a D 1=3, b D 1=2, c D 1=6, d D 0.
Therefore, if our initial guess at the form of the solution was correct, then the
summation is equal to n3 =3 C n2 =2 C n=6, which matches equation 14.14.
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564 Chapter 14 Sums and Asymptotics

The point is that if the desired formula turns out to be a polynomial, then once
you get an estimate of the degree of the polynomial, all the coefficients of the
polynomial can be found automatically.
Be careful! This method lets you discover formulas, but it doesn’t guarantee
they are right! After obtaining a formula by this method, it’s important to go back
and prove it by induction or some other method. If the initial guess at the solution
was not of the right form, then the resulting formula will be completely wrong! A
later chapter will describe a method based on generating functions that does not
require any guessing at all.

14.3 Approximating Sums
Unfortunately, it is not always possible to find a closed-form expression for a sum.
For example, no closed form is known for
n
X p
SD i:
i D1

In such cases, we need to resort to approximations for S if we want to have a
closed form. The good news is that there is a general method to find closed-form
upper and lower bounds that works well for many sums. Even better, the method
is simple and easy to remember. It works by replacing the sum by an integral and
then adding either the first or last term in the sum.
Definition 14.3.1. A function f W RC ! RC is strictly increasing when
x < y IMPLIES f .x/ < f .y/;
and it is weakly increasing3 when
x < y IMPLIES f .x/ f .y/:
Similarly, f is strictly decreasing when
x < y IMPLIES f .x/ > f .y/;
and it is weakly decreasing4 when
x < y IMPLIES f .x/ f .y/:
3 Weakly increasing functions are usually called nondecreasing functions. We will avoid this

terminology to prevent confusion between being a nondecreasing function and the much weaker
property of not being a decreasing function.
4 Weakly decreasing functions are usually called nonincreasing.
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14.3. Approximating Sums 565

f.n/
f.n�1/


f.3/
f.2/
f.1/

0 1 2 3  n�2 n�1 n

Figure 14.1 The area of the i th rectangle is f .i /. The shaded region has area
P n
i D1 f .i /.

p
For example, 2x and x are strictly increasing functions, while maxfx; 2g and
dxe are weakly increasing functions. The functions 1=x and 2 x are strictly de-
creasing, while minf1=x; 1=2g and b1=xc are weakly decreasing.

Theorem 14.3.2. Let f W RC ! RC be a weakly increasing function. Define
n
X
S WWD f .i / (14.15)
i D1

and Z n
I WWD f .x/ dx:
1
Then
I C f .1/ S I C f .n/: (14.16)
Similarly, if f is weakly decreasing, then

I C f .n/ S I C f .1/:

Proof. Suppose f W RC ! RC is weakly increasing. The value of the sum S
in (14.15) is the sum of the areas of n unit-width rectangles of heights f .1/; f .2/; : : : ; f .n/.
This area of these rectangles is shown shaded in Figure 14.1.
The value of Z n
I D f .x/ dx
1
is the shaded area under the curve of f .x/ from 1 to n shown in Figure 14.2.
“mcs” — 2017/3/10 — 22:22 — page 566 — #574

566 Chapter 14 Sums and Asymptotics

f.n/
f.n�1/
f.x/


f.3/
f.2/
f.1/

0 1 2 3  n�2 n�1 n

Figure R14.2 The shaded area under the curve of f .x/ from 1 to n (shown in bold)
n
is I D 1 f .x/ dx.

Comparing the shaded regions in Figures 14.1 and 14.2 shows that S is at least
I plus the area of the leftmost rectangle. Hence,

S I C f .1/ (14.17)

This is the lower bound for S given in (14.16).
To derive the upper bound for S given in (14.16), we shift the curve of f .x/
from 1 to n one unit to the left as shown in Figure 14.3.
Comparing the shaded regions in Figures 14.1 and 14.3 shows that S is at most
I plus the area of the rightmost rectangle. That is,

S I C f .n/;

which is the upper bound for S given in (14.16).
The very similar argument for the weakly decreasing case is left to Problem 14.10.

Theorem 14.3.2 provides good bounds for most sums. At worst, the bounds will
be off by the largest term in the sum. For example, we can use Theorem 14.3.2 to
bound the sum
n
X p
SD i
i D1

as follows.
“mcs” — 2017/3/10 — 22:22 — page 567 — #575

14.3. Approximating Sums 567

f.n/
f.n�1/ f.xC1/


f.3/
f.2/
f.1/

0 1 2 3  n�2 n�1 n

Figure 14.3 This curve is the same as the curve in Figure 14.2 shifted left by 1.

We begin by computing
Z np
I D x dx
1
ˇn
x 3=2 ˇˇ
D
3=2 ˇ
ˇ
1
2
D .n3=2 1/:
3
We then apply Theorem 14.3.2 to conclude that
2 3=2 2 3=2 p
.n 1/ C 1 S .n 1/ C n
3 3
and thus that
2 3=2 1 2 p 2
n C S n3=2 C n :
3 3 3 3
In other words, the sum is very close to 23 n3=2 . We’ll define several ways that one
thing can be “very close to” something else at the end of this chapter.
As a first application of Theorem 14.3.2, we explain in the next section how it
helps in resolving a classic paradox in structural engineering.
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568 Chapter 14 Sums and Asymptotics

14.4 Hanging Out Over the Edge
Suppose you have a bunch of books and you want to stack them up, one on top
of another in some off-center way, so the top book sticks out past books below it
without falling over. If you moved the stack to the edge of a table, how far past
the edge of the table do you think you could get the top book to go? Could the top
book stick out completely beyond the edge of table? You’re not supposed to use
glue or any other support to hold the stack in place.
Most people’s first response to the Book Stacking Problem—sometimes also
their second and third responses—is “No, the top book will never get completely
past the edge of the table.” But in fact, you can get the top book to stick out as far
as you want: one booklength, two booklengths, any number of booklengths!

14.4.1 Formalizing the Problem
We’ll approach this problem recursively. How far past the end of the table can we
get one book to stick out? It won’t tip as long as its center of mass is over the table,
so we can get it to stick out half its length, as shown in Figure 14.4.

center of mass
of book

1
2
table

Figure 14.4 One book can overhang half a book length.

Now suppose we have a stack of books that will not tip over if the bottom book
rests on the table—call that a stable stack. Let’s define the overhang of a stable
stack to be the horizontal distance from the center of mass of the stack to the furthest
edge of the top book. So the overhang is purely a property of the stack, regardless
of its placement on the table. If we place the center of mass of the stable stack at
the edge of the table as in Figure 14.5, the overhang is how far we can get the top
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14.4. Hanging Out Over the Edge 569

center of mass overhang
of the whole stack

table

Figure 14.5 Overhanging the edge of the table.

book in the stack to stick out past the edge.
In general, a stack of n books will be stable if and only if the center of mass of
the top i books sits over the .i C 1/st book for i D 1, 2, . . . , n 1.
So we want a formula for the maximum possible overhang Bn achievable with a
stable stack of n books.
We’ve already observed that the overhang of one book is 1/2 a book length. That
is,
1
B1 D :
2
Now suppose we have a stable stack of n C 1 books with maximum overhang.
If the overhang of the n books on top of the bottom book was not maximum, we
could get a book to stick out further by replacing the top stack with a stack of n
books with larger overhang. So the maximum overhang BnC1 of a stack of n C 1
books is obtained by placing a maximum overhang stable stack of n books on top
of the bottom book. And we get the biggest overhang for the stack of n C 1 books
by placing the center of mass of the n books right over the edge of the bottom book
as in Figure 14.6.
So we know where to place the n C 1st book to get maximum overhang. In fact,
the reasoning above actually shows that this way of stacking n C 1 books is the
unique way to build a stable stack where the top book extends as far as possible.
All we have to do is calculate what this extension is.
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570 Chapter 14 Sums and Asymptotics

center of mass
of all n+1 books
center of mass
of top n books
} top n books

}
1
table 2( n+1)

Figure 14.6 Additional overhang with n C 1 books.

The simplest way to do that is to let the center of mass of the top n books be the
origin. That way the horizontal coordinate of the center of mass of the whole stack
of n C 1 books will equal the increase in the overhang. But now the center of mass
of the bottom book has horizontal coordinate 1=2, so the horizontal coordinate of
center of mass of the whole stack of n C 1 books is
0 n C .1=2/ 1 1
D :
nC1 2.n C 1/
In other words,
1
BnC1 D Bn C ; (14.18)
2.n C 1/
as shown in Figure 14.6.
Expanding equation (14.18), we have
1 1
BnC1 D Bn 1 C C
2n 2.n C 1/
1 1 1
D B1 C C C C
22 2n 2.n C 1/
nC1
1X1
D : (14.19)
2 i
i D1

So our next task is to examine the behavior of Bn as n grows.
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14.4. Hanging Out Over the Edge 571

14.4.2 Harmonic Numbers
Definition 14.4.1. The nth harmonic number Hn is
n
X 1
Hn WWD :
i
i D1

So (14.19) means that
Hn
Bn D:
2
The first few harmonic numbers are easy to compute. For example, H4 D 1 C
1 1 1 25
2 C 3 C 4 D 12 > 2. The fact that H4 is greater than 2 has special significance:
it implies that the total extension of a 4-book stack is greater than one full book!
This is the situation shown in Figure 14.7.

1/2
1/4
1/6
1/8
Table

Figure 14.7 Stack of four books with maximum overhang.

There is good news and bad news about harmonic numbers. The bad news is
that there is no known closed-form expression for the harmonic numbers. The
good news is that we can use Theorem 14.3.2 to get close upper and lower bounds
on Hn . In particular, since
Z n ˇn
1
dx D ln.x/ ˇ D ln.n/;
ˇ
1 x 1

Theorem 14.3.2 means that
1
ln.n/ C Hn ln.n/ C 1: (14.20)
n
In other words, the nth harmonic number is very close to ln.n/.
Because the harmonic numbers frequently arise in practice, mathematicians have
worked hard to get even better approximations for them. In fact, it is now known
that
1 1 .n/
Hn D ln.n/ C C C C (14.21)
2n 12n2 120n4
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572 Chapter 14 Sums and Asymptotics

Here is a value 0:577215664 : : : called Euler’s constant, and .n/ is between 0
and 1 for all n. We will not prove this formula.
We are now finally done with our analysis of the book stacking problem. Plug-
ging the value of Hn into (14.19), we find that the maximum overhang for n books
is very close to 1=2 ln.n/. Since ln.n/ grows to infinity as n increases, this means
that if we are given enough books we can get a book to hang out arbitrarily far
over the edge of the table. Of course, the number of books we need will grow as
an exponential function of the overhang; it will take 227 books just to achieve an
overhang of 3, never mind an overhang of 100.

Extending Further Past the End of the Table
The overhang we analyzed above was the furthest out the top book could extend
past the table. This leaves open the question of if there is some better way to build
a stable stack where some book other than the top stuck out furthest. For example,
Figure 14.8 shows a stable stack of two books where the bottom book extends
further out than the top book. Moreover, the bottom book extends 3/4 of a book
length past the end of the table, which is the same as the maximum overhang for
the top book in a two book stack.
Since the two book arrangement in Figure 14.8(a) ties the maximum overhang
stack in Figure 14.8(b), we could take the unique stable stack of n books where the
top book extends furthest, and switch the top two books to look like Figure 14.8(a).
This would give a stable stack of n books where the second from the top book
extends the same maximum overhang distance. So for n > 1, there are at least
two ways of building a stable stack of n books which both extend the maximum
overhang distance—one way where the top book is furthest out, and another way
where the second from the top book is furthest out.
It turns out that there is no way to beat these two ways of making stable stacks.
In fact, it’s not too hard to show that these are the only two ways to get a stable
stack of books that achieves maximum overhang.
But there is more to the story. All our reasoning above was about stacks in which
one book rests on another. It turns out that by building structures in which more
than one book rests on top of another book—think of an inverted pyramid—it is
p
possible to get a stack of n books to extend proportional to 3 n—much more than
ln n—book lengths without falling over. See [14], Maximum Overhang.

14.4.3 Asymptotic Equality
For cases like equation 14.21 where we understand the growth of a function like Hn
up to some (unimportant) error terms, we use a special notation, , to denote the
leading term of the function. For example, we say that Hn ln.n/ to indicate that
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14.4. Hanging Out Over the Edge 573

table

1=2 3=4

(a)

table

1=4 1=2

(b)

Figure 14.8 Figure (a) shows a stable stack of two books where the bottom book
extends the same amount past the end of the table as the maximum overhang two-
book stack shown in Figure (b).
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574 Chapter 14 Sums and Asymptotics

the leading term of Hn is ln.n/. More precisely:

Definition 14.4.2. For functions f; g W R ! R, we say f is asymptotically equal
to g, in symbols,
f .x/ g.x/
iff
lim f .x/=g.x/ D 1:
x!1

Although it is tempting to write Hn ln.n/ C to indicate the two leading
terms, this is not really right. According to Definition 14.4.2, Hn ln.n/ C c
where c is any constant. The correct way to indicate that is the second-largest
term is Hn ln.n/ .
The reason that the notation is useful is that often we do not care about lower
order terms. For example, if n D 100, then we can compute H.n/ to great precision
using only the two leading terms:
ˇ ˇ
ˇ 1 1 1 ˇ< 1 :
ˇ
jHn ln.n/ jˇ ˇ C 4
200 120000 120 100 ˇ 200
We will spend a lot more time talking about asymptotic notation at the end of the
chapter. But for now, let’s get back to using sums.

14.5 Products
We’ve covered several techniques for finding closed forms for sums but no methods
for dealing with products. Fortunately, we do not need to develop an entirely new
set of tools when we encounter a product such as
n
Y
nŠ WWD i: (14.22)
iD1

That’s because we can convert any product into a sum by taking a logarithm. For
example, if
Yn
P D f .i /;
i D1
then
n
X
ln.P / D ln.f .i //:
i D1
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14.5. Products 575

We can then apply our summing tools to find a closed form (or approximate closed
form) for ln.P / and then exponentiate at the end to undo the logarithm.
For example, let’s see how this works for the factorial function nŠ. We start by
taking the logarithm:

ln.nŠ/ D ln.1 2 3 .n 1/ n/
D ln.1/ C ln.2/ C ln.3/ C C ln.n 1/ C ln.n/
n
X
D ln.i /:
i D1

Unfortunately, no closed form for this sum is known. However, we can apply
Theorem 14.3.2 to find good closed-form bounds on the sum. To do this, we first
compute
Z n ˇn
ln.x/ dx D x ln.x/ x ˇ
ˇ
1 1
D n ln.n/ n C 1:

Plugging into Theorem 14.3.2, this means that
n
X
n ln.n/ nC1 ln.i / n ln.n/ n C 1 C ln.n/:
i D1

Exponentiating then gives
nn nnC1
nŠ : (14.23)
en 1 en 1
This means that nŠ is within a factor of n of nn =e n 1.

14.5.1 Stirling’s Formula
The most commonly used product in discrete mathematics is probably nŠ, and
mathematicians have workedto find tight closed-form bounds on its value. The
most useful bounds are given in Theorem 14.5.1.
Theorem 14.5.1 (Stirling’s Formula). For all n 1,
p n n
nŠ D 2 n e .n/
e
where
1 1
.n/ :
12n C 1 12n
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576 Chapter 14 Sums and Asymptotics

Theorem 14.5.1 can be proved by induction (with some pain), and there are lots
of proofs using elementary calculus, but we won’t go into them.
There are several important things to notice about Stirling’s Formula. First, .n/
is always positive. This means that
p n n
nŠ > 2 n (14.24)
e
for all n 2 NC .
Second, .n/ tends to zero as n gets large. This means that
p n n
nŠ 2 n (14.25)
e
which is impressive. After all, who would expect both and e to show up in a
closed-form expression that is asymptotically equal to nŠ?
Third, .n/ is small even for small values of n. This means that Stirling’s For-
mula provides good approximations for nŠ for most all values of n. For example, if
we use p n n
2 n
e
as the approximation for nŠ, as many people do, we are guaranteed to be within a
factor of
1
e .n/ e 12n
of the correct value. For n 10, this means we will be within 1% of the correct
value. For n 100, the error will be less than 0.1%.
If we need an even closer approximation for nŠ, then we could use either
p n n
2 n e 1=12n
e
or p n n
2 n e 1=.12nC1/
e
depending on whether we want an upper, or a lower, bound. By Theorem 14.5.1,
we know that both bounds will be within a factor of
1 1 1
e 12n 12nC1 D e 144n2 C12n

of the correct value. For n 10, this means that either bound will be within 0.01%
of the correct value. For n 100, the error will be less than 0.0001%.
For quick future reference, these facts are summarized in Corollary 14.5.2 and
Table 14.1.
“mcs” — 2017/3/10 — 22:22 — page 577 — #585

14.6. Double Trouble 577

Approximation n1 n 10 n 100 n 1000
p n
2 n ne < 10% < 1% < 0.1% < 0.01%
p n
2 n ne e 1=12n < 1% < 0.01% < 0.0001% < 0.000001%

Table 14.1 Error bounds on common approximations
p for nŠ from Theo-
n n
rem 14.5.1. For example, if n 100, then 2 n e approximates nŠ to
within 0.1%.

Corollary 14.5.2.
8
ˆ1:09 for n 1;
p n n <

nŠ < 2 n 1:009 for n 10;
e ˆ
1:0009 for n 100:
:

14.6 Double Trouble
Sometimes we have to evaluate sums of sums, otherwise known as double summa-
tions. This sounds hairy, and sometimes it is. But usually, it is straightforward—
you just evaluate the inner sum, replace it with a closed form, and then evaluate the
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578 Chapter 14 Sums and Asymptotics

outer sum (which no longer has a summation inside it). For example,5
1 n 1
!
x nC1

n1
X X X
n i
y x D y equation 14.2
1 x
nD0 i D0 nD0
1
X 1
X
1 n 1
D y y n x nC1
1 x 1 x
nD0 nD0
1
1 x X
D .xy/n Theorem 14.1.1
.1 x/.1 y/ 1 x
nD0
1 x
D Theorem 14.1.1
.1 x/.1 y/ .1 x/.1 xy/
.1 xy/ x.1 y/
D
.1 x/.1 y/.1 xy/
1 x
D
.1 x/.1 y/.1 xy/
1
D :
.1 y/.1 xy/
When there’s no obvious closed form for the inner sum, a special trick that is
often useful is to try exchanging the order of summation. For example, suppose we
want to compute the sum of the first n harmonic numbers
n n X
k
X X 1
Hk D (14.26)
j
kD1 kD1 j D1

For intuition about this sum, we can apply Theorem 14.3.2 to equation 14.20 to
conclude that the sum is close to
Z n ˇn
ln.x/ dx D x ln.x/ x ˇ D n ln.n/ n C 1:
ˇ
1 1

Now let’s look for an exact answer. If we think about the pairs .k; j / over which
5 OK, so maybe this one is a little hairy, but it is also fairly straightforward. Wait till you see the

next one!
“mcs” — 2017/3/10 — 22:22 — page 579 — #587

14.6. Double Trouble 579

we are summing, they form a triangle:

j
1 2 3 4 5 ::: n
k 1 1
2 1 1=2
3 1 1=2 1=3
4 1 1=2 1=3 1=4
:::
n 1 1=2 ::: 1=n

The summation in equation 14.26 is summing each row and then adding the row
sums. Instead, we can sum the columns and then add the column sums. Inspecting
the table we see that this double sum can be written as
n n X
k
X X 1
Hk D
j
kD1 kD1 j D1
n
n X
X 1
D
j
j D1 kDj
n n
X 1 X
D 1
j
j D1 kDj
n
X 1
D .n j C 1/
j
j D1
n n
X nC1 X j
D
j j
j D1 j D1
n n
X 1 X
D .n C 1/ 1
j
j D1 j D1

D .n C 1/Hn n: (14.27)
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580 Chapter 14 Sums and Asymptotics

14.7 Asymptotic Notation
Asymptotic notation is a shorthand used to give a quick measure of the behavior
of a function f .n/ as n grows large. For example, the asymptotic notation of
Definition 14.4.2 is a binary relation indicating that two functions grow at the same
rate. There is also a binary relation “little oh” indicating that one function grows at
a significantly slower rate than another and “Big Oh” indicating that one function
grows not much more rapidly than another.

14.7.1 Little O
Definition 14.7.1. For functions f; g W R ! R, with g nonnegative, we say f is
asymptotically smaller than g, in symbols,

f .x/ D o.g.x//;

iff
lim f .x/=g.x/ D 0:
x!1

For example, 1000x 1:9 D o.x 2 / because 1000x 1:9 =x 2 D 1000=x 0:1 and since
x 0:1goes to infinity with x and 1000 is constant, we have limx!1 1000x 1:9 =x 2 D
0. This argument generalizes directly to yield

Lemma 14.7.2. x a D o.x b / for all nonnegative constants a < b.

Using the familiar fact that log x < x for all x > 1, we can prove

Lemma 14.7.3. log x D o.x / for all > 0.

Proof. Choose > ı > 0 and let x D z ı in the inequality log x < x. This implies

log z < z ı =ı D o.z / by Lemma 14.7.2: (14.28)

Corollary 14.7.4. x b D o.ax / for any a; b 2 R with a > 1.

Lemma 14.7.3 and Corollary 14.7.4 can also be proved using l’Hôpital’s Rule or
the Maclaurin Series for log x and e x . Proofs can be found in most calculus texts.
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14.7. Asymptotic Notation 581

14.7.2 Big O
“Big Oh” is the most frequently used asymptotic notation. It is used to give an
upper bound on the growth of a function, such as the running time of an algorithm.
There is a standard definition of Big Oh given below in 14.7.9, but we’ll begin with
an alternative definition that makes apparent several basic properties of Big Oh.

Definition 14.7.5. Given functions f; g W R ! R with g nonnegative, we say that

f D O.g/

iff
lim sup jf .x/j =g.x/ < 1:
x!1

Here we’re using the technical notion of limit superior6 instead of just limit. But
because limits and lim sup’s are the same when limits exist, this formulation makes
it easy to check basic properties of Big Oh. We’ll take the following Lemma for
granted.

Lemma 14.7.6. If a function f W R ! R has a finite or infinite limit as its argument
approaches infinity, then its limit and limit superior are the same.

Now Definition 14.7.5 immediately implies:

Lemma 14.7.7. If f D o.g/ or f g, then f D O.g/.

Proof. lim f =g D 0 or lim f =g D 1 implies lim f =g < 1, so by Lemma 14.7.6,
lim sup f =g < 1.

Note that the converse of Lemma 14.7.7 is not true. For example, 2x D O.x/,
but 2x 6 x and 2x ¤ o.x/.
We also have:

Lemma 14.7.8. If f D o.g/, then it is not true that g D O.f /.

Proof.
g.x/ 1 1
lim D D D 1;
x!1 f .x/ limx!1 f .x/=g.x/ 0
so by Lemma 14.7.6, g ¤ O.f /.
6 The precise definition of lim sup is

lim sup h.x/ WWD lim lubyx h.y/;
x!1 x!1

where “lub” abbreviates “least upper bound.”
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582 Chapter 14 Sums and Asymptotics

We need lim sup’s in Definition 14.7.5 to cover cases when limits don’t exist. For
example, if f .x/=g.x/ oscillates between 3 and 5 as x grows, then limx!1 f .x/=g.x/
does not exist, but f D O.g/ because lim supx!1 f .x/=g.x/ D 5.
An equivalent, more usual formulation of big O does not mention lim sup’s:
Definition 14.7.9. Given functions f; g W R ! R with g nonnegative, we say

f D O.g/

iff there exists a constant c 0 and an x0 such that for all x x0 , jf .x/j cg.x/.
This definition is rather complicated, but the idea is simple: f .x/ D O.g.x//
means f .x/ is less than or equal to g.x/, except that we’re willing to ignore a
constant factor, namely c and to allow exceptions for small x, namely, x < x0 . So
in the case that f .x/=g.x/ oscillates between 3 and 5, f D O.g/ according to
Definition 14.7.9 because f 5g.
Proposition 14.7.10. 100x 2 D O.x 2 /.
Proof. ˇChooseˇ c D 100 and x0 D 1. Then the proposition holds, since for all
x 1, ˇ100x 2 ˇ 100x 2 .

Proposition 14.7.11. x 2 C 100x C 10 D O.x 2 /.
Proof. .x 2 C100x C10/=x 2 D 1C100=x C10=x 2 and so its limit as x approaches
infinity is 1C0C0 D 1. So in fact, x 2 C100xC10 x 2 , and therefore x 2 C100xC
10 D O.x 2 /. Indeed, it’s conversely true that x 2 D O.x 2 C 100x C 10/.

Proposition 14.7.11 generalizes to an arbitrary polynomial:
Proposition 14.7.12. ak x k C ak 1x
k 1 C C a1 x C a0 D O.x k /.
We’ll omit the routine proof.
Big O notation is especially useful when describing the running time of an al-
gorithm. For example, the usual algorithm for multiplying n n matrices uses a
number of operations proportional to n3 in the worst case. This fact can be ex-
pressed concisely by saying that the running time is O.n3 /. So this asymptotic
notation allows the speed of the algorithm to be discussed without reference to
constant factors or lower-order terms that might be machine specific. It turns out
that there is another matrix multiplication procedure that uses O.n2:55 / operations.
The fact that this procedure is asymptotically faster indicates that it involves new
ideas that go beyond a simply more efficient implementation of the O.n3 / method.
Of course the asymptotically faster procedure will also definitely be much more
efficient on large enough matrices, but being asymptotically faster does not mean
“mcs” — 2017/3/10 — 22:22 — page 583 — #591

14.7. Asymptotic Notation 583

that it is a better choice. The O.n2:55 /-operation multiplication procedure is almost
never used in practice because it only becomes more efficient than the usual O.n3 /
procedure on matrices of impractical size.7

14.7.3 Theta
Sometimes we want to specify that a running time T .n/ is precisely quadratic up to
constant factors (both upper bound and lower bound). We could do this by saying
that T .n/ D O.n2 / and n2 D O.T .n//, but rather than say both, mathematicians
have devised yet another symbol ‚ to do the job.

Definition 14.7.13.

f D ‚.g/ iff f D O.g/ and g D O.f /:

The statement f D ‚.g/ can be paraphrased intuitively as “f and g are equal
to within a constant factor.”
The Theta notation allows us to highlight growth rates and suppress distracting
factors and low-order terms. For example, if the running time of an algorithm is

T .n/ D 10n3 20n2 C 1;

then we can more simply write

T .n/ D ‚.n3 /:

In this case, we would say that T is of order n3 or that T .n/ grows cubically, which
is often the main thing we really want to know. Another such example is

.2:7x 113 C x 9 86/4
2 3x 7
C p 1:083x D ‚.3x /:
x

Just knowing that the running time of an algorithm is ‚.n3 /, for example, is
useful, because if n doubles we can predict that the running time will by and large8
increase by a factor of at most 8 for large n. In this way, Theta notation preserves in-
formation about the scalability of an algorithm or system. Scalability is, of course,
a big issue in the design of algorithms and systems.
7 It is even conceivable that there is an O.n2 / matrix multiplication procedure, but none is known.
8 Since ‚.n3 / only implies that the running time T .n/ is between cn3 and d n3 for constants

0 < c < d , the time T .2n/ could regularly exceed T .n/ by a factor as large as 8d=c. The factor is
sure to be close to 8 for all large n only if T .n/ n3 .
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584 Chapter 14 Sums and Asymptotics

14.7.4 Pitfalls with Asymptotic Notation
There is a long list of ways to make mistakes with asymptotic notation. This section
presents some of the ways that big O notation can lead to trouble. With minimal
effort, you can cause just as much chaos with the other symbols.

The Exponential Fiasco
Sometimes relationships involving big O are not so obvious. For example, one
might guess that 4x D O.2x / since 4 is only a constant factor larger than 2. This
reasoning is incorrect, however; 4x actually grows as the square of 2x .

Constant Confusion
Every constant is O.1/. For example, 17 D O.1/. This is true because if we let
f .x/ D 17 and g.x/ D 1, then there exists a c > 0 and an x0 such that jf .x/j
cg.x/. In particular, we could choose c = 17 and x0 D 1, since j17j 17 1 for all
x 1. We can construct a false theorem that exploits this fact.

False Theorem 14.7.14.
n
X
i D O.n/
i D1

Bogus proof. Define f .n/ D niD1 i D 1 C 2 C 3 C C n. Since we have shown
P
that every constant i is O.1/, f .n/ D O.1/ C O.1/ C C O.1/ D O.n/.
Pn
Of course in reality i D1 i D n.n C 1/=2 ¤ O.n/.
The error stems from confusion over what is meant in the statement i D O.1/.
For any constant i 2 N it is true that i D O.1/. More precisely, if f is any constant
function, then f D O.1/. But in this False Theorem, i is not constant—it ranges
over a set of values 0; 1; : : : ; n that depends on n.
And anyway, we should not be adding O.1/’s as though they were numbers. We
never even defined what O.g/ means by itself; it should only be used in the context
“f D O.g/” to describe a relation between functions f and g.

Equality Blunder
The notation f D O.g/ is too firmly entrenched to avoid, but the use of “=” is
regrettable. For example, if f D O.g/, it seems quite reasonable to write O.g/ D
f . But doing so might tempt us to the following blunder: because 2n D O.n/, we
can say O.n/ D 2n. But n D O.n/, so we conclude that n D O.n/ D 2n, and
therefore n D 2n. To avoid such nonsense, we will never write “O.f / D g.”
“mcs” — 2017/3/10 — 22:22 — page 585 — #593

14.7. Asymptotic Notation 585

Similarly, you will often see statements like

1
Hn D ln.n/ C CO
n
or p n n
nŠ D .1 C o.1// 2 n
e
In such cases, the true meaning is

Hn D ln.n/ C C f .n/

for some f .n/ where f .n/ D O.1=n/, and
p n n
nŠ D .1 C g.n// 2 n
e
where g.n/ D o.1/. These last transgressions are OK as long as you (and your
reader) know what you mean.

Operator Application Blunder
It’s tempting to assume that familiar operations preserve asymptotic relations, but
it ain’t necessarily so. For example, f g in general does not even imply that
3f D ‚ .3g /. On the other hand, some operations preserve and even strengthen
asymptotic relations, for example,

f D ‚.g/ IMPLIES ln f ln g:

See Problem 14.25.

14.7.5 Omega (Optional)
Sometimes people incorrectly use Big Oh in the context of a lower bound. For
example, they might say, “The running time T .n/ is at least O.n2 /.” This is another
blunder! Big Oh can only be used for upper bounds. The proper way to express the
lower bound would be
n2 D O.T .n//:
The lower bound can also be described with another special notation “big Omega.”
Definition 14.7.15. Given functions f; g W R ! R with f nonnegative, define

f D .g/

to mean
g D O.f /:
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586 Chapter 14 Sums and Asymptotics

p
For example, x 2 D .x/, 2x D .x 2 / and x=100 D .100x C x/.
So if the running time of your algorithm on inputs of size n is T .n/, and you
want to say it is at least quadratic, say

T .n/ D .n2 /:

There is a similar “little omega” notation for lower bounds corresponding to little
o:

Definition 14.7.16. For functions f; g W R ! R with f nonnegative, define

f D !.g/

to mean
g D o.f /:
p
For example, x 1:5 D !.x/ and x D !.ln2 .x//.
The little omega symbol is not as widely used as the other asymptotic symbols
we defined.

Problems for Section 14.1
Class Problems
Problem 14.1.
We begin with two large glasses. The first glass contains a pint of water, and the
second contains a pint of wine. We pour 1/3 of a pint from the first glass into the
second, stir up the wine/water mixture in the second glass, and then pour 1/3 of
a pint of the mix back into the first glass and repeat this pouring back-and-forth
process a total of n times.
(a) Describe a closed-form formula for the amount of wine in the first glass after
n back-and-forth pourings.

(b) What is the limit of the amount of wine in each glass as n approaches infinity?

Problem 14.2.
“mcs” — 2017/3/10 — 22:22 — page 587 — #595

14.7. Asymptotic Notation 587

You’ve seen this neat trick for evaluating a geometric sum:
S D 1 C z C z2 C : : : C zn
zS D z C z 2 C : : : C z n C z nC1
S zS D 1 z nC1
1 z nC1
SD (where z ¤ 1/
1 z
Use the same approach to find a closed-form expression for this sum:
T D 1z C 2z 2 C 3z 3 C : : : C nz n

Problem 14.3.
Sammy the Shark is a financial service provider who offers loans on the following
terms.

Sammy loans a client m dollars in the morning. This puts the client m dollars
in debt to Sammy.
Each evening, Sammy first charges a service fee which increases the client’s
debt by f dollars, and then Sammy charges interest, which multiplies the
debt by a factor of p. For example, Sammy might charge a “modest” ten
cent service fee and 1% interest rate per day, and then f would be 0:1 and p
would be 1:01.

(a) What is the client’s debt at the end of the first day?

(b) What is the client’s debt at the end of the second day?

(d) If you borrowed $10 from Sammy for a year, how much would you owe him?

Homework Problems
Problem 14.4.
Is a Harvard degree really worth more than an MIT degree? Let us say that a person
with a Harvard degree starts with $40,000 and gets a $20,000 raise every year after
graduation, whereas a person with an MIT degree starts with $30,000, but gets a
20% raise every year. Assume inflation is a fixed 8% every year. That is, $1.08 a
year from now is worth $1.00 today.
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588 Chapter 14 Sums and Asymptotics

(a) How much is a Harvard degree worth today if the holder will work for n years
following graduation?

(b) How much is an MIT degree worth in this case?

Problem 14.5.
Suppose you deposit $100 into your MIT Credit Union account today, $99 in one
month from now, $98 in two months from now, and so on. Given that the interest
rate is constantly 0.3% per month, how long will it take to save $5,000?

Problems for Section 14.2
Class Problems
Problem 14.6.
Find a closed form for each of the following sums:
(a)
n
X 1 1
:
i C 2012 i C 2013
i D1

(b) Assuming the following sum equals a polynomial in n, find the polynomial.
Then verify by induction that the sum equals the polynomial you find.
n
X
i3
i D1

Problems for Section 14.3
Practice Problems
Problem 14.7.
Let
5
X p
S WWD 3n :
nD1
“mcs” — 2017/3/10 — 22:22 — page 589 — #597

14.7. Asymptotic Notation 589

Using the Integral Method of Section 14.3, we can find integers a, b, c, d and a
real number e such that
Z b Z d
e
x dx S x e dx
a c

What are appropriate values for a; : : : ; e?

Class Problems
Problem 14.8.
Let f W R ! R be a continuous, weakly increasing function. Say that f grows
slowly when Z n
f .n/ D o f .x/ dx :
1
(a) Prove that the function fa .n/ WWD a
n grows slowly for any a > 0.

(b) Prove that the function e n does not grow slowly.

Exam Problems
Problem 14.9.
Assume n is an integer larger than 1. Circle all the correct inequalities below.
Explanations are not required, but partial credit for wrong answers will not be
given without them. Hint: You may find the graphs in Figure 14.9 helpful.
n
X Z n
ln.i C 1/ ln 2 C ln.x C 1/dx
i D1 1

n
X Z n
ln.i C 1/ ln.x C 2/dx
i D1 0

n n
1 1
X Z
dx
i 0 xC1
i D1
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590 Chapter 14 Sums and Asymptotics

2.5

y = ln(x+1)
2

1.5 y = ln(x+2)

0.5

0
0 1 2 3 4 5 6 7 8

0.8

0.6 y = 1/x

0.4 y = 1/(x+1)

0.2

0
0 1 2 3 4 5 6 7 8

Figure 14.9 Integral bounds for two sums
“mcs” — 2017/3/10 — 22:22 — page 591 — #599

14.7. Asymptotic Notation 591

Homework Problems
Problem 14.10.
Let f W RC ! RC be a weakly decreasing function. Define
n
X
S WWD f .i /
i D1

and Z n
I WWD f .x/ dx:
1
Prove that
I C f .n/ S I C f .1/:
(Proof by very clear picture is OK.)

Problem 14.11.
Use integration to find upper and lower bounds that differ by at most 0.1 for the
following sum. (You may need to add the first few terms explicitly and then use
integrals to bound the sum of the remaining terms.)
1
X 1
.2i C 1/2
iD1

Problems for Section 14.4
Class Problems
Problem 14.12.
An explorer is trying to reach the Holy Grail, which she believes is located in a
desert shrine d days walk from the nearest oasis. In the desert heat, the explorer
must drink continuously. She can carry at most 1 gallon of water, which is enough
for 1 day. However, she is free to make multiple trips carrying up to a gallon each
time to create water caches out in the desert.
For example, if the shrine were 2=3 of a day’s walk into the desert, then she could
recover the Holy Grail after two days using the following strategy. She leaves the
oasis with 1 gallon of water, travels 1=3 day into the desert, caches 1=3 gallon, and
then walks back to the oasis—arriving just as her water supply runs out. Then she
picks up another gallon of water at the oasis, walks 1=3 day into the desert, tops off
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592 Chapter 14 Sums and Asymptotics

her water supply by taking the 1=3 gallon in her cache, walks the remaining 1=3
day to the shrine, grabs the Holy Grail, and then walks for 2=3 of a day back to the
oasis—again arriving with no water to spare.
But what if the shrine were located farther away?
(a) What is the most distant point that the explorer can reach and then return to
the oasis, with no water precached in the desert, if she takes a total of only 1 gallon
from the oasis?

(b) What is the most distant point the explorer can reach and still return to the
oasis if she takes a total of only 2 gallons from the oasis? No proof is required; just
do the best you can.

(c) The explorer will travel using a recursive strategy to go far into the desert and
back, drawing a total of n gallons of water from the oasis. Her strategy is to build
up a cache of n 1 gallons, plus enough to get home, a certain fraction of a day’s
distance into the desert. On the last delivery to the cache, instead of returning home,
she proceeds recursively with her n 1 gallon strategy to go farther into the desert
and return to the cache. At this point, the cache has just enough water left to get
her home.
Prove that with n gallons of water, this strategy will get her Hn =2 days into the
desert and back, where Hn is the nth Harmonic number:
1 1 1 1
Hn WWD C C C C :
1 2 3 n
Conclude that she can reach the shrine, however far it is from the oasis.

(d) Suppose that the shrine is d D 10 days walk into the desert. Use the asymp-
totic approximation Hn ln n to show that it will take more than a million years
for the explorer to recover the Holy Grail.

Problem 14.13.
There is a number a such that 1 p
P
i D1 i converges iff p < a. What is the value of
a?
Hint: Find a value for a you think that works, then apply the integral bound.

Homework Problems
Problem 14.14.
There is a bug on the edge of a 1-meter rug. The bug wants to cross to the other
side of the rug. It crawls at 1 cm per second. However, at the end of each second,
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14.7. Asymptotic Notation 593

a malicious first-grader named Mildred Anderson stretches the rug by 1 meter. As-
sume that her action is instantaneous and the rug stretches uniformly. Thus, here’s
what happens in the first few seconds:

The bug walks 1 cm in the first second, so 99 cm remain ahead.

Mildred stretches the rug by 1 meter, which doubles its length. So now there
are 2 cm behind the bug and 198 cm ahead.

The bug walks another 1 cm in the next second, leaving 3 cm behind and 197
cm ahead.

Then Mildred strikes, stretching the rug from 2 meters to 3 meters. So there
are now 3 .3=2/ D 4:5 cm behind the bug and 197 .3=2/ D 295:5 cm
ahead.

The bug walks another 1 cm in the third second, and so on.

Your job is to determine this poor bug’s fate.
(a) During second i , what fraction of the rug does the bug cross?

(b) Over the first n seconds, what fraction of the rug does the bug cross altogether?
Express your answer in terms of the Harmonic number Hn .

(c) The known universe is thought to be about 3 1010 light years in diameter.
How many universe diameters must the bug travel to get to the end of the rug?
(This distance is NOT the inflated distance caused by the stretching but only the
actual walking done by the bug).

Exam Problems
Problem 14.15.
Show that
1
X
ip
iD1

converges to a finite value iff p < 1.
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594 Chapter 14 Sums and Asymptotics

Problems for Section 14.7
Practice Problems
Problem 14.16.
Find the least nonnegative integer n such that f .x/ is O.x n / when f is defined by
each of the expressions below.
(a) 2x 3 C .log x/x 2

(b) 2x 2 + .log x/x 3

(d) .0:1/x

(e) .x 4 C x 2 C 1/=.x 3 C 1/

(f) .x 4 C 5 log x/=.x 4 C 1/
2/
(g) 2.3 log2 x

Problem 14.17.
Let f .n/ D n3 . For each function g.n/ in the table below, indicate which of the
indicated asymptotic relations hold.

g.n/ f D O.g/ f D o.g/ g D O.f / g D o.f /
6 5n 4n C 3n 2 3

n3 log n
.sin . n=2/ C 2/ n3
nsin. n=2/C2
log nŠ
e 0:2n 100n3

Problem 14.18.
Circle each of the true statements below.
Explanations are not required, but partial credit for wrong answers will not be
given without them.
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14.7. Asymptotic Notation 595

n2 n2 C n

3n D O 2n

nsin.n=2/C1 D o n2

3n3

nD‚
.n C 1/.n 1/

Problem 14.19.
Show that
ln.n2 Š/ D ‚.n2 ln n/
Hint: Stirling’s formula for .n2 /Š.

Problem 14.20.
The quantity

.2n/Š
(14.29)
22n .nŠ/2
will come up later in the course (it is the probability that in 22n flips of a fair coin,
1
exactly n will be Heads). Show that it is asymptotically equal to p .
n

Problem 14.21.
Suppose let f and g be real-valued functions.
(a) Give an example of f; g such that

lim sup fg < lim sup f lim sup g;

and all the lim sup’s are finite.

(b) Give an example of f; g such that

lim sup fg > lim sup f lim sup g:

and all the lim sup’s are finite.
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596 Chapter 14 Sums and Asymptotics

Homework Problems
Problem 14.22. (a) Prove that log x < x for all x > 1 (requires elementary calcu-
lus).

(b) Prove that the relation R on functions such that f R g iff g D o.f / is a strict
partial order.

Problem 14.23.
Indicate which of the following holds for each pair of functions .f .n/; g.n// in
the table below. Assume k 1, > 0, and c > 1 are constants. Pick the four
table entries you consider to be the most challenging or interesting and justify your
answers to these.

f .n/ g.n/ f D O.g/ f D o.g/ g D O.f / g D o.f / f D ‚.g/ f g
2n 2n=2
p
n nsin.n=2/
log.nŠ/ log.nn /
nk cn
logk n n

Problem 14.24.
Arrange the following functions in a sequence f1 , f2 , ... f24 so that fi D O.fi C1 /.
Additionally, if fi D ‚.fi C1 /, indicate that too:

1. n log n

2. 2100 n

3. n 1

4. n 1=2

5. .log n/=n
n
6. 64

7. nŠ
“mcs” — 2017/3/10 — 22:22 — page 597 — #605

14.7. Asymptotic Notation 597

100
8. 22
n
9. 22

10. 2n

11. 3n

12. n2n

13. 2nC1

14. 2n

15. 3n

16. log .nŠ/

17. log2 n

18. log10 n
p
19. 2:1 n

20. 22n

21. 4n

22. n64

23. n65

24. nn

Problem 14.25.
Let f , g be nonnegative real-valued functions such that limx!1 f .x/ D 1 and
f g.
(a) Give an example of f; g such that NOT.2f 2g /.

(b) Prove that log f log g.

log.nŠ/ n log n
“mcs” — 2017/3/10 — 22:22 — page 598 — #606

598 Chapter 14 Sums and Asymptotics

Problem 14.26.
Determine which of these choices
‚.n/; ‚.n2 log n/; ‚.n2 /; ‚.1/; ‚.2n /; ‚.2n ln n /; none of these
describes each function’s asymptotic behavior. Full proofs are not required, but
briefly explain your answers.
(a)
n C ln n C .ln n/2

(b)
n2 C 2n 3
n2 7
(c)
n
X
22iC1
i D0

(d)
ln.n2 Š/

(e)
n
X 1
k 1
2k
kD1

Problem 14.27. (a) Either prove or disprove each of the following statements.
nŠ D O..n C 1/Š/
.n C 1/Š D O.nŠ/
nŠ D ‚..n C 1/Š/
nŠ D o..n C 1/Š/
.n C 1/Š D o.nŠ/
nCe
(b) Show that n3 D o.nŠ/.

Problem 14.28.
Prove that
n
X
k 6 D ‚.n7 /:
kD1
“mcs” — 2017/3/10 — 22:22 — page 599 — #607

14.7. Asymptotic Notation 599

Class Problems
Problem 14.29.
Give an elementary proof (without appealing to Stirling’s formula) that log.nŠ/ D
‚.n log n/.

Problem 14.30.
Suppose f; g W NC ! NC and f g.
(a) Prove that 2f 2g.

(b) Prove that f 2 g 2 .

Problem 14.31.
Recall that for functions f; g on N, f D O.g/ iff

9c 2 N 9n0 2 N 8n n0 c g.n/ jf .n/j : (14.30)

For each pair of functions below, determine whether f D O.g/ and whether
g D O.f /. In cases where one function is O() of the other, indicate the smallest
nonnegative integer c and for that smallest c, the smallest corresponding nonnega-
tive integer n0 ensuring that condition (14.30) applies.
(a) f .n/ D n2 ; g.n/ D 3n.
f D O.g/ YES NO If YES, c D , n0 =
g D O.f / YES NO If YES, c D , n0 =

(b) f .n/ D .3n 7/=.n C 4/; g.n/ D 4
f D O.g/ YES NO If YES, c D , n0 =
g D O.f / YES NO If YES, c D , n0 =

Problem 14.32.
“mcs” — 2017/3/10 — 22:22 — page 600 — #608

600 Chapter 14 Sums and Asymptotics

False Claim.
2n D O.1/: (14.31)
Explain why the claim is false. Then identify and explain the mistake in the
following bogus proof.

Bogus proof. The proof is by induction on n where the induction hypothesis P .n/
is the assertion (14.31).
base case: P .0/ holds trivially.
inductive step: We may assume P .n/, so there is a constant c > 0 such that
2n c 1. Therefore,
2nC1 D 2 2n .2c/ 1;
which implies that 2nC1 D O.1/. That is, P .n C 1/ holds, which completes the
proof of the inductive step.
We conclude by induction that 2n D O.1/ for all n. That is, the exponential
function is bounded by a constant.

Problem 14.33. (a) Prove that the relation R on functions such that f R g iff
f D o.g/ is a strict partial order.

(b) Describe two functions f; g that are incomparable under big Oh:

f ¤ O.g/ AND g ¤ O.f /:

Conclude that R is not a linear order. How about three such functions?

Exam Problems
Problem 14.34.
Give an example of a pair of strictly increasing total functions, f W NC ! NC and
g W NC ! NC , that satisfy f g but not 3f D O .3g /.

Problem 14.35.
Let f; g be real-valued functions such that f D ‚.g/ and limx!1 f .x/ D 1.
Prove that
ln f ln g:
“mcs” — 2017/3/10 — 22:22 — page 601 — #609

14.7. Asymptotic Notation 601

Problem 14.36. (a) Show that

.an/b=n 1:

where a; b are positive constants and denotes asymptotic equality. Hint: an D
a2log2 n .

(b) You may assume that if f .n/ 1 and g.n/ 1 for all n, then f g !
1 1
f n g n . Show that p
n
nŠ D ‚.n/:

Problem 14.37.

(a) Define a function f .n/ such that f D ‚.n2 / and NOT.f n2 /.

f .n/ D

(b) Define a function g.n/ such that g D O.n2 /, g ¤ ‚.n2 /, g ¤ o.n2 /, and
n D O.g/.

g.n/ D

Problem 14.38. (a) Show that

.an/b=n 1:

where a; b are positive constants and denotes asymptotic equality. Hint: an D
a2log2 n .

(b) Show that p
n
nŠ D ‚.n/:

Problem 14.39.

(a) Indicate which of the following asymptotic relations below on the set of non-
negative real-valued functions are equivalence relations (E), strict partial orders (S),
weak partial orders (W), or none of the above (N).
“mcs” — 2017/3/10 — 22:22 — page 602 — #610

602 Chapter 14 Sums and Asymptotics

f g, the “asymptotically equal” relation.
f D o.g/, the “little Oh” relation.
f D O.g/, the “big Oh” relation.
f D ‚.g/, the “Theta” relation.
f D O.g/ AND NOT.g D O.f //.

(b) Indicate the implications among the assertions in part (a). For example,

f D o.g/ IMPLIES f D O.g/:

Problem 14.40.
Recall that if f and g are nonnegative real-valued functions on ZC , then f D O.g/
iff there exist c; n0 2 ZC such that

8n n0 : f .n/ cg.n/:

For each pair of functions f and g below, indicate the smallest c 2 ZC , and
for that smallest c, the smallest corresponding n0 2 ZC , that would establish
f D O.g/ by the definition given above. If there is no such c, write 1.
(a) f .n/ D 21 ln n2 ; g.n/ D n. cD , n0 =

(b) f .n/ D n; g.n/ D n ln n. cD , n0 =

.n 1/
(d) f .n/ D 3 sin C 2; g.n/ D 0:2. cD , n0 =
100

Problem 14.41.
Let f; g be positive real-valued functions on finite, connected, simple graphs. We
will extend the O./ notation to such graph functions as follows: f D O.g/ iff
there is a constant c > 0 such that

f .G/ c g.G/ for all connected simple graphs G with more than one vertex:

For each of the following assertions, state whether it is True or False and briefly
explain your answer. You are not expected to offer a careful proof or detailed
counterexample.
Reminder: V .G/ is the set of vertices and E.G/ is the set of edges of G, and G
is connected.
“mcs” — 2017/3/10 — 22:22 — page 603 — #611

14.7. Asymptotic Notation 603

(a) jV .G/j D O.jE.G/j/.

(b) jE.G/j D O.jV .G/j/.

(d) .G/ D O.jV .G/j/.
“mcs” — 2017/3/10 — 22:22 — page 604 — #612
“mcs” — 2017/3/10 — 22:22 — page 605 — #613

15 Cardinality Rules

15.1 Counting One Thing by Counting Another
How do you count the number of people in a crowded room? You could count
heads, since for each person there is exactly one head. Alternatively, you could
count ears and divide by two. Of course, you might have to adjust the calculation if
someone lost an ear in a pirate raid or someone was born with three ears. The point
here is that you can often count one thing by counting another, though some fudg-
ing may be required. This is a central theme of counting, from the easiest problems
to the hardest. In fact, we’ve already seen this technique used in Theorem 4.5.5,
where the number of subsets of an n-element set was proved to be the same as the
number of length-n bit-strings, by describing a bijection between the subsets and
the bit-strings.
The most direct way to count one thing by counting another is to find a bijection
between them, since if there is a bijection between two sets, then the sets have the
same size. This important fact is commonly known as the Bijection Rule. We’ve
already seen it as the Mapping Rules bijective case (4.7).

15.1.1 The Bijection Rule
The Bijection Rule acts as a magnifier of counting ability; if you figure out the size
of one set, then you can immediately determine the sizes of many other sets via
bijections. For example, let’s look at the two sets mentioned at the beginning of
Part III:
A D all ways to select a dozen donuts when five varieties are available
B D all 16-bit sequences with exactly 4 ones
An example of an element of set A is:
00
„ƒ‚… „ƒ‚… 0„ 0 ƒ‚
0 0 0 …0 00
„ƒ‚… 00
„ƒ‚…
chocolate lemon-filled sugar glazed plain

Here, we’ve depicted each donut with a 0 and left a gap between the different
varieties. Thus, the selection above contains two chocolate donuts, no lemon-filled,
six sugar, two glazed, and two plain. Now let’s put a 1 into each of the four gaps:
00
„ƒ‚… 1 „ƒ‚… 1 0 0 ƒ‚
„ 0 0 0 …0 1 00
„ƒ‚… 1 00
„ƒ‚…
chocolate lemon-filled sugar glazed plain
“mcs” — 2017/3/10 — 22:22 — page 606 — #614

606 Chapter 15 Cardinality Rules

and close up the gaps:
0011000000100100 :
We’ve just formed a 16-bit number with exactly 4 ones—an element of B!
This example suggests a bijection from set A to set B: map a dozen donuts
consisting of:

c chocolate, l lemon-filled, s sugar, g glazed, and p plain

to the sequence:

„0 ƒ‚
: : : 0… 1 „0 ƒ‚
: : : 0… 1 „0 ƒ‚
: : : 0… 1 „0 ƒ‚
: : : 0… 1 „0 ƒ‚
: : : 0…
c l s g p

The resulting sequence always has 16 bits and exactly 4 ones, and thus is an
element of B. Moreover, the mapping is a bijection: every such bit sequence comes
from exactly one order of a dozen donuts. Therefore, jAj D jBj by the Bijection
Rule. More generally,

Lemma 15.1.1. The number of ways to select n donuts when k flavors are available
is the same as the number of binary sequences with exactly n zeroes and k 1 ones.

This example demonstrates the power of the bijection rule. We managed to prove
that two very different sets are actually the same size—even though we don’t know
exactly how big either one is. But as soon as we figure out the size of one set, we’ll
immediately know the size of the other.
This particular bijection might seem frighteningly ingenious if you’ve not seen
it before. But you’ll use essentially this same argument over and over, and soon
you’ll consider it routine.

15.2 Counting Sequences
The Bijection Rule lets us count one thing by counting another. This suggests a
general strategy: get really good at counting just a few things, then use bijections
to count everything else! This is the strategy we’ll follow. In particular, we’ll get
really good at counting sequences. When we want to determine the size of some
other set T , we’ll find a bijection from T to a set of sequences S . Then we’ll
use our super-ninja sequence-counting skills to determine jSj, which immediately
gives us jT j. We’ll need to hone this idea somewhat as we go along, but that’s
pretty much it!
“mcs” — 2017/3/10 — 22:22 — page 607 — #615

15.2. Counting Sequences 607

15.2.1 The Product Rule
The Product Rule gives the size of a product of sets. Recall that if P1 ; P2 ; : : : ; Pn
are sets, then
P1 P2 Pn
is the set of all sequences whose first term is drawn from P1 , second term is drawn
from P2 and so forth.
Rule 15.2.1 (Product Rule). If P1 ; P2 ; : : : Pn are finite sets, then:
jP1 P2 Pn j D jP1 j jP2 j jPn j
For example, suppose a daily diet consists of a breakfast selected from set B, a
lunch from set L, and a dinner from set D where:
B D fpancakes; bacon and eggs; bagel; Doritosg
L D fburger and fries; garden salad; Doritosg
D D fmacaroni; pizza; frozen burrito; pasta; Doritosg
Then B LD is the set of all possible daily diets. Here are some sample elements:
.pancakes; burger and fries; pizza/
.bacon and eggs; garden salad; pasta/
.Doritos; Doritos; frozen burrito/
The Product Rule tells us how many different daily diets are possible:
jB L Dj D jBj jLj jDj
D435
D 60:

15.2.2 Subsets of an n-element Set
The fact that there are 2n subsets of an n-element set was proved in Theorem 4.5.5
by setting up a bijection between the subsets and the length-n bit-strings. So the
original problem about subsets was tranformed into a question about sequences—
exactly according to plan! Now we can fill in the missing explanation of why there
are 2n length-n bit-strings: we can write the set of all n-bit sequences as a product
of sets:
f0; 1gn WWD f0; 1g f0; 1g f0; 1g :
„ ƒ‚ …
n terms
Then Product Rule gives the answer:
jf0; 1gn j D jf0; 1gjn D 2n :
“mcs” — 2017/3/10 — 22:22 — page 608 — #616

608 Chapter 15 Cardinality Rules

15.2.3 The Sum Rule
Bart allocates his little sister Lisa a quota of 20 crabby days, 40 irritable days,
and 60 generally surly days. On how many days can Lisa be out-of-sorts one way
or another? Let set C be her crabby days, I be her irritable days, and S be the
generally surly. In these terms, the answer to the question is jC [ I [ S j. Now
assuming that she is permitted at most one bad quality each day, the size of this
union of sets is given by the Sum Rule:
Rule 15.2.2 (Sum Rule). If A1 ; A2 ; : : : ; An are disjoint sets, then:

jA1 [ A2 [ : : : [ An j D jA1 j C jA2 j C : : : C jAn j

Thus, according to Bart’s budget, Lisa can be out-of-sorts for:

jC [ I [ S j D jC j C jI j C jS j
D 20 C 40 C 60
D 120 days

Notice that the Sum Rule holds only for a union of disjoint sets. Finding the size
of a union of overlapping sets is a more complicated problem that we’ll take up in
Section 15.9.

15.2.4 Counting Passwords
Few counting problems can be solved with a single rule. More often, a solution is
a flurry of sums, products, bijections, and other methods.
For solving problems involving passwords, telephone numbers, and license plates,
the sum and product rules are useful together. For example, on a certain computer
system, a valid password is a sequence of between six and eight symbols. The first
symbol must be a letter (which can be lowercase or uppercase), and the remain-
ing symbols must be either letters or digits. How many different passwords are
possible?
Let’s define two sets, corresponding to valid symbols in the first and subsequent
positions in the password.

F D fa; b; : : : ; z; A; B; : : : ; Zg
S D fa; b; : : : ; z; A; B; : : : ; Z; 0; 1; : : : ; 9g

In these terms, the set of all possible passwords is:1

.F S 5 / [ .F S 6 / [ .F S 7 /
1 The notation S 5 means S S S S S.
“mcs” — 2017/3/10 — 22:22 — page 609 — #617

15.3. The Generalized Product Rule 609

Thus, the length-six passwords are in the set F S 5 , the length-seven passwords
are in F S 6 , and the length-eight passwords are in F S 7 . Since these sets
are disjoint, we can apply the Sum Rule and count the total number of possible
passwords as follows:

j.F S 5 / [ .F S 6 / [ .F S 7 /j
D jF S 5 j C jF S 6 j C jF S 7 j Sum Rule
5 6 7
D jF j jSj C jF j jS j C jF j jS j Product Rule
D 52 625 C 52 626 C 52 627
1:8 1014 different passwords:

15.3 The Generalized Product Rule
In how many ways can, say, a Nobel prize, a Japan prize, and a Pulitzer prize be
awarded to n people? This is easy to answer using our strategy of translating the
problem about awards into a problem about sequences. Let P be the set of n people
taking the course. Then there is a bijection from ways of awarding the three prizes
to the set P 3 WWD P P P . In particular, the assignment:

“Barack wins a Nobel, George wins a Japan, and Bill wins a Pulitzer prize”

maps to the sequence .Barack; George; Bill/. By the Product Rule, we have jP 3 j D
jP j3 D n3 , so there are n3 ways to award the prizes to a class of n people. Notice
that P 3 includes triples like .Barack; Bill; Barack/ where one person wins more
than one prize.
But what if the three prizes must be awarded to different students? As before,
we could map the assignment to the triple .Bill; George; Barack/ 2 P 3 . But this
function is no longer a bijection. For example, no valid assignment maps to the
triple .Barack; Bill; Barack/ because now we’re not allowing Barack to receive two
prizes. However, there is a bijection from prize assignments to the set:

S D f.x; y; z/ 2 P 3 j x, y and z are different peopleg

This reduces the original problem to a problem of counting sequences. Unfortu-
nately, the Product Rule does not apply directly to counting sequences of this type
because the entries depend on one another; in particular, they must all be different.
However, a slightly sharper tool does the trick.
“mcs” — 2017/3/10 — 22:22 — page 610 — #618

610 Chapter 15 Cardinality Rules

Prizes for truly exceptional Coursework
Given everyone’s hard work on this material, the instructors considered award-
ing some prizes for truly exceptional coursework. Here are three possible prize
categories:

Best Administrative Critique We asserted that the quiz was closed-book. On
the cover page, one strong candidate for this award wrote, “There is no
book.”

Awkward Question Award “Okay, the left sock, right sock, and pants are in
an antichain, but how—even with assistance—could I put on all three at
once?”

Best Collaboration Statement Inspired by a student who wrote “I worked alone”
on Quiz 1.

Rule 15.3.1 (Generalized Product Rule). Let S be a set of length-k sequences. If
there are:

n1 possible first entries,

n2 possible second entries for each first entry,
::
:

nk possible kth entries for each sequence of first k 1 entries,

then:
jS j D n1 n2 n3 nk

In the awards example, S consists of sequences .x; y; z/. There are n ways to
choose x, the recipient of prize #1. For each of these, there are n 1 ways to choose
y, the recipient of prize #2, since everyone except for person x is eligible. For each
combination of x and y, there are n 2 ways to choose z, the recipient of prize #3,
because everyone except for x and y is eligible. Thus, according to the Generalized
Product Rule, there are
jS j D n .n 1/ .n 2/
ways to award the 3 prizes to different people.
“mcs” — 2017/3/10 — 22:22 — page 611 — #619

15.3. The Generalized Product Rule 611

15.3.1 Defective Dollar Bills
A dollar bill is defective if some digit appears more than once in the 8-digit serial
number. If you check your wallet, you’ll be sad to discover that defective bills
are all-too-common. In fact, how common are nondefective bills? Assuming that
the digit portions of serial numbers all occur equally often, we could answer this
question by computing

jfserial #’s with all digits differentgj
fraction of nondefective bills D : (15.1)
jfserial numbersgj
Let’s first consider the denominator. Here there are no restrictions; there are 10
possible first digits, 10 possible second digits, 10 third digits, and so on. Thus, the
total number of 8-digit serial numbers is 108 by the Product Rule.
Next, let’s turn to the numerator. Now we’re not permitted to use any digit twice.
So there are still 10 possible first digits, but only 9 possible second digits, 8 possible
third digits, and so forth. Thus, by the Generalized Product Rule, there are
10Š
10 9 8 7 6 5 4 3 D D 1;814;400
2
serial numbers with all digits different. Plugging these results into Equation 15.1,
we find:
1;814;400
fraction of nondefective bills D D 1:8144%
100;000;000

15.3.2 A Chess Problem
In how many different ways can we place a pawn (P ), a knight (N ), and a bishop
(B) on a chessboard so that no two pieces share a row or a column? A valid con-
figuration is shown in Figure 15.1(a), and an invalid configuration is shown in Fig-
ure 15.1(b).
First, we map this problem about chess pieces to a question about sequences.
There is a bijection from configurations to sequences

.rP ; cP ; rN ; cN ; rB ; cB /

where rP , rN and rB are distinct rows and cP , cN and cB are distinct columns.
In particular, rP is the pawn’s row cP is the pawn’s column rN is the knight’s
row, etc. Now we can count the number of such sequences using the Generalized
Product Rule:

rP is one of 8 rows
“mcs” — 2017/3/10 — 22:22 — page 612 — #620

612 Chapter 15 Cardinality Rules

8
0Z0Z0Z0Z 8
0Z0Z0Z0Z
7
Z0Z0m0Z0 7
Z0Z0Z0Z0
6
0Z0Z0Z0Z 6
0Z0ZpZ0Z
5
Z0Z0Z0Z0 5
Z0Z0Z0Z0
4
0a0Z0Z0Z 4
0Z0Z0Z0Z
3
Z0Z0Z0Z0 3
Z0a0ZnZ0
2
0Z0Z0o0Z 2
0Z0Z0Z0Z
1
Z0Z0Z0Z0 1
Z0Z0Z0Z0
a b c d e f g h a b c d e f g h

(a) valid (b) invalid

Figure 15.1 Two ways of placing a pawn (p), a knight (N), and a bishop (B) on
a chessboard. The configuration shown in (b) is invalid because the bishop and the
knight are in the same row.

cP is one of 8 columns
rN is one of 7 rows (any one but rP )
cN is one of 7 columns (any one but cP )
rB is one of 6 rows (any one but rP or rN )
cB is one of 6 columns (any one but cP or cN )

Thus, the total number of configurations is .8 7 6/2 .

15.3.3 Permutations
A permutation of a set S is a sequence that contains every element of S exactly
once. For example, here are all the permutations of the set fa; b; cg:

.a; b; c/ .a; c; b/ .b; a; c/
.b; c; a/ .c; a; b/ .c; b; a/

How many permutations of an n-element set are there? Well, there are n choices
for the first element. For each of these, there are n 1 remaining choices for the
second element. For every combination of the first two elements, there are n 2
ways to choose the third element, and so forth. Thus, there are a total of

n .n 1/ .n 2/ 3 2 1 D nŠ

permutations of an n-element set. In particular, this formula says that there are
“mcs” — 2017/3/10 — 22:22 — page 613 — #621

15.4. The Division Rule 613

3Š D 6 permutations of the 3-element set fa; b; cg, which is the number we found
above.
Permutations will come up again in this course approximately 1.6 bazillion times.
In fact, permutations are the reason why factorial comes up so often and why we
taught you Stirling’s approximation:
p n n
nŠ 2 n :
e

15.4 The Division Rule
Counting ears and dividing by two is a silly way to count the number of people in
a room, but this approach is representative of a powerful counting principle.
A k-to-1 function maps exactly k elements of the domain to every element of
the codomain. For example, the function mapping each ear to its owner is 2-to-1.
Similarly, the function mapping each finger to its owner is 10-to-1, and the function
mapping each finger and toe to its owner is 20-to-1. The general rule is:

Rule 15.4.1 (Division Rule). If f W A ! B is k-to-1, then jAj D k jBj.

For example, suppose A is the set of ears in the room and B is the set of people.
There is a 2-to-1 mapping from ears to people, so by the Division Rule, jAj D
2 jBj. Equivalently, jBj D jAj=2, expressing what we knew all along: the number
of people is half the number of ears. Unlikely as it may seem, many counting
problems are made much easier by initially counting every item multiple times and
then correcting the answer using the Division Rule. Let’s look at some examples.

15.4.1 Another Chess Problem
In how many different ways can you place two identical rooks on a chessboard
so that they do not share a row or column? A valid configuration is shown in
Figure 15.2(a), and an invalid configuration is shown in Figure 15.2(b).
Let A be the set of all sequences

.r1 ; c1 ; r2 ; c2 /

where r1 and r2 are distinct rows and c1 and c2 are distinct columns. Let B be the
set of all valid rook configurations. There is a natural function f from set A to set
B; in particular, f maps the sequence .r1 ; c1 ; r2 ; c2 / to a configuration with one
rook in row r1 , column c1 and the other rook in row r2 , column c2 .
“mcs” — 2017/3/10 — 22:22 — page 614 — #622

614 Chapter 15 Cardinality Rules

8
0Z0Z0Z0s 8
0Z0Z0Z0Z
7
Z0Z0Z0Z0 7
Z0Z0Z0Z0
6
0Z0Z0Z0Z 6
0Z0s0Z0Z
5
Z0Z0Z0Z0 5
Z0Z0Z0Z0
4
0Z0Z0Z0Z 4
0Z0Z0Z0Z
3
Z0Z0Z0Z0 3
Z0Z0Z0Z0
2
0Z0Z0Z0Z 2
0Z0Z0Z0Z
1
s0Z0Z0Z0 1
Z0ZrZ0Z0
a b c d e f g h a b c d e f g h

(a) valid (b) invalid

Figure 15.2 Two ways to place 2 rooks (R) on a chessboard. The configuration
in (b) is invalid because the rooks are in the same column.

But now there’s a snag. Consider the sequences:

.1; a; 8; h/ and .8; h; 1; a/

The first sequence maps to a configuration with a rook in the lower-left corner and
a rook in the upper-right corner. The second sequence maps to a configuration with
a rook in the upper-right corner and a rook in the lower-left corner. The problem is
that those are two different ways of describing the same configuration! In fact, this
arrangement is shown in Figure 15.2(a).
More generally, the function f maps exactly two sequences to every board con-
figuration; f is a 2-to-1 function. Thus, by the quotient rule, jAj D 2 jBj. Rear-
ranging terms gives:

jAj .8 7/2
jBj D D :
2 2
On the second line, we’ve computed the size of A using the General Product Rule
just as in the earlier chess problem.

15.4.2 Knights of the Round Table
In how many ways can King Arthur arrange to seat his n different knights at his
round table? A seating defines who sits where. Two seatings are considered to be
the same arrangement if each knight sits between the same two knights in both
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15.4. The Division Rule 615

seatings. An equivalent way to say this is that two seatings yield the same arrange-
ment when they yield the same sequence of knights starting at knight number 1
and going clockwise around the table. For example, the following two seatings
determine the same arrangement:

k
#1 #3 k

k4 k2 k2 k4
"! "!
k3 k1

A seating is determined by the sequence of knights going clockwise around the
table starting at the top seat. So seatings correspond to permutations of the knights,
and there are nŠ of them. For example,

#2k

.k2 ; k4 ; k1 ; k3 / ! k3 k4
"!
k1

Two seatings determine the same arrangement if they are the same when the
table is rotated so knight 1 is at the top seat. For example with n D 4, there are 4
different sequences that correspond to the seating arrangement:

.k2 ; k4 ; k1 ; k3 / #1k
.k4 ; k1 ; k3 ; k2 /
! k4 k3
.k1 ; k3 ; k2 ; k4 /
"!
.k3 ; k2 ; k4 ; k1 / k2

This mapping from seating to arrangments is actually an n-to-1 function, since all n
cyclic shifts of the sequence of knights in the seating map to the same arrangement.
Therefore, by the division rule, the number of circular seating arrangements is:

# seatings nŠ
D D .n 1/Š :
n n
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616 Chapter 15 Cardinality Rules

15.5 Counting Subsets
How many k-element subsets of an n-element set are there? This question arises
all the time in various guises:

In how many ways can I select 5 books from my collection of 100 to bring
on vacation?

How many different 13-card bridge hands can be dealt from a 52-card deck?

In how many ways can I select 5 toppings for my pizza if there are 14 avail-
able toppings?

This number comes up so often that there is a special notation for it:
!
n
WWD the number of k-element subsets of an n-element set.
k

The expression kn is read “n choose k.” Now we can immediately express the

answers to all three questions above:

I can select 5 books from 100 in 100

5 ways.

There are 52

13 different bridge hands.

There are 14

5 different 5-topping pizzas, if 14 toppings are available.

15.5.1 The Subset Rule
We can derive a simple formula for the n choose k number using the Division Rule.
We do this by mapping any permutation of an n-element set fa1 ; : : : ; an g into a k-
element subset simply by taking the first k elements of the permutation. That is,
the permutation a1 a2 : : : an will map to the set fa1 ; a2 ; : : : ; ak g.
Notice that any other permutation with the same first k elements a1 ; : : : ; ak in
any order and the same remaining elements n k elements in any order will also
map to this set. What’s more, a permutation can only map to fa1 ; a2 ; : : : ; ak g
if its first k elements are the elements a1 ; : : : ; ak in some order. Since there are
kŠ possible permutations of the first k elements and .n k/Š permutations of the
remaining elements, we conclude from the Product Rule that exactly kŠ.n k/Š
permutations of the n-element set map to the particular subset S . In other words,
the mapping from permutations to k-element subsets is kŠ.n k/Š-to-1.
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15.5. Counting Subsets 617

But we know there are nŠ permutations of an n-element set, so by the Division
Rule, we conclude that !
n
nŠ D kŠ.n k/Š
k
which proves:

Rule 15.5.1 (Subset Rule). The number of k-element subsets of an n-element set is
!
n nŠ
D :
k kŠ .n k/Š

Notice that this works even for 0-element subsets: nŠ=0ŠnŠ D 1. Here we use the
fact that 0Š is a product of 0 terms, which by convention2 equals 1.

15.5.2 Bit Sequences
How many n-bit sequences contain exactly k ones? We’ve already seen the straight-
forward bijection between subsets of an n-element set and n-bit sequences. For
example, here is a 3-element subset of fx1 ; x2 ; : : : ; x8 g and the associated 8-bit
sequence:
f x1 ; x4 ; x5 g
. 1; 0; 0; 1; 1; 0; 0; 0 /
Notice that this sequence has exactly 3 ones, each corresponding to an element
of the 3-element subset. More generally, the n-bit sequences corresponding to a
k-element subset will have exactly k ones. So by the Bijection Rule,
!
n
Corollary 15.5.2. The number of n-bit sequences with exactly k ones is .
k

Also, the bijection between selections of flavored donuts and bit sequences of
Lemma 15.1.1 now implies,

Corollary 15.5.3. The number of ways to select n donuts when k flavors are avail-
able is !
n C .k 1/
:
n
2 We don’t use it here, but a sum of zero terms equals 0.
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618 Chapter 15 Cardinality Rules

15.6 Sequences with Repetitions
15.6.1 Sequences of Subsets
Choosing a k-element subset of an n-element set is the same as splitting the set
into a pair of subsets: the first subset of size k and the second subset consisting of
the remaining n k elements. So, the Subset Rule can be understood as a rule for
counting the number of such splits into pairs of subsets.
We can generalize this to a way to count splits into more than two subsets. Let
A be an n-element set and k1 ; k2 ; : : : ; km be nonnegative integers whose sum is n.
A .k1 ; k2 ; : : : ; km /-split of A is a sequence

.A1 ; A2 ; : : : ; Am /

where the Ai are disjoint subsets of A and jAi j D ki for i D 1; : : : ; m.
To count the number of splits we take the same approach as for the Subset
Rule. Namely, we map any permutation a1 a2 : : : an of an n-element set A into
a .k1 ; k2 ; : : : ; km /-split by letting the 1st subset in the split be the first k1 elements
of the permutation, the 2nd subset of the split be the next k2 elements, . . . , and the
mth subset of the split be the final km elements of the permutation. This map is
a k1 Š k2 Š km Š-to-1 function from the nŠ permutations to the .k1 ; k2 ; : : : ; km /-
splits of A, so from the Division Rule we conclude the Subset Split Rule:

Definition 15.6.1. For n; k1 ; : : : ; km 2 N, such that k1 Ck2 C Ckm D n, define
the multinomial coefficient
!
n nŠ
WWD :
k1 ; k2 ; : : : ; km k1 Š k2 Š : : : km Š

Rule 15.6.2 (Subset Split Rule). The number of .k1 ; k2 ; : : : ; km /-splits of an n-
element set is !
n
:
k1 ; : : : ; km

15.6.2 The Bookkeeper Rule
We can also generalize our count of n-bit sequences with k ones to counting se-
quences of n letters over an alphabet with more than two letters. For example,
how many sequences can be formed by permuting the letters in the 10-letter word
BOOKKEEPER?
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15.6. Sequences with Repetitions 619

Notice that there are 1 B, 2 O’s, 2 K’s, 3 E’s, 1 P, and 1 R in BOOKKEEPER. This
leads to a straightforward bijection between permutations of BOOKKEEPER and
(1,2,2,3,1,1)-splits of f1; 2; : : : ; 10g. Namely, map a permutation to the sequence
of sets of positions where each of the different letters occur.
For example, in the permutation BOOKKEEPER itself, the B is in the 1st posi-
tion, the O’s occur in the 2nd and 3rd positions, K’s in 4th and 5th, the E’s in the
6th, 7th and 9th, P in the 8th, and R is in the 10th position. So BOOKKEEPER
maps to
.f1g; f2; 3g; f4; 5g; f6; 7; 9g; f8g; f10g/:
From this bijection and the Subset Split Rule, we conclude that the number of ways
to rearrange the letters in the word BOOKKEEPER is:
total letters
‚…„ƒ
10Š
1Š „ƒ‚…
„ƒ‚… 2Š „ƒ‚…
2Š „ƒ‚…
3Š „ƒ‚…
1Š „ƒ‚…
1Š
B’s O’s K’s E’s P’s R’s

This example generalizes directly to an exceptionally useful counting principle
which we will call the

Rule 15.6.3 (Bookkeeper Rule). Let l1 ; : : : ; lm be distinct elements. The number
of sequences with k1 occurrences of l1 , and k2 occurrences of l2 , . . . , and km
occurrences of lm is !
k1 C k2 C C km
:
k1 ; : : : ; km

For example, suppose you are planning a 20-mile walk, which should include 5
northward miles, 5 eastward miles, 5 southward miles, and 5 westward miles. How
many different walks are possible?
There is a bijection between such walks and sequences with 5 N’s, 5 E’s, 5 S’s,
and 5 W’s. By the Bookkeeper Rule, the number of such sequences is:

20Š
:
.5Š/4
A Word about Words
Someday you might refer to the Subset Split Rule or the Bookkeeper Rule in front
of a roomful of colleagues and discover that they’re all staring back at you blankly.
This is not because they’re dumb, but rather because we made up the name “Book-
keeper Rule.” However, the rule is excellent and the name is apt, so we suggest
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620 Chapter 15 Cardinality Rules

that you play through: “You know? The Bookkeeper Rule? Don’t you guys know
anything?”
The Bookkeeper Rule is sometimes called the “formula for permutations with
indistinguishable objects.” The size k subsets of an n-element set are sometimes
called k-combinations. Other similar-sounding descriptions are “combinations with
repetition, permutations with repetition, r-permutations, permutations with indis-
tinguishable objects,” and so on. However, the counting rules we’ve taught you are
sufficient to solve all these sorts of problems without knowing this jargon, so we
won’t burden you with it.

15.6.3 The Binomial Theorem
Counting gives insight into one of the basic theorems of algebra. A binomial is a
sum of two terms, such as a C b. Now consider its fourth power .a C b/4 .
By repeatedly using distributivity of products over sums to multiply out this 4th
power expression completely, we get

.a C b/4 D aaaa C aaab C aaba C aabb
C abaa C abab C abba C abbb
C baaa C baab C baba C babb
C bbaa C bbab C bbba C bbbb

Notice that there is one term for every sequence of a’s and b’s. So there are 24
terms, and the number of terms with k copies of b and n k copies of a is:
!
nŠ n
D
kŠ .n k/Š k

by the Bookkeeper Rule. Hence, the coefficient of an k b k is kn . So for n D 4,

this means:
! ! ! ! !
4 4 4 4 4
.a C b/4 D a4 b 0 C a3 b 1 C a2 b 2 C a1 b 3 C a0 b 4
0 1 2 3 4

In general, this reasoning gives the Binomial Theorem:

Theorem 15.6.4 (Binomial Theorem). For all n 2 N and a; b 2 R:
n
!
n
X n n k k
.a C b/ D a b
k
kD0
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15.7. Counting Practice: Poker Hands 621

The Binomial Theorem explains why the n choose k number is called a binomial
coefficient.
This reasoning about binomials extends nicely to multinomials, which are sums
of two or more terms. For example, suppose we wanted the coefficient of

bo2 k 2 e 3 pr

in the expansion of .b C o C k C e C p C r/10 . Each term in this expansion is a
product of 10 variables where each variable is one of b, o, k, e, p or r. Now, the
coefficient of bo2 k 2 e 3 pr is the number of those terms with exactly 1 b, 2 o’s, 2
k’s, 3 e’s, 1 p and 1 r. And the number of such terms is precisely the number of
rearrangements of the word BOOKKEEPER:
!
10 10Š
D :
1; 2; 2; 3; 1; 1 1Š 2Š 2Š 3Š 1Š 1Š

This reasoning extends to a general theorem:

Theorem 15.6.5 (Multinomial Theorem). For all n 2 N,
!
n
z1k1 z2k2 zm
X
n km
.z1 C z2 C C zm / D :
k1 ; k2 ; : : : ; km
k1 ;:::;km 2N
k1 CCkm Dn

But you’ll be better off remembering the reasoning behind the Multinomial The-
orem rather than this cumbersome formal statement.

15.7 Counting Practice: Poker Hands
Five-Card Draw is a card game in which each player is initially dealt a hand con-
sisting of 5 cards from a deck of 52 cards.3 The number of different hands in
3 There are 52 cards in a standard deck. Each card has a suit and a rank. There are four suits:

(spades) ~ (hearts) | (clubs) } (diamonds)

And there are 13 ranks, listed here from lowest to highest:

Ace Jack Queen King
A; 2; 3; 4; 5; 6; 7; 8; 9; J ; Q ; K :

Thus, for example, 8~ is the 8 of hearts and A is the ace of spades.
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622 Chapter 15 Cardinality Rules

Five-Card Draw is the number of 5-element subsets of a 52-element set, which is
!
52
D 2; 598; 960:
5

Let’s get some counting practice by working out the number of hands with various
special properties.

15.7.1 Hands with a Four-of-a-Kind
A Four-of-a-Kind is a set of four cards with the same rank. How many different
hands contain a Four-of-a-Kind? Here are a couple examples:

f8; 8}; Q~; 8~; 8|g
fA|; 2|; 2~; 2}; 2g

As usual, the first step is to map this question to a sequence-counting problem. A
hand with a Four-of-a-Kind is completely described by a sequence specifying:

1. The rank of the four cards.

2. The rank of the extra card.

3. The suit of the extra card.

Thus, there is a bijection between hands with a Four-of-a-Kind and sequences con-
sisting of two distinct ranks followed by a suit. For example, the three hands above
are associated with the following sequences:

.8; Q; ~/ $ f 8; 8}; 8~; 8|; Q~g
.2; A; |/ $ f2|; 2~; 2}; 2; A|g

Now we need only count the sequences. There are 13 ways to choose the first rank,
12 ways to choose the second rank, and 4 ways to choose the suit. Thus, by the
Generalized Product Rule, there are 13 12 4 D 624 hands with a Four-of-a-Kind.
This means that only 1 hand in about 4165 has a Four-of-a-Kind. Not surprisingly,
Four-of-a-Kind is considered to be a very good poker hand!
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15.7. Counting Practice: Poker Hands 623

15.7.2 Hands with a Full House
A Full House is a hand with three cards of one rank and two cards of another rank.
Here are some examples:

f2; 2|; 2}; J |; J }g
f5}; 5|; 5~; 7~; 7|g

Again, we shift to a problem about sequences. There is a bijection between Full
Houses and sequences specifying:

1. The rank of the triple, which can be chosen in 13 ways.

2. The suits of the triple, which can be selected in 43 ways.

3. The rank of the pair, which can be chosen in 12 ways.

4. The suits of the pair, which can be selected in 42 ways.

The example hands correspond to sequences as shown below:

.2; f; |; }g; J; f|; }g/ $ f2; 2|; 2}; J |; J }g
.5; f}; |; ~g; 7; f~; |g/ $ f5}; 5|; 5~; 7~; 7|g

By the Generalized Product Rule, the number of Full Houses is:
! !
4 4
13 12 :
3 2

We’re on a roll—but we’re about to hit a speed bump.

15.7.3 Hands with Two Pairs
How many hands have Two Pairs; that is, two cards of one rank, two cards of
another rank, and one card of a third rank? Here are examples:

f3}; 3; Q}; Q~; A|g
f9~; 9}; 5~; 5|; Kg

Each hand with Two Pairs is described by a sequence consisting of:

1. The rank of the first pair, which can be chosen in 13 ways.

2. The suits of the first pair, which can be selected 42 ways.

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624 Chapter 15 Cardinality Rules

3. The rank of the second pair, which can be chosen in 12 ways.

4. The suits of the second pair, which can be selected in 42 ways.

5. The rank of the extra card, which can be chosen in 11 ways.

6. The suit of the extra card, which can be selected in 41 D 4 ways.

Thus, it might appear that the number of hands with Two Pairs is:
! !
4 4
13 12 11 4:
2 2

Wrong answer! The problem is that there is not a bijection from such sequences to
hands with Two Pairs. This is actually a 2-to-1 mapping. For example, here are the
pairs of sequences that map to the hands given above:

.3; f}; g; Q; f}; ~g; A; |/ &
f3}; 3; Q}; Q~; A|g
.Q; f}; ~g; 3; f}; g; A; |/ %

.9; f~; }g; 5; f~; |g; K; / &
f9~; 9}; 5~; 5|; Kg
.5; f~; |g; 9; f~; }g; K; / %

The problem is that nothing distinguishes the first pair from the second. A pair of
5’s and a pair of 9’s is the same as a pair of 9’s and a pair of 5’s. We avoided this
difficulty in counting Full Houses because, for example, a pair of 6’s and a triple of
kings is different from a pair of kings and a triple of 6’s.
We ran into precisely this difficulty last time, when we went from counting ar-
rangements of different pieces on a chessboard to counting arrangements of two
identical rooks. The solution then was to apply the Division Rule, and we can do
the same here. In this case, the Division rule says there are twice as many sequences
as hands, so the number of hands with Two Pairs is actually:

13 42 12 42 11 4

:
2
Another Approach
The preceding example was disturbing! One could easily overlook the fact that the
mapping was 2-to-1 on an exam, fail the course, and turn to a life of crime. You
can make the world a safer place in two ways:
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15.7. Counting Practice: Poker Hands 625

1. Whenever you use a mapping f W A ! B to translate one counting problem
to another, check that the same number of elements in A are mapped to each
element in B. If k elements of A map to each of element of B, then apply
the Division Rule using the constant k.

2. As an extra check, try solving the same problem in a different way. Multiple
approaches are often available—and all had better give the same answer!
(Sometimes different approaches give answers that look different, but turn
out to be the same after some algebra.)

We already used the first method; let’s try the second. There is a bijection be-
tween hands with two pairs and sequences that specify:
1. The ranks of the two pairs, which can be chosen in 13

2 ways.

2. The suits of the lower-rank pair, which can be selected in 42 ways.

3. The suits of the higher-rank pair, which can be selected in 42 ways.

4. The rank of the extra card, which can be chosen in 11 ways.

5. The suit of the extra card, which can be selected in 41 D 4 ways.

For example, the following sequences and hands correspond:

.f3; Qg; f}; g; f}; ~g; A; |/ $ f3}; 3; Q}; Q~; A|g
.f9; 5g; f~; |g; f~; }g; K; / $ f9~; 9}; 5~; 5|; Kg

Thus, the number of hands with two pairs is:
! ! !
13 4 4
11 4:
2 2 2

This is the same answer we got before, though in a slightly different form.

15.7.4 Hands with Every Suit
How many hands contain at least one card from every suit? Here is an example of
such a hand:
f7}; K|; 3}; A~; 2g
Each such hand is described by a sequence that specifies:

1. The ranks of the diamond, the club, the heart, and the spade, which can be
selected in 13 13 13 13 D 134 ways.
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626 Chapter 15 Cardinality Rules

2. The suit of the extra card, which can be selected in 4 ways.

3. The rank of the extra card, which can be selected in 12 ways.

For example, the hand above is described by the sequence:

.7; K; A; 2; }; 3/ $ f7}; K|; A~; 2; 3}g:

Are there other sequences that correspond to the same hand? There is one more!
We could equally well regard either the 3} or the 7} as the extra card, so this
is actually a 2-to-1 mapping. Here are the two sequences corresponding to the
example hand:

.7; K; A; 2; }; 3/ &
f7}; K|; A~; 2; 3}g
.3; K; A; 2; }; 7/ %

Therefore, the number of hands with every suit is:

134 4 12
:
2

15.8 The Pigeonhole Principle
Here is an old puzzle:

A drawer in a dark room contains red socks, green socks, and blue
socks. How many socks must you withdraw to be sure that you have a
matching pair?

For example, picking out three socks is not enough; you might end up with one
red, one green, and one blue. The solution relies on the

Pigeonhole Principle
If there are more pigeons than holes they occupy, then at least two
pigeons must be in the same hole.
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15.8. The Pigeonhole Principle 627

A f B

1st sock red

2nd sock green

3rd sock blue

4th sock

Figure 15.3 One possible mapping of four socks to three colors.

What pigeons have to do with selecting footwear under poor lighting conditions
may not be immediately obvious, but if we let socks be pigeons and the colors be
three pigeonholes, then as soon as you pick four socks, there are bound to be two
in the same hole, that is, with the same color. So four socks are enough to ensure
a matched pair. For example, one possible mapping of four socks to three colors is
shown in Figure 15.3.
A rigorous statement of the Principle goes this way:

Rule 15.8.1 (Pigeonhole Principle). If jAj > jBj, then for every total function
f W A ! B, there exist two different elements of A that are mapped by f to the
same element of B.

Stating the Principle this way may be less intuitive, but it should now sound
familiar: it is simply the contrapositive of the Mapping Rules injective case (4.6).
Here, the pigeons form set A, the pigeonholes are the set B, and f describes which
hole each pigeon occupies.
Mathematicians have come up with many ingenious applications for the pigeon-
hole principle. If there were a cookbook procedure for generating such arguments,
we’d give it to you. Unfortunately, there isn’t one. One helpful tip, though: when
you try to solve a problem with the pigeonhole principle, the key is to clearly iden-
tify three things:

1. The set A (the pigeons).

2. The set B (the pigeonholes).

3. The function f (the rule for assigning pigeons to pigeonholes).
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628 Chapter 15 Cardinality Rules

15.8.1 Hairs on Heads
There are a number of generalizations of the pigeonhole principle. For example:

Rule 15.8.2 (Generalized Pigeonhole Principle). If jAj > k jBj, then every total
function f W A ! B maps at least k C1 different elements of A to the same element
of B.

For example, if you pick two people at random, surely they are extremely un-
likely to have exactly the same number of hairs on their heads. However, in the
remarkable city of Boston, Massachusetts, there is a group of three people who
have exactly the same number of hairs! Of course, there are many completely bald
people in Boston, and they all have zero hairs. But we’re talking about non-bald
people; say a person is non-bald if they have at least ten thousand hairs on their
head.
Boston has about 500,000 non-bald people, and the number of hairs on a person’s
head is at most 200,000. Let A be the set of non-bald people in Boston, let B D
f10; 000; 10; 001; : : : ; 200; 000g, and let f map a person to the number of hairs on
his or her head. Since jAj > 2jBj, the Generalized Pigeonhole Principle implies
that at least three people have exactly the same number of hairs. We don’t know
who they are, but we know they exist!

15.8.2 Subsets with the Same Sum
For your reading pleasure, we have displayed ninety 25-digit numbers in Fig-
ure 15.4. Are there two different subsets of these 25-digit numbers that have the
same sum? For example, maybe the sum of the last ten numbers in the first column
is equal to the sum of the first eleven numbers in the second column?
Finding two subsets with the same sum may seem like a silly puzzle, but solving
these sorts of problems turns out to be useful in diverse applications such as finding
good ways to fit packages into shipping containers and decoding secret messages.
It turns out that it is hard to find different subsets with the same sum, which
is why this problem arises in cryptography. But it is easy to prove that two such
subsets exist. That’s where the Pigeonhole Principle comes in.
Let A be the collection of all subsets of the 90 numbers in the list. Now the sum
of any subset of numbers is at most 90 1025 , since there are only 90 numbers and
every 25-digit number is less than 1025 . So let B be the set of integers f0; 1; : : : ; 90
1025 g, and let f map each subset of numbers (in A) to its sum (in B).
We proved that an n-element set has 2n different subsets in Section 15.2. There-
fore:
jAj D 290 1:237 1027
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15.8. The Pigeonhole Principle 629

0020480135385502964448038 3171004832173501394113017
5763257331083479647409398 8247331000042995311646021
0489445991866915676240992 3208234421597368647019265
5800949123548989122628663 8496243997123475922766310
1082662032430379651370981 3437254656355157864869113
6042900801199280218026001 8518399140676002660747477
1178480894769706178994993 3574883393058653923711365
6116171789137737896701405 8543691283470191452333763
1253127351683239693851327 3644909946040480189969149
6144868973001582369723512 8675309258374137092461352
1301505129234077811069011 3790044132737084094417246
6247314593851169234746152 8694321112363996867296665
1311567111143866433882194 3870332127437971355322815
6814428944266874963488274 8772321203608477245851154
1470029452721203587686214 4080505804577801451363100
6870852945543886849147881 8791422161722582546341091
1578271047286257499433886 4167283461025702348124920
6914955508120950093732397 9062628024592126283973285
1638243921852176243192354 4235996831123777788211249
6949632451365987152423541 9137845566925526349897794
1763580219131985963102365 4670939445749439042111220
7128211143613619828415650 9153762966803189291934419
1826227795601842231029694 4815379351865384279613427
7173920083651862307925394 9270880194077636406984249
1843971862675102037201420 4837052948212922604442190
7215654874211755676220587 9324301480722103490379204
2396951193722134526177237 5106389423855018550671530
7256932847164391040233050 9436090832146695147140581
2781394568268599801096354 5142368192004769218069910
7332822657075235431620317 9475308159734538249013238
2796605196713610405408019 5181234096130144084041856
7426441829541573444964139 9492376623917486974923202
2931016394761975263190347 5198267398125617994391348
7632198126531809327186321 9511972558779880288252979
2933458058294405155197296 5317592940316231219758372
7712154432211912882310511 9602413424619187112552264
3075514410490975920315348 5384358126771794128356947
7858918664240262356610010 9631217114906129219461111
8149436716871371161932035 3157693105325111284321993
3111474985252793452860017 5439211712248901995423441
7898156786763212963178679 9908189853102753335981319
3145621587936120118438701 5610379826092838192760458
8147591017037573337848616 9913237476341764299813987
3148901255628881103198549 5632317555465228677676044
5692168374637019617423712 8176063831682536571306791

Figure 15.4 Ninety 25-digit numbers. Can you find two different subsets of these
numbers that have the same sum?
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630 Chapter 15 Cardinality Rules

On the other hand:
jBj D 90 1025 C 1 0:901 1027 :
Both quantities are enormous, but jAj is a bit greater than jBj. This means that f
maps at least two elements of A to the same element of B. In other words, by the
Pigeonhole Principle, two different subsets must have the same sum!
Notice that this proof gives no indication which two sets of numbers have the
same sum. This frustrating variety of argument is called a nonconstructive proof.

The $100 prize for two same-sum subsets
To see if it was possible to actually find two different subsets of the ninety 25-digit
numbers with the same sum, we offered a $100 prize to the first student who did it.
We didn’t expect to have to pay off this bet, but we underestimated the ingenuity
and initiative of the students. One computer science major wrote a program that
cleverly searched only among a reasonably small set of “plausible” sets, sorted
them by their sums, and actually found a couple with the same sum. He won the
prize. A few days later, a math major figured out how to reformulate the sum
problem as a “lattice basis reduction” problem; then he found a software package
implementing an efficient basis reduction procedure, and using it, he very quickly
found lots of pairs of subsets with the same sum. He didn’t win the prize, but he
got a standing ovation from the class—staff included.

The $500 Prize for Sets with Distinct Subset Sums
How can we construct a set of n positive integers such that all its subsets have
distinct sums? One way is to use powers of two:

f1; 2; 4; 8; 16g

This approach is so natural that one suspects all other such sets must involve
larger numbers. (For example, we could safely replace 16 by 17, but not by 15.)
Remarkably, there are examples involving smaller numbers. Here is one:

f6; 9; 11; 12; 13g

One of the top mathematicians of the Twentieth Century, Paul Erdős, conjectured
in 1931 that there are no such sets involving significantly smaller numbers. More
precisely, he conjectured that the largest number in such a set must be greater
than c2n for some constant c > 0. He offered $500 to anyone who could prove
or disprove his conjecture, but the problem remains unsolved.
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15.8. The Pigeonhole Principle 631

15.8.3 A Magic Trick
A Magician sends an Assistant into the audience with a deck of 52 cards while the
Magician looks away.
Five audience members each select one card from the deck. The Assistant then
gathers up the five cards and holds up four of them so the Magician can see them.
The Magician concentrates for a short time and then correctly names the secret,
fifth card!
Since we don’t really believe the Magician can read minds, we know the Assis-
tant has somehow communicated the secret card to the Magician. Real Magicians
and Assistants are not to be trusted, so we expect that the Assistant would secretly
signal the Magician with coded phrases or body language, but for this trick they
don’t have to cheat. In fact, the Magician and Assistant could be kept out of sight
of each other while some audience member holds up the 4 cards designated by the
Assistant for the Magician to see.
Of course, without cheating, there is still an obvious way the Assistant can com-
municate to the Magician: he can choose any of the 4Š D 24 permutations of the
4 cards as the order in which to hold up the cards. However, this alone won’t
quite work: there are 48 cards remaining in the deck, so the Assistant doesn’t have
enough choices of orders to indicate exactly what the secret card is (though he
could narrow it down to two cards).

15.8.4 The Secret
The method the Assistant can use to communicate the fifth card exactly is a nice
application of what we know about counting and matching.
The Assistant has a second legitimate way to communicate: he can choose which
of the five cards to keep hidden. Of course, it’s not clear how the Magician could
determine which of these five possibilities the Assistant selected by looking at the
four visible cards, but there is a way, as we’ll now explain.
The problem facing the Magician and Assistant is actually a bipartite matching
problem. Each vertex on the left will correspond to the information available to the
Assistant, namely, a set of 5 cards. So the set X of left-hand vertices will have 525
elements.
Each vertex on the right will correspond to the information available to the Ma-
gician, namely, a sequence of 4 distinct cards. So the set Y of right-hand vertices
will have 52 51 50 49 elements. When the audience selects a set of 5 cards, then
the Assistant must reveal a sequence of 4 cards from that hand. This constraint is
represented by having an edge between a set of 5 cards on the left and a sequence
of 4 cards on the right precisely when every card in the sequence is also in the
set. This specifies the bipartite graph. Some edges are shown in the diagram in
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632 Chapter 15 Cardinality Rules

yDall
sequences of 4
xDall distinct cards
sets of 
5 cards 
 

 f8~;K;Q;2}g
f8~;K;Q;2};6}g fK;8~;Q;2}g

 fK;8~;6};Qg


f8~;K;Q;9|;6}g 
 


Figure 15.5 The bipartite graph where the nodes on the left correspond to sets
of 5 cards and the nodes on the right correspond to sequences of 4 cards. There is
an edge between a set and a sequence whenever all the cards in the sequence are
contained in the set.

Figure 15.5.
For example,
f8~; K; Q; 2}; 6}g (15.2)
is an element of X on the left. If the audience selects this set of 5 cards, then
there are many different 4-card sequences on the right in set Y that the Assis-
tant could choose to reveal, including .8~; K; Q; 2}/, .K; 8~; Q; 2}/ and
.K; 8~; 6}; Q/.
What the Magician and his Assistant need to perform the trick is a matching for
the X vertices. If they agree in advance on some matching, then when the audience
selects a set of 5 cards, the Assistant reveals the matching sequence of 4 cards. The
Magician uses the matching to find the audience’s chosen set of 5 cards, and so he
can name the one not already revealed.
For example, suppose the Assistant and Magician agree on a matching containing
the two bold edges in Figure 15.5. If the audience selects the set

f8~; K; Q; 9|; 6}g; (15.3)

then the Assistant reveals the corresponding sequence

.K; 8~; 6}; Q/: (15.4)
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15.8. The Pigeonhole Principle 633

Using the matching, the Magician sees that the hand (15.3) is matched to the se-
quence (15.4), so he can name the one card in the corresponding set not already
revealed, namely, the 9|. Notice that the fact that the sets are matched, that is,
that different sets are paired with distinct sequences, is essential. For example, if
the audience picked the previous hand (15.2), it would be possible for the Assistant
to reveal the same sequence (15.4), but he better not do that; if he did, then the
Magician would have no way to tell if the remaining card was the 9| or the 2}.
So how can we be sure the needed matching can be found? The answer is that
each vertex on the left has degree 54Š D 120, since there are five ways to select the
card kept secret and there are 4Š permutations of the remaining 4 cards. In addition,
each vertex on the right has degree 48, since there are 48 possibilities for the fifth
card. So this graph is degree-constrained according to Definition 12.5.5, and so has
a matching by Theorem 12.5.6.
In fact, this reasoning shows that the Magician could still pull off the trick if 120
cards were left instead of 48, that is, the trick would work with a deck as large as
124 different cards—without any magic!

15.8.5 The Real Secret
But wait a minute! It’s all very well in principle to have the Magician and his
Assistant agree on a matching, but how are they supposed to remember a matching
with 525 D 2; 598; 960 edges? For the trick to work in practice, there has to be a
way to match hands and card sequences mentally and on the fly.
We’ll describe one approach. As a running example, suppose that the audience
selects:
10~ 9} 3~ Q J }:

The Assistant picks out two cards of the same suit. In the example, the
assistant might choose the 3~ and 10~. This is always possible because of
the Pigeonhole Principle—there are five cards and 4 suits so two cards must
be in the same suit.

The Assistant locates the ranks of these two cards on the cycle shown in Fig-
ure 15.6. For any two distinct ranks on this cycle, one is always between 1
and 6 hops clockwise from the other. For example, the 3~ is 6 hops clock-
wise from the 10~.

The more counterclockwise of these two cards is revealed first, and the other
becomes the secret card. Thus, in our example, the 10~ would be revealed,
and the 3~ would be the secret card. Therefore:
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634 Chapter 15 Cardinality Rules

A
K 2

Q 3

J 4

10 5

9 6
8 7

Figure 15.6 The 13 card ranks arranged in cyclic order.

– The suit of the secret card is the same as the suit of the first card re-
vealed.
– The rank of the secret card is between 1 and 6 hops clockwise from the
rank of the first card revealed.

All that remains is to communicate a number between 1 and 6. The Magician
and Assistant agree beforehand on an ordering of all the cards in the deck
from smallest to largest such as:

A| A} A~ A 2| 2} 2~ 2 : : : K~ K

The order in which the last three cards are revealed communicates the num-
ber according to the following scheme:

. small; medium; large / = 1
. small; large; medium / = 2
. medium; small; large / = 3
. medium; large; small / = 4
. large; small; medium / = 5
. large; medium; small / = 6

In the example, the Assistant wants to send 6 and so reveals the remaining
three cards in large, medium, small order. Here is the complete sequence that
the Magician sees:
10~ Q J } 9}
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15.9. Inclusion-Exclusion 635

The Magician starts with the first card 10~ and hops 6 ranks clockwise to
reach 3~, which is the secret card!

So that’s how the trick can work with a standard deck of 52 cards. On the other
hand, Hall’s Theorem implies that the Magician and Assistant can in principle per-
form the trick with a deck of up to 124 cards. It turns out that there is a method
which they could actually learn to use with a reasonable amount of practice for a
124-card deck, but we won’t explain it here.4

15.8.6 The Same Trick with Four Cards?
Suppose that the audience selects only four cards and the Assistant reveals a se-
quence of three to the Magician. Can the Magician determine the fourth card?
Let X be all the sets of four cards that the audience might select, and let Y be all
the sequences of three cards that the Assistant might reveal. Now, on one hand, we
have !
52
jXj D D 270; 725
4
by the Subset Rule. On the other hand, we have

jY j D 52 51 50 D 132; 600

by the Generalized Product Rule. Thus, by the Pigeonhole Principle, the Assistant
must reveal the same sequence of three cards for at least

270; 725
D3
132; 600

different four-card hands. This is bad news for the Magician: if he sees that se-
quence of three, then there are at least three possibilities for the fourth card which
he cannot distinguish. So there is no legitimate way for the Assistant to communi-
cate exactly what the fourth card is!

15.9 Inclusion-Exclusion
How big is a union of sets? For example, suppose there are 60 math majors, 200
EECS majors, and 40 physics majors. How many students are there in these three
4 See The Best Card Trick by Michael Kleber for more information.
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636 Chapter 15 Cardinality Rules

departments? Let M be the set of math majors, E be the set of EECS majors, and
P be the set of physics majors. In these terms, we’re asking for jM [ E [ P j.
The Sum Rule says that if M , E and P are disjoint, then the sum of their sizes
is
jM [ E [ P j D jM j C jEj C jP j:
However, the sets M , E and P might not be disjoint. For example, there might be a
student majoring in both math and physics. Such a student would be counted twice
on the right side of this equation, once as an element of M and once as an element
of P . Worse, there might be a triple-major5 counted three times on the right side!
Our most-complicated counting rule determines the size of a union of sets that
are not necessarily disjoint. Before we state the rule, let’s build some intuition by
considering some easier special cases: unions of just two or three sets.

15.9.1 Union of Two Sets
For two sets, S1 and S2 , the Inclusion-Exclusion Rule is that the size of their union
is:
jS1 [ S2 j D jS1 j C jS2 j jS1 \ S2 j (15.5)
Intuitively, each element of S1 is accounted for in the first term, and each element
of S2 is accounted for in the second term. Elements in both S1 and S2 are counted
twice—once in the first term and once in the second. This double-counting is cor-
rected by the final term.

15.9.2 Union of Three Sets
So how many students are there in the math, EECS, and physics departments? In
other words, what is jM [ E [ P j if:

jM j D 60
jEj D 200
jP j D 40:

The size of a union of three sets is given by a more complicated Inclusion-Exclusion
formula:

jS1 [ S2 [ S3 j D jS1 j C jS2 j C jS3 j
jS1 \ S2 j jS1 \ S3 j jS2 \ S3 j
C jS1 \ S2 \ S3 j:
5 . . . though not at MIT anymore.
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15.9. Inclusion-Exclusion 637

Remarkably, the expression on the right accounts for each element in the union of
S1 , S2 and S3 exactly once. For example, suppose that x is an element of all three
sets. Then x is counted three times (by the jS1 j, jS2 j and jS3 j terms), subtracted
off three times (by the jS1 \ S2 j, jS1 \ S3 j and jS2 \ S3 j terms), and then counted
once more (by the jS1 \ S2 \ S3 j term). The net effect is that x is counted just
once.
If x is in two sets (say, S1 and S2 ), then x is counted twice (by the jS1 j and
jS2 j terms) and subtracted once (by the jS1 \ S2 j term). In this case, x does not
contribute to any of the other terms, since x … S3 .
So we can’t answer the original question without knowing the sizes of the various
intersections. Let’s suppose that there are:
4 math - EECS double majors
3 math - physics double majors
11 EECS - physics double majors
2 triple majors

Then jM \Ej D 4C2, jM \P j D 3C2, jE \P j D 11C2, and jM \E \P j D 2.
Plugging all this into the formula gives:

jM [ E [ P j D jM j C jEj C jP j jM \ Ej jM \ P j jE \ P j
C jM \ E \ P j
D 60 C 200 C 40 6 5 13 C 2
D 278

15.9.3 Sequences with 42, 04, or 60
In how many permutations of the set f0; 1; 2; : : : ; 9g do either 4 and 2, 0 and 4, or
6 and 0 appear consecutively? For example, none of these pairs appears in:

.7; 2; 9; 5; 4; 1; 3; 8; 0; 6/:

The 06 at the end doesn’t count; we need 60. On the other hand, both 04 and 60
appear consecutively in this permutation:

.7; 2; 5; 6; 0; 4; 3; 8; 1; 9/:

Let P42 be the set of all permutations in which 42 appears. Define P60 and P04
similarly. Thus, for example, the permutation above is contained in both P60 and
P04 , but not P42 . In these terms, we’re looking for the size of the set P42 [ P04 [
P60 .
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638 Chapter 15 Cardinality Rules

First, we must determine the sizes of the individual sets, such as P60 . We can use
a trick: group the 6 and 0 together as a single symbol. Then there is an immediate
bijection between permutations of f0; 1; 2; : : : 9g containing 6 and 0 consecutively
and permutations of:
f60; 1; 2; 3; 4; 5; 7; 8; 9g:
For example, the following two sequences correspond:

.7; 2; 5; 6; 0; 4; 3; 8; 1; 9/ ! .7; 2; 5; 60; 4; 3; 8; 1; 9/:

There are 9Š permutations of the set containing 60, so jP60 j D 9Š by the Bijection
Rule. Similarly, jP04 j D jP42 j D 9Š as well.
Next, we must determine the sizes of the two-way intersections, such as P42 \
P60 . Using the grouping trick again, there is a bijection with permutations of the
set:
f42; 60; 1; 3; 5; 7; 8; 9g:
Thus, jP42 \ P60 j D 8Š. Similarly, jP60 \ P04 j D 8Š by a bijection with the set:

f604; 1; 2; 3; 5; 7; 8; 9g:

And jP42 \ P04 j D 8Š as well by a similar argument. Finally, note that jP60 \
P04 \ P42 j D 7Š by a bijection with the set:

f6042; 1; 3; 5; 7; 8; 9g:

Plugging all this into the formula gives:

jP42 [ P04 [ P60 j D 9Š C 9Š C 9Š 8Š 8Š 8Š C 7Š:

15.9.4 Union of n Sets
The size of a union of n sets is given by the following rule.

Rule 15.9.1 (Inclusion-Exclusion).

jS1 [ S2 [ [ Sn j D

the sum of the sizes of the individual sets
minus the sizes of all two-way intersections
plus the sizes of all three-way intersections
minus the sizes of all four-way intersections
plus the sizes of all five-way intersections, etc.
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15.9. Inclusion-Exclusion 639

The formulas for unions of two and three sets are special cases of this general
rule.
This way of expressing Inclusion-Exclusion is easy to understand and nearly
as precise as expressing it in mathematical symbols, but we’ll need the symbolic
version below, so let’s work on deciphering it now.
We already have a concise notation for the sum of sizes of the individual sets,
namely,
Xn
jSi j:
i D1

A “two-way intersection” is a set of the form Si \ Sj for i ¤ j . We regard Sj \ Si
as the same two-way intersection as Si \ Sj , so we can assume that i < j . Now
we can express the sum of the sizes of the two-way intersections as
X
jSi \ Sj j:
1i <j n

Similarly, the sum of the sizes of the three-way intersections is
X
jSi \ Sj \ Sk j:
1i <j <kn

These sums have alternating signs in the Inclusion-Exclusion formula, with the
sum of the k-way intersections getting the sign . 1/k 1 . This finally leads to a
symbolic version of the rule:

Rule (Inclusion-Exclusion).
ˇ n ˇ n
ˇ[ ˇ X
S D jSi j
ˇ ˇ
ˇ iˇ
ˇ ˇ
iD1 i D1
X
jSi \ Sj j
1i <j n
X
C jSi \ Sj \ Sk j C
1i <j <kn
ˇ n ˇ
ˇ\ ˇ
C . 1/n 1
Si ˇ :
ˇ ˇ
ˇ
ˇ ˇ
i D1

While it’s often handy express the rule in this way as a sum of sums, it is not
necessary to group the terms by how many sets are in the intersections. So another
way to state the rule is:
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640 Chapter 15 Cardinality Rules

Rule (Inclusion-Exclusion-II).
ˇ n ˇ ˇ ˇ
ˇ[ ˇ X ˇ\ ˇ
jI jC1 ˇ
Si ˇ D . 1/ ˇ Si ˇ (15.6)
ˇ ˇ ˇ
ˇ
ˇ ˇ ˇ ˇ
i D1 ;¤I f1;:::;ng i 2I

A proof of these rules using just highschool algebra is given in Problem 15.58.

15.9.5 Computing Euler’s Function
We can also use Inclusion-Exclusion to derive the explicit formula for Euler’s func-
tion claimed in Corollary 9.10.11: if the prime factorization of n is p1e1 pm
em
for
distinct primes pi , then
m
Y 1
.n/ D n 1 : (15.7)
pi
i D1

To begin, let S be the set of integers in Œ0::n/ that are not relatively prime to n.
So .n/ D n jS j. Next, let Ca be the set of integers in Œ0::n/ that are divisible by
a:
Ca WWD fk 2 Œ0::n/ j a j kg:
So the integers in S are precisely the integers in Œ0::n/ that are divisible by at least
one of the pi ’s. Namely,
[m
SD Cpi : (15.8)
i D1

We’ll be able to find the size of this union using Inclusion-Exclusion because the
intersections of the Cpi ’s are easy to count. For example, Cp \ Cq \ Cr is the set
of integers in Œ0::n/ that are divisible by each of p, q and r. But since the p; q; r
are distinct primes, being divisible by each of them is the same as being divisible
by their product. Now if k is a positive divisor of n, then there are exactly n=k
multiples of k in Œ0::n/. So exactly n=pqr of the integers in Œ0::n/ are divisible by
all three primes p, q, r. In other words,
n
jCp \ Cq \ Cr j D :
pqr
This reasoning extends to arbitrary intersections of Cp ’s, namely,
ˇ ˇ
ˇ\ ˇ
ˇ ˇ n
ˇ Cpj ˇˇ D Q ; (15.9)
j 2I pj
ˇ
ˇ j 2I ˇ
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15.10. Combinatorial Proofs 641

for any nonempty set I Œ1::m. This lets us calculate:
ˇm ˇ
ˇ[ ˇ
jS j D ˇ Cpi ˇ (by (15.8))
ˇ ˇ
ˇ ˇ
iD1
ˇ ˇ
X ˇ\ ˇ
D . 1/jI jC1 ˇ Cpi ˇ (by Inclusion-Exclusion (15.6))
ˇ ˇ
ˇ ˇ
;¤I Œ1::m i 2I
X n
D . 1/jI jC1 Q (by (15.9))
;¤I Œ1::m j 2I pj
X 1
D n Q
;¤I Œ1::m j 2I . pj /
m !
Y 1
D n 1 C n;
pi
i D1

so
m
Y 1
.n/ D n jSj D n 1 ;
pi
i D1
which proves (15.7).
Yikes! That was pretty hairy. Are you getting tired of all that nasty algebra? If
so, then good news is on the way. In the next section, we will show you how to
prove some heavy-duty formulas without using any algebra at all. Just a few words
and you are done. No kidding.

15.10 Combinatorial Proofs
Suppose you have n different T-shirts, but only want to keep k. You could equally
well select the k shirts you want to keep or select the complementary set of n k
shirts you want to throw out. Thus, the number of ways to select k shirts from
among n must be equal to the number of ways to select n k shirts from among n.
Therefore: ! !
n n
D :
k n k
This is easy to prove algebraically, since both sides are equal to:
nŠ
:
kŠ .n k/Š
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642 Chapter 15 Cardinality Rules

But we didn’t really have to resort to algebra; we just used counting principles.
Hmmm.. . .

15.10.1 Pascal’s Triangle Identity
Bob, famed Math for Computer Science Teaching Assistant, has decided to try out
for the US Olympic boxing team. After all, he’s watched all of the Rocky movies
and spent hours in front of a mirror sneering, “Yo, you wanna piece a’ me?!” Bob
figures that n people (including himself) are competing for spots on the team and
only k will be selected. As part of maneuvering for a spot on the team, he needs to
work out how many different teams are possible. There are two cases to consider:

Bob is selected for the team, and his k 1 teammates are selected from
among the other n 1 competitors. The number of different teams that can
be formed in this way is: !
n 1
:
k 1

Bob is not selected for the team, and all k team members are selected from
among the other n 1 competitors. The number of teams that can be formed
this way is: !
n 1
:
k

All teams of the first type contain Bob, and no team of the second type does;
therefore, the two sets of teams are disjoint. Thus, by the Sum Rule, the total
number of possible Olympic boxing teams is:
! !
n 1 n 1
C :
k 1 k

Ted, equally-famed Teaching Assistant, thinks Bob isn’t so tough and so he
might as well also try out. He reasons that n people (including himself) are try-
ing out for k spots. Thus, the number of ways to select the team is simply:
!
n
:
k

Ted and Bob each correctly counted the number of possible boxing teams. Thus,
their answers must be equal. So we know:
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15.10. Combinatorial Proofs 643

Lemma 15.10.1 (Pascal’s Triangle Identity).
! ! !
n n 1 n 1
D C : (15.10)
k k 1 k

We proved Pascal’s Triangle Identity without any algebra! Instead, we relied
purely on counting techniques.

15.10.2 Giving a Combinatorial Proof
A combinatorial proof is an argument that establishes an algebraic fact by relying
on counting principles. Many such proofs follow the same basic outline:

1. Define a set S .

2. Show that jSj D n by counting one way.

3. Show that jSj D m by counting another way.

4. Conclude that n D m.

In the preceding example, S was the set of all possible Olympic boxing teams. Bob
computed ! !
n 1 n 1
jSj D C
k 1 k
by counting one way, and Ted computed
!
n
jS j D
k

by counting another way. Equating these two expressions gave Pascal’s Identity.

Checking a Combinatorial Proof
Combinatorial proofs are based on counting the same thing in different ways. This
is fine when you’ve become practiced at different counting methods, but when in
doubt, you can fall back on bijections and sequence counting to check such proofs.
For example, let’s take a closer look at the combinatorial proof of Pascal’s Iden-
tity (15.10). In this case, the set S of things to be counted is the collection of all
size-k subsets of integers in the interval Œ1::n.
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644 Chapter 15 Cardinality Rules

Now we’ve already counted S one way, via the Bookkeeper Rule, and found
n
jS j D k . The other “way” corresponds to defining a bijection between S and the
disjoint union of two sets A and B where,

A WWD f.1; X / j X Œ2; n AND jX j D k 1g
B WWD f.0; Y / j Y Œ2; n AND jY j D kg:

Clearly A and B are disjoint since the pairs in the two sets have different first
coordinates, so jA [ Bj D jAj C jBj. Also,
!
n 1
jAj D # specified sets X D ;
k 1
!
n 1
jBj D # specified sets Y D :
k

Now finding a bijection f W .A [ B/ ! S will prove the identity (15.10). In
particular, we can define
(
X [ f1g if c D .1; X /;
f .c/ WWD
Y if c D .0; Y /:

It should be obvious that f is a bijection.

15.10.3 A Colorful Combinatorial Proof
The set that gets counted in a combinatorial proof in different ways is usually de-
fined in terms of simple sequences or sets rather than an elaborate story about
Teaching Assistants. Here is another colorful example of a combinatorial argu-
ment.

Theorem 15.10.2.
n
! ! !
X n 2n 3n
D
r n r n
rD0

Proof. We give a combinatorial proof. Let S be all n-card hands that can be dealt
from a deck containing n different red cards and 2n different black cards. First,
note that every 3n-element set has
!
3n
jS j D
n
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15.11. References 645

n-element subsets.
From another perspective, the number of hands with exactly r red cards is
! !
n 2n
r n r

since there are nr ways to choose the r red cards and n2nr ways to choose the

n r black cards. Since the number of red cards can be anywhere from 0 to n, the
total number of n-card hands is:
n
! !
X n 2n
jS j D :
r n r
rD0

Equating these two expressions for jS j proves the theorem.

Finding a Combinatorial Proof
Combinatorial proofs are almost magical. Theorem 15.10.2 looks pretty scary, but
we proved it without any algebraic manipulations at all. The key to constructing a
combinatorial proof is choosing the set S properly, which can be tricky. Generally,
the simpler side of the equation should provide some guidance. For example, the
3n
right side of Theorem 15.10.2 is n , which suggests that it will be helpful to
choose S to be all n-element subsets of some 3n-element set.

15.11 References
[5], [15]

Problems for Section 15.2
Practice Problems
Problem 15.1.
Alice is thinking of a number between 1 and 1000.
What is the least number of yes/no questions you could ask her and be guaranteed
to discover what it is? (Alice always answers truthfully.)
(a)
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646 Chapter 15 Cardinality Rules

Problem 15.2.
In how many different ways is it possible to answer the next chapter’s practice
problems if:

the first problem has four true/false questions,

the second problem requires choosing one of four alternatives, and

the answer to the third problem is an integer 15 and 20?

Problem 15.3.
How many total functions are there from set A to set B if jAj D 3 and jBj D 7?

Problem 15.4.
Let X be the six element set fx1 ; x2 ; x3 ; x4 ; x5 ; x6 g.
(a) How many subsets of X contain x1 ?

(b) How many subsets of X contain x2 and x3 but do not contain x6 ?

Class Problems
Problem 15.5.
A license plate consists of either:

3 letters followed by 3 digits (standard plate)

5 letters (vanity plate)

2 characters—letters or numbers (big shot plate)

Let L be the set of all possible license plates.
(a) Express L in terms of

A D fA; B; C; : : : ; Zg
D D f0; 1; 2; : : : ; 9g

using unions ([) and set products ().

(b) Compute jLj, the number of different license plates, using the sum and product
rules.
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15.11. References 647

Problem 15.6. (a) How many of the billion numbers in the range from 1 to 109
contain the digit 1? (Hint: How many don’t?)

(b) There are 20 books arranged in a row on a shelf. Describe a bijection between
ways of choosing 6 of these books so that no two adjacent books are selected and
15-bit strings with exactly 6 ones.

Problem 15.7.

(a) Let Sn;k be the possible nonnegative integer solutions to the inequality

x1 C x2 C C xk n: (15.11)

That is
Sn;k WWD f.x1 ; x2 ; : : : ; xk / 2 Nk j (15.11) is trueg:
Describe a bijection between Sn;k and the set of binary strings with n zeroes and k
ones.

(b) Let Ln;k be the length k weakly increasing sequences of nonnegative integers
n. That is

Ln;k WWD f.y1 ; y2 ; : : : ; yk / 2 Nk j y1 y2 yk ng:

Describe a bijection between Ln;k and Sn;k .

Problem 15.8.
An n-vertex numbered tree is a tree whose vertex set is f1; 2; : : : ; ng for some
n > 2. We define the code of the numbered tree to be a sequence of n 2 integers
from 1 to n obtained by the following recursive process:6

If there are more than two vertices left, write down the father of the largest leaf,
delete this leaf, and continue this process on the resulting smaller tree. If there
are only two vertices left, then stop—the code is complete.

For example, the codes of a couple of numbered trees are shown in the Fig-
ure 15.7.
6 The necessarily unique node adjacent to a leaf is called its father.
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648 Chapter 15 Cardinality Rules

tree code
1 5 4

2 6 65622

3 7

1 2 3 4 5 432

Figure 15.7

(a) Describe a procedure for reconstructing a numbered tree from its code.

(b) Conclude there is a bijection between the n-vertex numbered trees and f1; : : : ; ngn 2,

and state how many n-vertex numbered trees there are.

Problem 15.9.
Let X and Y be finite sets.
(a) How many binary relations from X to Y are there?

(b) Define a bijection between the set ŒX ! Y of all total functions from X to
Y and the set Y jX j . (Recall Y n is the Cartesian product of Y with itself n times.)
Based on that, what is j ŒX ! Y j?

(c) Using the previous part, how many functions, not necessarily total, are there
from X to Y ? How does the fraction of functions vs. total functions grow as the
size of X grows? Is it O.1/, O.jXj/, O.2jXj /,. . . ?

(d) Show a bijection between the powerset pow.X / and the set ŒX ! f0; 1g of
0-1-valued total functions on X .

(e) Let X be a set of size n and BX be the set of all bijections from X to X.
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15.11. References 649

Describe a bijection from BX to the set of permutations of X .7 This implies that
there are how may bijections from X to X?

Problems for Section 15.4
Class Problems
Problem 15.10.
Use induction to prove that there are 2n subsets of an n-element set (Theorem 4.5.5).

Homework Problems
Problem 15.11.
Fermat’s Little Theorem 9.10.88 asserts that

ap a .mod p/ (15.12)

for all primes p and nonnegative integers a. This is immediate for a D 0; 1 so we
assume that a 2.
This problem offers a proof of (15.12) by counting strings over a fixed alphabet
with a characters.
(a) How many length-k strings are there over an a-character alphabet? How
many of these are strings use more than one character?

Let z be a length-k string. The length-n rotation of z is the string yx, where
z D xy and the length jxj of x is remainder.n; k/.
(b) Verify that if u is a length-n rotation of z, and v is a length-m rotation of u,
then v is a length-(n C m) rotation of z.

vz IFF v is a length-n rotation of z

for some n 2 N. Prove that is an equivalence relation.
7 A sequence in which all the elements of a set X appear exactly once is called a permutation of

X (see Section 15.3.3).
8 This Theorem is usually stated as

ap 1
1 .mod p/;

for all primes p and integers a not divisible by p. This follows immediately from (15.12) by canceling
a.
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650 Chapter 15 Cardinality Rules

(d) Prove that if xy D yx then x and y each consist of repetitions of some string
u. That is, if xy D yx, then x; y 2 u for some string u.
Hint: By induction on the length jxyj of xy.

(e) Conclude that if p is prime and z is a length-p string containing at least two
different characters, then z is equivalent under to exactly p strings (counting
itself).

(f) Conclude from parts (a) and (e) that p j .ap a), which proves Fermat’s Little
Theorem (15.12).

Problems for Section 15.5
Practice Problems
Problem 15.12.
Eight students—Anna, Brian, Caine,. . . —are to be seated around a circular table
in a circular room. Two seatings are regarded as defining the same arrangement if
each student has the same student on his or her right in both seatings: it does not
matter which way they face. We’ll be interested in counting how many arrange-
ments there are of these 8 students, given some restrictions.
(a) As a start, how many different arrangements of these 8 students around the
table are there without any restrictions?

(b) How many arrangements of these 8 students are there with Anna sitting next
to Brian?

(d) How many arrangements are there with Brian sitting next to Anna OR Caine?

Problem 15.13.
How many different ways are there to select three dozen colored roses if red, yellow,
pink, white, purple and orange roses are available?

Problem 15.14.
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15.11. References 651

Suppose n books are lined up on a shelf. The number of selections of m of the
books so that selected books are separated by at least three unselected books is the
same as the number of all length k binary strings with exactly m ones.
(a) What is the value of k?

(b) Describe a bijection between between the set of all length k binary strings with
exactly m ones and such book selections.

Problem 15.15.
Six women and nine men are on the faculty of a school’s EECS department. The
individuals are distinguishable. How many ways are there to select a committee of
5 members if at least 1 woman must be on the committee?

Class Problems
Problem 15.16.
Your class tutorial has 12 students, who are supposed to break up into 4 groups of
3 students each. Your Teaching Assistant (TA) has observed that the students waste
too much time trying to form balanced groups, so he decided to pre-assign students
to groups and email the group assignments to his students.
(a) Your TA has a list of the 12 students in front of him, so he divides the list into
consecutive groups of 3. For example, if the list is ABCDEFGHIJKL, the TA would
define a sequence of four groups to be .fA; B; C g; fD; E; F g; fG; H; I g; fJ; K; Lg/.
This way of forming groups defines a mapping from a list of twelve students to a
sequence of four groups. This is a k-to-1 mapping for what k?

(b) A group assignment specifies which students are in the same group, but not
any order in which the groups should be listed. If we map a sequence of 4 groups,

.fA; B; C g; fD; E; F g; fG; H; I g; fJ; K; Lg/;

into a group assignment

ffA; B; C g; fD; E; F g; fG; H; I g; fJ; K; Lgg;

this mapping is j -to-1 for what j ?

(d) In how many ways can 3n students be broken up into n groups of 3?
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652 Chapter 15 Cardinality Rules

Problem 15.17.
A pizza house is having a promotional sale. Their commercial reads:

We offer 9 different toppings for your pizza! Buy 3 large pizzas at
the regular price, and you can get each one with as many different
toppings as you wish, absolutely free. That’s 22; 369; 621 different
ways to choose your pizzas!

The ad writer was a former Harvard student who had evaluated the formula .29 /3 =3Š
on his calculator and gotten close to 22; 369; 621. Unfortunately, .29 /3 =3Š can’t be
an integer, so clearly something is wrong. What mistaken reasoning might have
led the ad writer to this formula? Explain how to fix the mistake and get a correct
formula.

Problem 15.18.
Answer the following quesions using the Generalized Product Rule.
(a) Next week, I’m going to get really fit! On day 1, I’ll exercise for 5 minutes.
On each subsequent day, I’ll exercise 0, 1, 2, or 3 minutes more than the previous
day. For example, the number of minutes that I exercise on the seven days of next
week might be 5, 6, 9, 9, 9, 11, 12. How many such sequences are possible?

(b) An r-permutation of a set is a sequence of r distinct elements of that set. For
example, here are all the 2-permutations of fa; b; c; d g:

.a; b/ .a; c/ .a; d /
.b; a/ .b; c/ .b; d /
.c; a/ .c; b/ .c; d /
.d; a/ .d; b/ .d; c/

How many r-permutations of an n-element set are there? Express your answer
using factorial notation.

Problem 15.19. (a) There are 30 books arranged in a row on a shelf. In how many
ways can eight of these books be selected so that there are at least two unselected
books between any two selected books?
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15.11. References 653

(b) How many nonnegative integer solutions are there for the following equality?

x1 C x2 C C xm D k: (15.13)

(d) How many length m weakly increasing sequences of nonnegative integers k
are there?

Homework Problems
Problem 15.20.
This problem is about binary relations on the set of integers in the interval Œ1::n
and digraphs whose vertex set is Œ1::n.
(a) How many digraphs are there?

(b) How many simple graphs are there?

(d) How many linear strict partial orders are there?

Problem 15.21.
Answer the following questions with a number or a simple formula involving fac-
torials and binomial coefficients. Briefly explain your answers.
(a) How many ways are there to order the 26 letters of the alphabet so that no two
of the vowels a, e, i, o, u appear consecutively and the last letter in the ordering
is not a vowel?
Hint: Every vowel appears to the left of a consonant.

(b) How many ways are there to order the 26 letters of the alphabet so that there
are at least two consonants immediately following each vowel?

(d) Two n-digit sequences of digits 0,1,. . . ,9 are said to be of the same type if the
digits of one are a permutation of the digits of the other. For n D 8, for example,
the sequences 03088929 and 00238899 are the same type. How many types of
n-digit sequences are there?
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654 Chapter 15 Cardinality Rules

Problem 15.22.
In a standard 52-card deck, each card has one of thirteen ranks in the set R and one
of four suits in the set S where

R WWD fA; 2; : : : ; 10; J; Q; Kg;
S WWD f|; }; ~; g:

A 5-card hand is a set of five distinct cards from the deck.
For each part describe a bijection between a set that can easily be counted using
the Product and Sum Rules of Ch. 15.1, and the set of hands matching the specifi-
cation. Give bijections, not numerical answers.
For instance, consider the set of 5-card hands containing all 4 suits. Each such
hand must have 2 cards of one suit. We can describe a bijection between such hands
and the set S R2 R3 where R2 is the set of two-element subsets of R. Namely,
an element
.s; fr1 ; r2 g; .r3 ; r4 ; r5 // 2 S R2 R3
indicates

1. the repeated suit s 2 S,

2. the set fr1 ; r2 g 2 R2 of ranks of the cards of suit s and

3. the ranks .r3 ; r4 ; r5 / of the remaining three cards, listed in increasing suit
order where
| } ~ :

For example,

.|; f10; Ag; .J; J; 2// ! fA|; 10|; J }; J ~; 2g:

(a) A single pair of the same rank (no 3-of-a-kind, 4-of-a-kind, or second pair).

(b) Three or more aces.

Problem 15.23.
Suppose you have seven dice—each a different color of the rainbow; otherwise
the dice are standard, with faces numbered 1 to 6. A roll is a sequence specify-
ing a value for each die in rainbow (ROYGBIV) order. For example, one roll is
.3; 1; 6; 1; 4; 5; 2/ indicating that the red die showed a 3, the orange die showed 1,
the yellow 6,. . . .
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15.11. References 655

For the problems below, describe a bijection between the specified set of rolls
and another set that is easily counted using the Product, Generalized Product, and
similar rules. Then write a simple arithmetic formula, possibly involving factorials
and binomial coefficients, for the size of the set of rolls. You do not need to prove
that the correspondence between sets you describe is a bijection, and you do not
need to simplify the expression you come up with.
For example, let A be the set of rolls where 4 dice come up showing the same
number, and the other 3 dice also come up the same, but with a different number.
Let R be the set of seven rainbow colors and S WWD Œ1; 6 be the set of dice values.
Define B WWD PS;2 R3 , where PS;2 is the set of 2-permutations of S and R3
is the set of size-3 subsets of R. Then define a bijection from A to B by mapping
a roll in A to the sequence in B whose first element is a pair consisting of the
number that came up three times followed by the number that came up four times,
and whose second element is the set of colors of the three matching dice.
For example, the roll
.4; 4; 2; 2; 4; 2; 4/ 2 A
maps to
..2; 4/; fyellow,green,indigog/ 2 B:
Now by the Bijection rule jAj D jBj, and by the Generalized Product and Subset
rules, !
7
jBj D 6 5 :
3
(a) For how many rolls do exactly two dice have the value 6 and the remaining
five dice all have different values? Remember to describe a bijection and write a
simple arithmetic formula.
Example: .6; 2; 6; 1; 3; 4; 5/ is a roll of this type, but .1; 1; 2; 6; 3; 4; 5/ and .6; 6; 1; 2; 4; 3; 4/
are not.

(b) For how many rolls do two dice have the same value and the remaining five
dice all have different values? Remember to describe a bijection and write a simple
arithmetic formula.
Example: .4; 2; 4; 1; 3; 6; 5/ is a roll of this type, but .1; 1; 2; 6; 1; 4; 5/ and .6; 6; 1; 2; 4; 3; 4/
are not.

(c) For how many rolls do two dice have one value, two different dice have a
second value, and the remaining three dice a third value? Remember to describe a
bijection and write a simple arithmetic formula.
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656 Chapter 15 Cardinality Rules

Example: .6; 1; 2; 1; 2; 6; 6/ is a roll of this type, but .4; 4; 4; 4; 1; 3; 5/ and .5; 5; 5; 6; 6; 1; 2/
are not.

Problem 15.24 (Counting trees).
What is the number Tn of different trees that can be formed from a set of n distinct
vertices?9 Cayley’s formula gives the answer

Tn D nn 2
:

One way to derive this appears in Problem 15.8. This and three additional deriva-
tions are given by Aigner & Ziegler (1998), who comment that “the most beautiful
of them all” is a counting argument due to Jim Pitman that we now describe.
Pitman’s derivation counts in two different ways the number of different se-
quences of edges that can be added to an empty graph on n vertices to form a
rooted tree. One way to form such a sequence is to start with one of the Tn possible
unrooted trees, choose one of its n vertices as root, and choose one of the .n 1/Š
possible sequences in which to add its n 1 edges. Therefore, the total number of
sequences that can be formed in this way is

Tn n.n 1/Š D Tn nŠ :

Another way to count these edge sequences is to start with the empty graph and
build up a spanning forest of rooted trees by adding edges in sequence. When
n k edges have been added, the graph with these edges will be a spanning forest
consisting of k rooted trees. To add the next edge, we choose any vertex to be the
root of a new tree. Then we add an edge between this new root and the root of any
one of the k 1 subtrees that did not include the chosen vertex. So the next edge
can be chosen in n.k 1/ ways to form a new spanning forest consisting of k 1
rooted trees.
Therefore, if one multiplies together the number of choices from the first step,
the second step, etc., the total number of choices is
n
Y
n.k 1/ D nn 1
.n 1/Š D nn 2
nŠ :
kD2

Equating these two formulas for the number of edge sequences, we get Tn nŠ D
nn 2 nŠ, and cancelling nŠ we arrive at Cayley’s formula

Tn D nn 2
:
9 From Double counting, wikipedia, Aug. 30, 2014. See also Prüfer Sequences
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15.11. References 657

Generalize Pitman’s derivation to count the number of spanning forests consist-
ing of k rooted trees on n vertices.

Exam Problems
Problem 15.25.
Suppose that two identical 52-card decks are mixed together. Write a simple for-
mula for the number of distinct permutations of the 104 cards.

Problems for Section 15.6
Class Problems
Problem 15.26.
The Tao of BOOKKEEPER: we seek enlightenment through contemplation of the
word BOOKKEEPER.
(a) In how many ways can you arrange the letters in the word POKE?

(b) In how many ways can you arrange the letters in the word BO1 O2 K? Observe
that we have subscripted the O’s to make them distinct symbols.

(c) Suppose we map arrangements of the letters in BO1 O2 K to arrangements
of the letters in BOOK by erasing the subscripts. Indicate with arrows how the
arrangements on the left are mapped to the arrangements on the right.

O2 BO1 K
KO2 BO1
BOOK
O1 BO2 K
OBOK
KO1 BO2
KOBO
BO1 O2 K
:::
BO2 O1 K
:::
(d) What kind of mapping is this, young grasshopper?

(e) In light of the Division Rule, how many arrangements are there of BOOK?

(f) Very good, young master! How many arrangements are there of the letters in
KE1 E2 PE3 R?
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658 Chapter 15 Cardinality Rules

(g) Suppose we map each arrangement of KE1 E2 PE3 R to an arrangement of
KEEPER by erasing subscripts. List all the different arrangements of KE1 E2 PE3 R
that are mapped to REPEEK in this way.

(h) What kind of mapping is this?

(i) So how many arrangements are there of the letters in KEEPER?
Now you are ready to face the BOOKKEEPER!
(j) How many arrangements of BO1 O2 K1 K2 E1 E2 PE3 R are there?

(k) How many arrangements of BOOK1 K2 E1 E2 PE3 R are there?

(l) How many arrangements of BOOKKE1 E2 PE3 R are there?

(m) How many arrangements of BOOKKEEPER are there?

Remember well what you have learned: subscripts on, subscripts off.
This is the Tao of Bookkeeper.

(n) How many arrangements of VOODOODOLL are there?

(o) How many length 52 sequences of digits contain exactly 17 two’s, 23 fives,
and 12 nines?

Problems for Section 15.6
Practice Problems
Problem 15.27.
How many different permutations are there of the sequence of letters in “MISSIS-
SIPPI”?

Class Problems
Problem 15.28.
Find the coefficients of
(a) x 5 in .1 C x/11

(b) x 8 y 9 in .3x C 2y/17

15.11. References 659

Problem 15.29.
Let p be a prime number.
(a) Explain why the multinomial coefficient
!
p
k1 ; k2 ; : : : ; kn

is divisible by p if all the ki ’s are nonnegative integers less than p.

(b) Conclude from part (a) that
p p
.x1 C x2 C C xn /p x1 C x2 C C xnp .mod p/: (15.15)

(Do not prove this using Fermat’s “little” Theorem. The point of this problem is to
offer an independent proof of Fermat’s theorem.)

np 1
1 .mod p/

when n is not a multiple of p.

Homework Problems
Problem 15.30.
The degree sequence of a simple graph is the weakly decreasing sequence of de-
grees of its vertices. For example, the degree sequence for the 5-vertex numbered
tree pictured in the Figure 15.7 in Problem 15.8 is .2; 2; 2; 1; 1/ and for the 7-vertex
tree it is .3; 3; 2; 1; 1; 1; 1/.
We’re interested in counting how many numbered trees there are with a given
degree sequence. We’ll do this using the bijection defined in Problem 15.8 between
n-vertex numbered trees and length n 2 code words whose characters are integers
between 1 and n.
The occurrence number for a character in a word is the number of times that
the character occurs in the word. For example, in the word 65622, the occurrence
number for 6 is two, and the occurrence number for 5 is one. The occurrence
sequence of a word is the weakly decreasing sequence of occurrence numbers of
characters in the word. The occurrence sequence for this word is .2; 2; 1/ because
it has two occurrences of each of the characters 6 and 2, and one occurrence of 5.
(a) There is a simple relationship between the degree sequence of an n-vertex
numbered tree and the occurrence sequence of its code. Describe this relationship
and explain why it holds. Conclude that counting n-vertex numbered trees with a
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660 Chapter 15 Cardinality Rules

given degree sequence is the same as counting the number of length n 2 code
words with a given occurrence sequence.
Hint: How many times does a vertex of degree d occur in the code?
For simplicity, let’s focus on counting 9-vertex numbered trees with a given de-
gree sequence. By part (a), this is the same as counting the number of length 7 code
words with a given occurrence sequence.
Any length 7 code word has a pattern, which is another length 7 word over the
alphabet a,b,c,d,e,f,g that has the same occurrence sequence.
(b) How many length 7 patterns are there with three occurrences of a, two occur-
rences of b, and one occurrence of c and d?
(c) How many ways are there to assign occurrence numbers to integers 1; 2; : : : ; 9
so that a code word with those occurrence numbers would have the occurrence
sequence 3; 2; 1; 1; 0; 0; 0; 0; 0?
In general, to find the pattern of a code word, list its characters in decreasing order
by number of occurrences, and list characters with the same number of occurrences
in decreasing order. Then replace successive characters in the list by successive
letters a,b,c,d,e,f,g. The code word 2468751, for example, has the pattern
fecabdg, which is obtained by replacing its characters 8,7,6,5,4,2,1 by
a,b,c,d,e,f,g, respectively. The code word 2449249 has pattern caabcab,
which is obtained by replacing its characters 4,9,2 by a,b,c, respectively.
(d) What length 7 code word has three occurrences of 7, two occurrences of 8,
one occurrence each of 2 and 9, and pattern abacbad?

(e) Explain why the number of 9-vertex numbered trees with degree sequence
.4; 3; 2; 2; 1; 1; 1; 1; 1/ is the product of the answers to parts (b) and (c).

Problem 15.31.
Let G be a simple graph with 6 vertices and an edge between every pair of vertices
(that is, G is a complete graph). A length-3 cycle in G is called a triangle.
A set of two edges that share a vertex is called an incident pair (i.p.); the shared
vertex is called the center of the i.p. That is, an i.p. is a set,

fhu—vi ; hv—wig;

where u; v and w are distinct vertices, and its center is v.
(a) How many triangles are there?

(b) How many incident pairs are there?
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15.11. References 661

Now suppose that every edge in G is colored either red or blue. A triangle or i.p.
is called multicolored when its edges are not all the same color.
(c) Map the i.p.
fhu—vi ; hv—wig
to the triangle
fhu—vi ; hv—wi ; hu—wig:
Notice that multicolored i.p.’s map to multicolored triangles. Explain why this
mapping is 2-to-1 on these multicolored objects.

(d) Show that at most six multicolored i.p.’s can have the same center. Conclude
that there are at most 36 possible multicolored i.p.’s.
Hint: A vertex incident to r red edges and b blue edges is the center of r b different
multicolored i.p.’s.

(e) If two people are not friends, they are called strangers. If every pair of people
in a group are friends, or if every pair are strangers, the group is called uniform.
Explain why parts (a), (c), and (d) imply that

Every set of six people includes two uniform three-person groups.

Exam Problems
Problem 15.32.
There is a robot that steps between integer positions in 3-dimensional space. Each
step of the robot increments one coordinate and leaves the other two unchanged.
(a) How many paths can the robot follow going from the origin .0; 0; 0/ to .3; 4; 5/?

(b) How many paths can the robot follow going from the origin .i; j; k/ to .m; n; p/?

Problems for Section 15.7
Practice Problems
Problem 15.33.
Indicate how many 5-card hands there are of each of the following kinds.
(a) A Sequence is a hand consisting of five consecutive cards of any suit, such as

5~ 6~ 7 8 9|:
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662 Chapter 15 Cardinality Rules

Note that an ace may either be high (as in 10-J-Q-K-A), or low (as in A-2-3-4-5),
but can’t go “around the corner” (that is, Q-K-A-2-3 is not a sequence).
How many different Sequence hands are possible?

(b) A Matching Suit is a hand consisting of cards that are all of the same suit in
any order.
How many different Matching Suit hands are possible?

(c) A Straight Flush is a hand that is both a Sequence and a Matching Suit.
How many different Straight Flush hands are possible?

(d) A Straight is a hand that is a Sequence but not a Matching Suit.
How many possible Straights are there?

(e) A Flush is a hand that is a Matching Suit but not a Sequence.
How many possible Flushes are there?

Class Problems
Problem 15.34.
Here are the solutions to the next 7 short answer questions, in no particular order.
Indicate the solutions for the questions and briefly explain your answers.
!
nŠ nCm
1: 2: 3: .n m/Š 4: mn
.n m/Š m
! !
n 1Cm n 1Cm
5: 6: 7: 2mn 8: nm
m n
(a) How many length m words can be formed from an n-letter alphabet, if no letter
is used more than once?

(b) How many length m words can be formed from an n-letter alphabet, if letters
can be reused?

(d) How many total injective functions are there from set A to set B, where jAj D
m and jBj D n m?
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15.11. References 663

(e) How many ways are there to place a total of m distinguishable balls into n
distinguishable urns, with some urns possibly empty or with several balls?

(f) How many ways are there to place a total of m indistinguishable balls into n
distinguishable urns, with some urns possibly empty or with several balls?

(g) How many ways are there to put a total of m distinguishable balls into n dis-
tinguishable urns with at most one ball in each urn?

Exam Problems
Problem 15.35. (a) How many solutions over the positive integers are there to the
inequality:

x1 C x2 C : : : C x10 100

(b) In how many ways can Mr. and Mrs. Grumperson distribute 13 identical
pieces of coal to their three children for Christmas so that each child gets at least
one piece?

Problem 15.36.
Answer the following questions about finite simple graphs. You may answer with
formulas involving exponents, binomial coefficents, and factorials.
(a) How many edges are there in the complete graph K41 ?

(b) How many edges are there in a spanning tree of K41 ?

(d) What is the chromatic number .C41 /, of the cycle of length 41?

(e) Let H be the graph in Figure 15.8. How many distinct isomorphisms are there
from H to H ?
a b

d c

Figure 15.8 The graph H .
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664 Chapter 15 Cardinality Rules

(f) A graph G is created by adding a single edge to a tree with 41 vertices.
How many cycles does G have?

(g) What is the smallest number of leaves possible in a tree with 41 vertices?

(h) What is the largest number of leaves possible in a tree with 41 vertices?

(i) How many length-10 paths are there in K41 ?

(j) Let s be the number of length-10 paths in K41 —that is, s is the correct answer
to part (i).
In terms of s, how many length-11 cycles are in K41 ?
Hint: For vertices a; b; c; d; e, the sequences abcde, bcdea and edcba all describe
the same length-5 cycle, for example.

Problems for Section 15.8
Practice Problems
Problem 15.37.
Below is a list of properties that a group of people might possess.
For each property, either give the minimum number of people that must be in a
group to ensure that the property holds, or else indicate that the property need not
hold even for arbitrarily large groups of people.
(Assume that every year has exactly 365 days; ignore leap years.)
(a) At least 2 people were born on the same day of the year (ignore year of birth).

(b) At least 2 people were born on January 1.

(d) At least 4 people were born in the same month.

(e) At least 2 people were born exactly one week apart.

Class Problems
Problem 15.38.
Solve the following problems using the pigeonhole principle. For each problem,
try to identify the pigeons, the pigeonholes, and a rule assigning each pigeon to a
pigeonhole.
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15.11. References 665

(a) In a certain Institute of Technology, every ID number starts with a 9. Suppose
that each of the 75 students in a class sums the nine digits of their ID number.
Explain why two people must arrive at the same sum.

(b) In every set of 100 integers, there exist two whose difference is a multiple of
37.

(d) Show that if n C 1 numbers are selected from f1; 2; 3; : : : ; 2ng, two must be
consecutive, that is, equal to k and k C 1 for some k.

Problem 15.39. (a) Prove that every positive integer divides a number such as 70,
700, 7770, 77000, whose decimal representation consists of one or more 7’s fol-
lowed by one or more 0’s.
Hint: 7; 77; 777; 7777; : : :

(b) Conclude that if a positive number is not divisible by 2 or 5, then it divides a
number whose decimal representation is all 7’s.

Problem 15.40.
The aim of this problem is to prove that there exist a natural number n such that 3n
has at least 2013 consecutive zeros in its decimal expansion.
(a) Prove that there exist a nonnegative integer n such that

3n 1pmod 102014 :

Hint: Use pigeonhole principle or Euler’s theorem.

(b) Conclude that there exist a natural number n such that 3n has at least 2013
consecutive zeros.

Problem 15.41. (a) Show that the Magician could not pull off the trick with a deck
larger than 124 cards.
Hint: Compare the number of 5-card hands in an n-card deck with the number of
4-card sequences.
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666 Chapter 15 Cardinality Rules

(b) Show that, in principle, the Magician could pull off the Card Trick with a deck
of 124 cards.
Hint: Hall’s Theorem and degree-constrained (12.5.5) graphs.

Problem 15.42.
The Magician can determine the 5th card in a poker hand when his Assisant reveals
the other 4 cards. Describe a similar method for determining 2 hidden cards in a
hand of 9 cards when your Assisant reveals the other 7 cards.

Problem 15.43.
Suppose 2n C 1 numbers are selected from f1; 2; 3; : : : ; 4ng. Using the Pigeonhole
Principle, show that there must be two selected numbers whose difference is 2.
Clearly indicate what are the pigeons, holes, and rules for assigning a pigeon to a
hole.

Problem 15.44.
Let
k1 ; k2 ; : : : ; k101
be a sequence of 101 integers. A sequence
kmC1 ; kmC2 ; : : : ; kn
where 0 m < n 101 is called a subsequence. Prove that there is a subsequence
whose elements sum to a number divisible by 100.

Homework Problems
Problem 15.45. (a) Show that any odd integer x in the range 109 < x < 2 109
containing all ten digits 0; 1; : : : ; 9 must have consecutive even digits. Hint: What
can you conclude about the parities of the first and last digit?

(b) Show that there are 2 vertices of equal degree in any finite undirected graph
with n 2 vertices. Hint: Cases conditioned upon the existence of a degree zero
vertex.

Problem 15.46.
Suppose n C 1 numbers are selected from f1; 2; 3; : : : ; 2ng. Using the Pigeonhole
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15.11. References 667

Principle, show that there must be two selected numbers whose quotient is a power
of two. Clearly indicate what are the pigeons, holes, and rules for assigning a
pigeon to a hole.
Hint: Factor each number into the product of an odd number and a power of 2.

Problem 15.47. (a) Let R be an 82 4 rectangular matrix each of whose entries
are colored red, white or blue. Explain why at least two of the 82 rows in R must
have identical color patterns.

(b) Conclude that R contains four points with the same color that form the corners
of a rectangle.

(c) Now show that the conclusion from part (b) holds even when R has only 19
rows.
Hint: How many ways are there to pick two positions in a row of length four and
color them the same?

Problem 15.48.
Section 15.8.6 explained why it is not possible to perform a four-card variant of the
hidden-card magic trick with one card hidden. But the Magician and her Assistant
are determined to find a way to make a trick like this work. They decide to change
the rules slightly: instead of the Assistant lining up the three unhidden cards for
the Magician to see, he will line up all four cards with one card face down and the
other three visible. We’ll call this the face-down four-card trick.
For example, suppose the audience members had selected the cards 9~, 10},
A|, 5|. Then the Assistant could choose to arrange the 4 cards in any order so
long as one is face down and the others are visible. Two possibilities are:

A| ? 10} 5|

? 5| 9~ 10}

(a) Explain how to model this face-down four-card trick as a matching problem,
and show that there must be a bipartite matching which theoretically will allow the
Magician and Assistant to perform the trick.
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668 Chapter 15 Cardinality Rules

(b) There is actually a simple way to perform the face-down four-card trick.10

Case 1. there are two cards with the same suit: Say there are two cards. The
Assistant proceeds as in the original card trick: he puts one of the cards face
up as the first card. He will place the second card face down. He then uses a
permutation of the face down card and the remaining two face up cards to code
the offset of the face down card from the first card.
Case 2. all four cards have different suits: Assign numbers 0; 1; 2; 3 to the four
suits in some agreed upon way. The Assistant computes s the sum modulo 4 of
the ranks of the four cards, and chooses the card with suit s to be placed face
down as the first card. He then uses a permutation of the remaining three face-up
cards to code the rank of the face down card.

Explain how in Case 2. the Magician can determine the face down card from the
cards the Assistant shows her.

(c) Explain how any method for performing the face-down four-card trick can be
adapted to perform the regular (5-card hand, show 4 cards) with a 52-card deck
consisting of the usual 52 cards along with a 53rd card called the joker.

Problem 15.49.
Suppose 2n C 1 numbers are selected from f1; 2; 3; : : : ; 4ng. Using the Pigeonhole
Principle, show that for any positive integer j that divides 2n, there must be two
selected numbers whose difference is j . Clearly indicate what are the pigeons,
holes, and rules for assigning a pigeon to a hole.

Problem 15.50.
Let’s start by
p marking a point on a circle of length one. Next, mark the point that
is distance 2 clockwise around
p the circle. So you wrap around once and actually
mark the point at distance 2 1 clockwise from the start. Now repeat with the
newly marked point as the starting point. In other words, the marked points are
those at clockwise distances
p p p p
0; 2; 2 2; 3 2; : : : ; n 2; : : : ;

from the start.
10 This elegant method was devised in Fall ’09 by student Katie E Everett.
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15.11. References 669

We will use a pigeonhole argument to prove that marked points are dense on the
circle: for any point p on the circle, and any > 0, there is a marked point within
distance of p.
(a)
p Prove that
p no point gets marked twice. That is, the points at clockwise distance
k 2 and m 2 are the same iff k D m.

(b) Prove that among the first n > 1 marked points, there have to be two that are
at most distance 1=n from each other.

(c) Prove that every point on the circle is within 1=n of a marked point. This
implies the claim that the marked points are dense on the circle.

Exam Problems
Problem 15.51.
A standard 52 card deck has 13 cards of each suit. Use the Pigeonhole Principle to
determine the smallest k such that every set of k cards from the deck contains five
cards of the same suit (called a flush). Clearly indicate what are the pigeons, holes,
and rules for assigning a pigeon to a hole.

Problem 15.52.
Use the Pigeonhole Principle to determine the smallest nonnegative integer n such
that every set of n integers is guaranteed to contain three integers that are congruent
mod 211. Clearly indicate what are the pigeons, holes, and rules for assigning a
pigeon to a hole, and give the value of n.

Problems for Section 15.9
Practice Problems
Problem 15.53.
Let A1 , A2 , A3 be sets with jA1 j D 100, jA2 j D 1; 000, and jA3 j D 10; 000.
Determine jA1 [ A2 [ A3 j in each of the following cases:
(a) A1 A2 A3 .

(b) The sets are pairwise disjoint.

670 Chapter 15 Cardinality Rules

(d) There are two elements common to each pair of sets and one element in all
three sets.

Problem 15.54.
The working days in the next year can be numbered 1, 2, 3, . . . , 300. I’d like to
avoid as many as possible.

On even-numbered days, I’ll say I’m sick.

On days that are a multiple of 3, I’ll say I was stuck in traffic.

On days that are a multiple of 5, I’ll refuse to come out from under the
blankets.

In total, how many work days will I avoid in the coming year?

ein

Problem 15.55.
Twenty people work at CantorCorp, a small, unsuccessful start-up. A single six-
person committee is to be formed. (It will be charged with the sole task of working
to prove the Continuum Hypothesis.) Employees appointed to serve on the com-
mittee join as equals—they do not get assigned distinct roles or ranks.
(a) Let D denote the set of all possible committees. Find jDj.

(b) Two of the workers, Aleph and Beth, will be unhappy if they are to serve
together.
Let P denote the set of all possible committees on which Aleph and Beth would
serve together. Find jP j.

(c) Beth will also be unhappy if she has to serve with both Ferdinand and Georg.
Let Q denote the set of all possible committees on which Beth, Ferdinand, and
Georg would all serve together. Find jQj.

(d) Find jP \ Qj.

(e) Let S denote the set of all possible committees on which there is at least one
unhappy employee. Express S in terms of P and Q only.

(f) Find jS j.
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15.11. References 671

(g) If we want to form a committee with no unhappy employees, how many
choices do we have to choose from?

(h) Suddenly, we realize that it would be better to have two six-person committees
instead of one. (One committee would work on proving the Continuum Hypothesis,
while the other would work to disprove it!) Each employee can serve on at most
one committee. How many ways are there to form such a pair of committees, if
employee happiness is not taken into consideration?

Class Problems
Problem 15.56.
To ensure password security, a company requires their employees to choose a pass-
word. A length 10 word containing each of the characters:
a, d, e, f, i, l, o, p, r, s,
is called a cword. A password can be a cword which does not contain any of the
subwords “fails”, “failed”, or “drop.”
For example, the following two words are passwords: adefiloprs, srpolifeda,
but the following three cwords are not: adropeflis, failedrops, dropefails.
(a) How many cwords contain the subword “drop”?

(b) How many cwords contain both “drop” and “fails”?

(c) Use the Inclusion-Exclusion Principle to find a simple arithmetic formula in-
volving factorials for the number of passwords.

Problem 15.57.
We want to count step-by-step paths between points in the plane with integer coor-
dinates. Only two kinds of step are allowed: a right-step which increments the x
coordinate, and an up-step which increments the y coordinate.

(a) How many paths are there from .0; 0/ to .20; 30/?

(b) How many paths are there from .0; 0/ to .20; 30/ that go through the point
.10; 10/?

(c) How many paths are there from .0; 0/ to .20; 30/ that do not go through either
of the points .10; 10/ and .15; 20/?
Hint: Let P be the set of paths from .0; 0/ to .20; 30/, N1 be the paths in P that go
through .10; 10/ and N2 be the paths in P that go through .15; 20/.
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672 Chapter 15 Cardinality Rules

Problem 15.58.
Let’s develop a proof of the Inclusion-Exclusion formula using high school algebra.
(a) Most high school students will get freaked by the following formula, even
though they actually know the rule it expresses. How would you explain it to them?

n
Y X Y
.1 xi / D . 1/jI j xj : (15.16)
i D1 I f1;:::;ng j 2I

Hint: Show them an example.

Now to start proving (15.16), let MS be the membership function for any set S :
(
1 if x 2 S ;
MS .x/ D
0 if x … S :

Let S1 ; : : : ; Sn be a sequence of finite sets, and abbreviate MSi as Mi . Let the
domain of discourse D be the union of the Si ’s. That is, we let
n
[
D WWD Si ;
i D1

and take complements with respect to D, that is,

T WWD D T;

for T D.
(b) Verify that for T D and I f1; : : : ng,

MT D 1 MT ; (15.17)
Y
M.Ti2I Si / D Mi ; (15.18)
i 2I
Y
M.Si2I Si / D 1 .1 Mi /: (15.19)
i 2I

(Note that (15.18) holds when I is empty because, by convention, an empty product
equals 1, and an empty intersection equals the domain of discourse D.)

(c) Use (15.16) and (15.19) to prove
X Y
MD D . 1/jI jC1 Mj : (15.20)
;¤I f1;:::;ng j 2I
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15.11. References 673

(d) Prove that X
jT j D MT .u/: (15.21)
u2D

(e) Now use the previous parts to prove
ˇ ˇ
X ˇ\ ˇ
jI jC1 ˇ
jDj D . 1/ ˇ Si ˇ (15.22)
ˇ
ˇ ˇ
;¤I f1;:::;ng i 2I

(f) Finally, explain why (15.22) immediately implies the usual form of the Inclusion-
Exclusion Principle:
ˇ ˇ
n
X X ˇ\ ˇ
. 1/i C1
ˇ ˇ
jDj D ˇ
ˇ Sj ˇˇ : (15.23)
i D1 I f1;:::;ng ˇj 2I ˇ
jI jDi

Homework Problems
Problem 15.59.
A derangement is a permutation .x1 ; x2 ; : : : ; xn / of the set f1; 2; : : : ; ng such that
xi ¤ i for all i. For example, .2; 3; 4; 5; 1/ is a derangement, but .2; 1; 3; 5; 4/
is not because 3 appears in the third position. The objective of this problem is to
count derangements.
It turns out to be easier to start by counting the permutations that are not de-
rangements. Let Si be the set of all permutations .x1 ; x2 ; : : : ; xn / that are not
derangements because xi D i. So the set of non-derangements is
n
[
Si :
i D1

(a) What is jSi j?
ˇ ˇ
(b) What is ˇSi \ Sj ˇ where i ¤ j ?
ˇ ˇ
(c) What is ˇSi1 \ Si2 \ \ Sik ˇ where i1 ; i2 ; : : : ; ik are all distinct?

(d) Use the inclusion-exclusion formula to express the number of non-derangements
in terms of sizes of possible intersections of the sets S1 ; : : : ; Sn .

(e) How many terms in the expression in part (d) have the form
ˇ ˇ
ˇSi \ Si \ \ Si ˇ‹
1 2 k
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674 Chapter 15 Cardinality Rules

(f) Combine your answers to the preceding parts to prove the number of non-
derangements is:
1 1 1 1
nŠ C ˙ :
1Š 2Š 3Š nŠ
Conclude that the number of derangements is

1 1 1 1
nŠ 1 C C ˙ :
1Š 2Š 3Š nŠ
(g) As n goes to infinity, the number of derangements approaches a constant frac-
tion of all permutations. What is that constant? Hint:
x2 x3
ex D 1 C x C C C
2Š 3Š

Problem 15.60.
How many of the numbers 2; : : : ; n are prime? The Inclusion-Exclusion Principle
offers a useful way to calculate the answer when n is large. Actually, we will use
Inclusion-Exclusion to count the number of composite (nonprime) integers from 2
to n. Subtracting this from n 1 gives the number of primes.
Let Cn be the set of composites from 2 to n, and let Am be the set of numbers in
the range m C 1; : : : ; n that are divisible by m. Notice that by definition, Am D ;
for m n. So
n[1
Cn D Ai : (15.24)
i D2
(a) Verify that if m j k, then Am Ak .

(b) Explain why the right-hand side of (15.24) equals
[
Ap : (15.25)
p
primes p n

(d) Consider any two relatively prime numbers p; q n. What is the one number
in .Ap \ Aq / Apq ?

(e) Let P be a finite set of at least two primes. Give a simple formula for
ˇ ˇ
ˇ ˇ
ˇ \ ˇ
ˇ
ˇ Ap ˇˇ :
ˇp2P ˇ
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15.11. References 675

(f) Use the Inclusion-Exclusion principle to obtain a formula for jC150 j in terms
the sizes of intersections among the sets A2 ; A3 ; A5 ; A7 ; A11 . (Omit the intersec-
tions that are empty; for example, any intersection of more than three of these sets
must be empty.)

(g) Use this formula to find the number of primes up to 150.

Exam Problems
Problem 15.61.
We want to count the number of length-n binary strings in which the substring 011
occurs in various places. For example, the length-14 string
00100110011101;
has 011 in the 4th position and the 8th position. (Note that by convention, a length-
n string starts with position zero and ends with position n 1.) Assume n 7:
(a) Let r be the number of length-n binary strings in which 011 occurs starting at
the 4th position. Write a formula for r in terms of n.

(b) Let Ai be the set of length-n binary strings in which 011 occurs starting at
the i th position. (Ai is empty for i > n 3.) If i ¤ j , the intersection Ai \ Aj is
either empty or of size s. Write a formula for s in terms of n.

(c) Let t be the number of pairs .i; j / such that Ai \ Aj is nonempty, where
0 i < j . Write a binomial coefficient for t in terms of n.

(d) How many length 9 binary strings are there that contain the substring 011?
You should express your answer as an integer or as a simple expression which may
include the above constants r, s and t for n D 9.
ˇS ˇ
Hint: Inclusion-exclusion for ˇ 80 Ai ˇ.
ˇ ˇ

Problem 15.62.
There are 10 students A; B; : : : ; J who will be lined up left to right according to
the some rules below.
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676 Chapter 15 Cardinality Rules

Rule I: Student A must not be rightmost.
Rule II: Student B must be adjacent to C (directly to the left or right of C).
Rule III: Student D is always second.
You may answer the following questions with a numerical formula that may
involve factorials.
(a) How many possible lineups are there that satisfy all three of these rules?

(b) How many possible lineups are there that satisfy at least one of these rules?

Problem 15.63.
A robot on a point in the 3-D integer lattice can move a unit distance in one positive
direction at a time. That is, from position .x; y; z/, it can move to either .x C
1; y; z/, .x; y C 1; z/ or .x; y; z C 1/. For any two points P and Q in space, let
n.P; Q/ denote the number of distinct paths the spacecraft can follow to go from
P to Q.
Let

A D .0; 10; 20/; B D .30; 50; 70/; C D .80; 90; 100/; D D .200; 300; 400/:

(a) Express n.A; B/ as a single multinomial coefficient.
Answer the following questions with arithmetic expressions involving terms n.P; Q/
for P; Q 2 fA; B; C; Dg. Do not use numbers.
(b) How many paths from A to C go through B?

(d) How many paths from A to D go through neither B nor C ?

Problem 15.64.
In a standard 52-card deck (13 ranks and 4 suits), a hand is a 5-card subset of the set
of 52 cards. Express the answer to each part as a formula using factorial, binomial,
or multinomial notation.
(a) Let H be the set of all hands. What is jH j?

(b) Let HNP be the set of all hands that include no pairs; that is, no two cards in
the hand have the same rank. What is jHNP j?
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15.11. References 677

(c) Let HS be the set of all hands that are straights, that is, the ranks of the five
cards are consecutive. The order of the ranks is .A; 2; 3; 4; 5; 6; 7; 8; 9; 10; J; Q; K; A/;
note that A appears twice.
What is jHS j?

(d) Let HF be the set of all hands that are flushes, that is, the suits of the five cards
are identical. What is jHF j?

(e) Let HSF be the set of all straight flush hands, that is, the hand is both a straight
and a flush. What is jHSF j?

(f) Let HH C be the set of all high-card hands; that is, hands that do not include
pairs, are not straights, and are not flushes. Write a formula for jHH C j in terms of
jHNP j; jHS j; jHF j; jHSF j.

Problems for Section 15.10
Practice Problems
Problem 15.65.
Prove the following identity by algebraic manipulation and by giving a combinato-
rial argument: ! ! ! !
n r n n k
D
r k k r k

Problem 15.66.
Give a combinatorial proof for this identity:
!
X n
D 3n
i; j; k
i Cj CkDn
i;j;k0

Class Problems
Problem 15.67.
According to the Multinomial theorem, .w C x C y C z/n can be expressed as a
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678 Chapter 15 Cardinality Rules

sum of terms of the form
!
n
w r1 x r2 y r3 z r4 :
r1 ; r2 ; r3 ; r4

(a) How many terms are there in the sum?

(b) The sum of these multinomial coefficients has an easily expressed value. What
is it? !
X n
D‹ (15.26)
r1 ; r2 ; r3 ; r4
r1 Cr2 Cr3 Cr4 Dn;
ri 2N

Hint: How many terms are there when .w C x C y C z/n is expressed as a sum
of monomials in w; x; y; z before terms with like powers of these variables are
collected together under a single coefficient?

Problem 15.68.

(a) Give a combinatorial proof of the following identity by letting S be the set of
all length-n sequences of letters a, b and a single c and counting jSj is two different
ways.

n
!
X n
n2n 1
D k (15.27)
k
kD1

(b) Now prove (15.27) algebraically by applying the Binomial Theorem to .1 C
x/n and taking derivatives.

Problem 15.69.
What do the following expressions equal? Give both algebraic and combinatorial
proofs for your answers.
(a)
n
!
X n
i
i D0
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15.11. References 679

(b)
n
!
X n
. 1/i
i
i D0

Hint: Consider the bit strings with an even number of ones and an odd number of
ones.

Problem 15.70.
When an integer k occurs as the kth element of a sequence, we’ll say it is “in place”
in the sequence. For example, in the sequence

12453678

precisely the integers 1; 2; 6; 7 and 8 occur in place. We’re going to classify the
sequences of distinct integers from 1 to n, that is the permutations of Œ1::n, accord-
ing to which integers do not occur “in place.” Then we’ll use this classification to
prove the combinatorial identity11
n
X
nŠ D 1 C .k 1/ .k 1/Š : (15.28)
kD1

If is a permutation of Œ1::n, let mnp ./ be the maximum integer in Œ1::n that
does not occur in place in . For example, for n D 8,

mnp .12345687/ D 8;
mnp .21345678/ D 2;
mnp .23145678/ D 3:

(a) For how many permutations of Œ1::n is every element in place?

(b) How many permutations of Œ1::n have mnp ./ D 1?

(d) Conclude the equation (15.28).

11 Thisproblem is based on “Use of everywhere divergent generating function,” mathoverflow,
response 8,147 by Aaron Meyerowitz, Nov. 12, 2010.
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680 Chapter 15 Cardinality Rules

Problem 15.71.
Each day, an MIT student selects a breakfast from among b possibilities, lunch
from among l possibilities, and dinner from among d possibilities. In each case
one of the possibilities is Doritos. However, a legimate daily menu may include
Doritos for at most one meal. Give a combinatorial (not algebraic) proof based on
the number of legimate daily menus that

bld Œ.b 1/ C .l 1/ C .d 1/ C 1
D b.l 1/.d 1/ C .b 1/l.d 1/ C .b 1/.l 1/d
3.b 1/.l 1/.d 1/ C .b 1/.l 1/.d 1/

Hint: Let Mb be the number of menus where, if Doritos appear at all, they only
appear at breakfast; likewise, for Ml ; Md .

Homework Problems
Problem 15.72. (a) Find a combinatorial (not algebraic) proof that
n
!
X n
D 2n :
i
i D0

(b) Below is a combinatorial proof of an equation. What is the equation?

Proof. Stinky Peterson owns n newts, t toads, and s slugs. Conveniently, he lives
in a dorm with n C t C s other students. (The students are distinguishable, but
creatures of the same variety are not distinguishable.) Stinky wants to put one
creature in each neighbor’s bed. Let W be the set of all ways in which this can be
done.
On one hand, he could first determine who gets the slugs. Then, he could decide
who among his remaining neighbors has earned a toad. Therefore, jW j is equal to
the expression on the left.
On the other hand, Stinky could first decide which people deserve newts and slugs
and then, from among those, determine who truly merits a newt. This shows that
jW j is equal to the expression on the right.
Since both expressions are equal to jW j, they must be equal to each other.

(Combinatorial proofs are real proofs. They are not only rigorous, but also con-
vey an intuitive understanding that a purely algebraic argument might not reveal.
However, combinatorial proofs are usually less colorful than this one.)
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15.11. References 681

Problem 15.73.
Give a combinatorial proof for this identity:
n
! !
X kCi kCnC1
D
k kC1
i D0

Hint: Let Si be the set of binary sequences with exactly n zeroes, k C1 ones, and
a total of exactly i occurrences of zeroes appearing before the rightmost occurrence
of a one.

Problem 15.74.
According to the Multinomial Theorem 15.6.5, .x1 C x2 C C xk /n can be
expressed as a sum of terms of the form
!
n
x r1 x r2 : : : xkrk :
r1 ; r2 ; : : : ; rk 1 2

(a) How many terms are there in the sum?

(b) The sum of these multinomial coefficients has an easily expressed value:
!
X n
D kn (15.29)
r1 ; r2 ; : : : ; rk
r1 Cr2 CCrk Dn;
ri 2N

Give a combinatorial proof of this identity.
Hint: How many terms are there when .x1 C x2 C C xk /n is expressed as a sum
of monomials in xi before terms with like powers of these variables are collected
together under a single coefficient?

Problem 15.75.
You want to choose a team of m people for your startup company from a pool of n
applicants, and from these m people you want to choose k to be the team managers.
You took a Math for Computer Science subject, so you know you can do this in
! !
n m
m k
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682 Chapter 15 Cardinality Rules

ways. But your CFO, who went to Harvard Business School, comes up with the
formula ! !
n n k
:
k m k
Before doing the reasonable thing—dump on your CFO or Harvard Business School—
you decide to check his answer against yours.
(a) Give a combinatorial proof that your CFO’s formula agrees with yours.

(b) Verify this combinatorial proof by giving an algebraic proof of this same fact.

Exam Problems
Problem 15.76.
Each day, an MIT student selects a breakfast from among b possibilities, lunch
from among l possibilities, and dinner from among d possibilities. In each case
one of the possibilities is Doritos. However, a legimate daily menu may include
Doritos for at most one meal. Give a combinatorial (not algebraic) proof based on
the number of legimate daily menus that

bld Œ.b 1/ C .l 1/ C .d 1/ C 1
D .l 1/.d 1/ C .b 1/.d 1/ C .b 1/.l 1/ C .b 1/.l 1/.d 1/:

Problem 15.77.
Give a combinatorial proof of
!
nC1
1 2 C 2 3 C 3 4 C C .n 1/ n D 2
3

Hint: Classify sets of three numbers from the integer interval Œ0::n by their
maximum element.
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16 Generating Functions
Generating Functions are one of the most surprising and useful inventions in Dis-
crete Mathematics. Roughly speaking, generating functions transform problems
about sequences into problems about algebra. This is great because we’ve got piles
of algebraic rules. Thanks to generating functions, we can reduce problems about
sequences to checking properties of algebraic expressions. This will allow us to
use generating functions to solve all sorts of counting problems.
Several flavors of generating functions such as ordinary, exponential, and Dirich-
let come up regularly in combinatorial mathematics. In addition, Z-transforms,
which are closely related to ordinary generating functions, are important in control
theory and signal processing. But ordinary generating functions are enough to il-
lustrate the power of the idea, so we’ll stick to them. So from now on generating
function will mean the ordinary kind, and we will offer a taste of this large subject
by showing how generating functions can be used to solve certain kinds of count-
ing problems and how they can be used to find simple formulas for linear-recursive
functions.

16.1 Infinite Series
Informally, a generating function F .x/ is an infinite series

F .x/ D f0 C f1 x C f2 x 2 C f3 x 3 C : (16.1)

We use the notation Œx n F .x/ for the coefficient of x n in the generating function
F .x/. That is, Œx n F .x/ WWD fn .
We can analyze the behavior of any sequence of numbers f0 ; f1 : : : fn : : : by
regarding the elements of the sequence as successive coefficients of a generating
function. It turns out that properties of complicated sequences that arise from
counting, recursive definition, and programming problems are easy to explain by
treating them as generating functions.
Generating functions can produce noteworthy insights even when the sequence
of coefficients is trivial. For example, let G.x/ be the generating function for the
infinite sequence of ones 1; 1; : : : , namely, the geometric series.

G.x/ WWD 1 C x C x 2 C C x n C : (16.2)
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684 Chapter 16 Generating Functions

We’ll use typical generating function reasoning to derive a simple formula for
G.x/. The approach is actually an easy version of the perturbation method of
Section 14.1.2. Specifically,
G.x/ D 1 C x C x 2 C x 3 C C x n C
xG.x/ D x x2 x3 xn
G.x/ xG.x/ D 1:
Solving for G.x/ gives
1
1 X
D G.x/ WWD xn: (16.3)
1 x
nD0
In other words,
n 1
Œx D1
1 x
Continuing with this approach yields a nice formula for
N.x/ WWD 1 C 2x C 3x 2 C C .n C 1/x n C : (16.4)
Specifically,
N.x/ D 1 C 2x C 3x 2 C 4x 3 C C .n C 1/x n C
xN.x/ D x 2x 2 3x 3 nx n
N.x/ 2 3
xN.x/ D 1 C x C x C x C C x n C
D G.x/:
Solving for N.x/ gives
1
1 G.x/ X
D D N.x/ WWD .n C 1/x n : (16.5)
.1 x/2 1 x
nD0
On other words,
n 1
Œx D n C 1:
.1 x/2

16.1.1 Never Mind Convergence
Equations (16.3) and (16.5) hold numerically only when jxj < 1, because both
generating function series diverge when jxj 1. But in the context of generat-
ing functions, we regard infinite series as formal algebraic objects. Equations such
as (16.3) and (16.5) define symbolic identities that hold for purely algebraic rea-
sons. In fact, good use can be made of generating functions determined by infinite
series that don’t converge anywhere (besides x D 0). We’ll explain this further in
Section 16.5 at the end of this chapter, but for now, take it on faith that you don’t
need to worry about convergence.
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16.2. Counting with Generating Functions 685

16.2 Counting with Generating Functions
Generating functions are particularly useful for representing and counting the num-
ber of ways to select n things. For example, suppose there are two flavors of donuts,
chocolate and plain. Let dn be the number of ways to select n chocolate or plain
flavored donuts. dn D n C 1, because there are n C 1 such donut selections—all
chocolate, 1 plain and n 1 chocolate, 2 plain and n 2 chocolate,. . . , all plain.
We define a generating function D.x/ for counting these donut selections by letting
the coefficient of x n be dn . This gives us equation (16.5)
1
D.x/ D : (16.6)
.1 x/2

16.2.1 Apples and Bananas too
More generally, suppose we have two kinds of things—say, apples and bananas—
and some constraints on how many of each may be selected. Say there are an ways
to select n apples and bn ways to select n bananas. The generating function for
counting apples would be
X1
A.x/ WWD an x n ;
nD0
and for bananas would be
1
X
B.x/ WWD bn x n :
nD0
Now suppose apples come in baskets of 6, so there is no way to select 1 to 5
apples, one way to select 6 apples, no way to select 7, etc. In other words,
(
1 if n is a multiple of 6;
an D
0 otherwise:

In this case we would have

A.x/ D 1 C x 6 C x 12 C C x 6n C
D 1 C y C y2 C C yn C where y D x 6 ;
1 1
D D :
1 y 1 x6
Let’s also suppose there are two kinds of bananas—red and yellow. Now, bn D
n C 1 by the same reasoning used to count selections of n chocolate and plain
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686 Chapter 16 Generating Functions

donuts, and by (16.6) we have
1
B.x/ D :
.1 x/2
So how many ways are there to select a mix of n apples and bananas? First, we
decide how many apples to select. This can be any number k from 0 to n. We can
then select these apples in ak ways, by definition. This leaves n k bananas to
be selected, which by definition can be done in bn k ways. So the total number
of ways to select k apples and n k bananas is ak bn k . This means that the total
number of ways to select some size n mix of apples and bananas is

a0 bn C a1 bn 1 C a2 bn 2 C C an b0 : (16.7)

16.2.2 Products of Generating Functions
Now here’s the cool connection between counting and generating functions: ex-
pression (16.7) is equal to the coefficient of x n in the product A.x/B.x/.
In other words, we’re claiming that

Rule (Product).

Œx n .A.x/ B.x// D a0 bn C a1 bn 1 C a2 bn 2 C C an b0 : (16.8)

To explain the generating function Product Rule, we can think about evaluating
the product A.x/ B.x/ by using a table to identify all the cross-terms from the
product of the sums:

b0 x 0 b1 x 1 b2 x 2 b3 x 3 :::

a0 x 0 a0 b0 x 0 a0 b1 x 1 a0 b2 x 2 a0 b3 x 3 :::

a1 x 1 a1 b0 x 1 a1 b1 x 2 a1 b2 x 3 :::

a2 x 2 a2 b0 x 2 a2 b1 x 3 :::

a3 x 3 a3 b0 x 3 :::

::
: :::

In this layout, all the terms involving the same power of x lie on a 45-degree sloped
diagonal. So, the index-n diagonal contains all the x n -terms, and the coefficient of
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16.2. Counting with Generating Functions 687

x n in the product A.x/ B.x/ is the sum of all the coefficients of the terms on this
diagonal, namely, (16.7). The sequence of coefficients of the product A.x/ B.x/
is called the convolution of the sequences .a0 ; a1 ; a2 ; : : : / and .b0 ; b1 ; b2 ; : : : /. In
addition to their algebraic role, convolutions of sequences play a prominent role in
signal processing and control theory.
This Product Rule provides the algebraic justification for the fact that a geometric
series equals 1=.1 x/ regardless of convergence. Specifically, the constant 1
describes the generating function

1 D 1 C 0x C 0x 2 C C 0x n C :

Likewise, the expression 1 x describes the generating function

1 x D 1 C . 1/x C 0x 2 C C 0x n C :

So for the series G.x/ whose coefficients are all equal to 1, the Product Rule implies
in a purely formal way that

.1 x/ G.x/ D 1 C 0x C 0x 2 C C 0x n C D 1:

In other words, under the Product Rule, the geometric series G.x/ is the multiplica-
tive inverse 1=.1 x/ of 1 x.
Similar reasoning justifies multiplying a generating function by a constant term
by term. That is, a special case of the Product Rule is the
Rule (Constant Factor). For any constant c and generating function F .x/

Œx n .c F .x// D c Œx n F .x/: (16.9)

16.2.3 The Convolution Rule
We can summarize the discussion above with the
Rule (Convolution). Let A.x/ be the generating function for selecting items from
a set A, and let B.x/ be the generating function for selecting items from a set B
disjoint from A. The generating function for selecting items from the union A [ B
is the product A.x/ B.x/.
The Rule depends on a precise definition of what “selecting items from the union
A [ B” means. Informally, the idea is that the restrictions on the selection of items
from sets A and B carry over to selecting items from A [ B.1
1 Formally, the Convolution Rule applies when there is a bijection between n-element selections

from A [ B and ordered pairs of selections from the sets A and B containing a total of n elements.
We think the informal statement is clear enough.
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688 Chapter 16 Generating Functions

16.2.4 Counting Donuts with the Convolution Rule
We can use the Convolution Rule to derive in another way the generating function
D.x/ for the number of ways to select chocolate and plain donuts given in (16.6).
To begin, there is only one way to select exactly n chocolate donuts. That means
every coefficient of the generating function for selecting n chocolate donuts equals
one. So the generating function for chocolate donut selections is 1=.1 x/; likewise
for the generating function for selecting only plain donuts. Now by the Convolution
Rule, the generating function for the number of ways to select n donuts when both
chocolate and plain flavors are available is
1 1 1
D.x/ D D :
1 x 1 x .1 x/2
So we have derived (16.6) without appeal to (16.5).
Our application of the Convolution Rule for two flavors carries right over to the
general case of k flavors; the generating function for selections of donuts when k
flavors are available is 1=.1 x/k . We already derived the formula for the number
of ways to select a n donuts when k flavors are available, namely, nC.kn 1/ from

Corollary 15.5.3. So we have
!
1 n C .k 1/
Œx n D : (16.10)
.1 x/k n

Extracting Coefficients from Maclaurin’s Theorem
We’ve used a donut-counting argument to derive the coefficients of 1=.1 x/k ,
but it’s instructive to derive this coefficient algebraically, which we can do using
Maclaurin’s Theorem:

Theorem 16.2.1 (Maclaurin’s Theorem).

f 00 .0/ 2 f 000 .0/ 3 f .n/ .0/ n
f .x/ D f .0/ C f 0 .0/x C x C x C C x C :
2Š 3Š nŠ

This theorem says that the nth coefficient of 1=.1 x/k is equal to its nth deriva-
tive evaluated at 0 and divided by nŠ. Computing the nth derivative turns out not to
be very difficult

dn 1 .kCn/
n
D k.k C 1/ .k C n 1/.1 x/
d x .1 x/k
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16.2. Counting with Generating Functions 689

(see Problem 16.5), so
n
n 1 d 1 1
Œx k
D n k
.0/
.1 x/ d x .1 x/ nŠ
k.k C 1/ .k C n 1/.1 0/ .kCn/
D
! nŠ
n C .k 1/
D :
n

In other words, instead of using the donut-counting formula (16.10) to find the
coefficients of x n , we could have used this algebraic argument and the Convolution
Rule to derive the donut-counting formula.

16.2.5 The Binomial Theorem from the Convolution Rule
The Convolution Rule also provides a new perspective on the Binomial Theo-
rem 15.6.4. Here’s how: first, work with the single-element set fa1 g. The gen-
erating function for the number of ways to select n different elements from this set
is simply 1 C x: we have 1 way to select zero elements, 1 way to select the one
element, and 0 ways to select more than one element. Similarly, the number of
ways to select n elements from any single-element set fai g has the same generating
function 1 C x. Now by the Convolution Rule, the generating function for choosing
a subset of n elements from the set fa1 ; a2 ; : : : ; am g is the product .1 C x/m of the
generating functions for selecting from each of the m one-element sets. Since we
know that the number of ways to select n elements from a set of size m is m n , we
conclude that that !
m
Œx n .1 C x/m D ;
n
which is a restatement of the Binomial Theorem 15.6.4. Thus, we have proved
the Binomial Theorem without having to analyze the expansion of the expression
.1 C x/m into a sum of products.
These examples of counting donuts and deriving the binomial coefficients illus-
trate where generating functions get their power:

Generating functions can allow counting problems to be solved by algebraic
manipulation, and conversely, they can allow algebraic identities to be derived by
counting techniques.
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690 Chapter 16 Generating Functions

16.2.6 An Absurd Counting Problem
So far everything we’ve done with generating functions we could have done another
way. But here is an absurd counting problem—really over the top! In how many
ways can we fill a bag with n fruits subject to the following constraints?

The number of apples must be even.
The number of bananas must be a multiple of 5.
There can be at most four oranges.
There can be at most one pear.

For example, there are 7 ways to form a bag with 6 fruits:
Apples 6 4 4 2 2 0 0
Bananas 0 0 0 0 0 5 5
Oranges 0 2 1 4 3 1 0
Pears 0 0 1 0 1 0 1
These constraints are so complicated that getting a nice answer may seem impossi-
ble. But let’s see what generating functions reveal.
First, we’ll construct a generating function for choosing apples. We can choose a
set of 0 apples in one way, a set of 1 apple in zero ways (since the number of apples
must be even), a set of 2 apples in one way, a set of 3 apples in zero ways, and so
forth. So, we have:
1
A.x/ D 1 C x 2 C x 4 C x 6 C D
1 x2
Similarly, the generating function for choosing bananas is:
1
B.x/ D 1 C x 5 C x 10 C x 15 C D
1 x5
Now, we can choose a set of 0 oranges in one way, a set of 1 orange in one way,
and so on. However, we cannot choose more than four oranges, so we have the
generating function:
1 x5
O.x/ D 1 C x C x 2 C x 3 C x 4 D :
1 x
Here the right-hand expression is simply the formula (14.2) for a finite geometric
sum. Finally, we can choose only zero or one pear, so we have:

P .x/ D 1 C x
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16.3. Partial Fractions 691

The Convolution Rule says that the generating function for choosing from among
all four kinds of fruit is:
1 1 1 x5
A.x/B.x/O.x/P .x/ D .1 C x/
1 x2 1 x5 1 x
1
D
.1 x/2
D 1 C 2x C 3x 2 C 4x 3 C
Almost everything cancels! We’re left with 1=.1 x/2 , which we found a power
series for earlier: the coefficient of x n is simply n C 1. Thus, the number of ways to
form a bag of n fruits is just n C 1. This is consistent with the example we worked
out, since there were 7 different fruit bags containing 6 fruits. Amazing!
This example was contrived to seem complicated at first sight so we could high-
light the power of counting with generating functions. But the simple suggests that
there ought to be an elementary derivation without resort to generating functions,
and indeed there is (Problem 16.8).

16.3 Partial Fractions
We got a simple solution to the seemingly impossible counting problem of Sec-
tion 16.2.6 because its generating function simplified to the expression 1=.1 x/2 ,
whose power series coefficients we already knew. This problem was set up so the
answer would work out neatly, but other problems are not so neat. To solve more
general problems using generating functions, we need ways to find power series co-
efficients for generating functions given as formulas. Maclaurin’s Theorem 16.2.1
is a very general method for finding coefficients, but it only applies when formulas
for repeated derivatives can be found, which isn’t often. However, there is an auto-
matic way to find the power series coefficients for any formula that is a quotient of
polynomials, namely, the method of partial fractions from elementary calculus.
The partial fraction method is based on the fact that quotients of polynomials
can be expressed as sums of terms whose power series coefficients have nice for-
mulas. For example when the denominator polynomial has distinct nonzero roots,
the method rests on
Lemma 16.3.1. Let p.x/ be a polynomial of degree less than n and let ˛1 ; : : : ; ˛n
be distinct, nonzero numbers. Then there are constants c1 ; : : : ; cn such that
p.x/ c1 c2 cn
D C C C :
.1 ˛1 x/.1 ˛2 x/ .1 ˛n x/ 1 ˛1 x 1 ˛2 x 1 ˛n x
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692 Chapter 16 Generating Functions

Let’s illustrate the use of Lemma 16.3.1 by finding the power series coefficients
for the function
x
R.x/ WWD :
1 x x2
We can use the quadratic formula to find the roots r1 ; r2 of the denominator 1
x x2. p p
1 5 1C 5
r1 D ; r2 D :
2 2
So
1 x x 2 D .x r1 /.x r2 / D r1 r2 .1 x=r1 /.1 x=r2 /:
With a little algebra, we find that
x
R.x/ D
.1 ˛1 x/.1 ˛2 x/
where
p
1C 5
˛1 D
2p
1 5
˛2 D :
2
Next we find c1 and c2 which satisfy:
x c1 c2
D C (16.11)
.1 ˛1 x/.1 ˛2 x/ 1 ˛1 x 1 ˛2 x
In general, we can do this by plugging in a couple of values for x to generate two
linear equations in c1 and c2 and then solve the equations for c1 and c2 . A simpler
approach in this case comes from multiplying both sides of (16.11) by the left-hand
denominator to get
x D c1 .1 ˛2 x/ C c2 .1 ˛1 x/:
Now letting x D 1=˛2 we obtain

1=˛2 1 1
c2 D D D p ;
1 ˛1 =˛2 ˛2 ˛1 5
and similarly, letting x D 1=˛1 we obtain
1
c1 D p :
5
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16.3. Partial Fractions 693

Plugging these values for c1 ; c2 into equation (16.11) finally gives the partial frac-
tion expansion

x 1 1 1
R.x/ D Dp
1 x x2 5 1 ˛1 x 1 ˛2 x
Each term in the partial fractions expansion has a simple power series given by the
geometric sum formula:
1
D 1 C ˛1 x C ˛12 x 2 C
1 ˛1 x
1
D 1 C ˛2 x C ˛22 x 2 C
1 ˛2 x
Substituting in these series gives a power series for the generating function:
1
R.x/ D p .1 C ˛1 x C ˛12 x 2 C / .1 C ˛2 x C ˛22 x 2 C / ;

5
so
˛n ˛n
Œx n R.x/ D 1p 2
5
p !n p !n !
1 1C 5 1 5
Dp (16.12)
5 2 2

16.3.1 Partial Fractions with Repeated Roots
Lemma 16.3.1 generalizes to the case when the denominator polynomial has a re-
peated nonzero root with multiplicity m by expanding the quotient into a sum a
terms of the form
c
.1 ˛x/k
where ˛ is the reciprocal of the root and k m. A formula for the coefficients of
such a term follows from the donut formula (16.10).
!
n c n n C .k 1/
Œx D c˛ : (16.13)
.1 ˛x/k n
When ˛ D 1, this follows from the donut formula (16.10) and termwise multipli-
cation by the constant c. The case for arbitrary ˛ follows by substituting ˛x for x
in the power series; this changes x n into .˛x/n and so has the effect of multiplying
the coefficient of x n by ˛ n .2
2 In other words,
Œx n F .˛x/ D ˛ n Œx n F .x/:
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694 Chapter 16 Generating Functions

16.4 Solving Linear Recurrences
16.4.1 A Generating Function for the Fibonacci Numbers
The Fibonacci numbers f0 ; f1 ; : : : ; fn ; : : : are defined recursively as follows:

f0 WWD 0
f1 WWD 1
fn D WWDfn 1 C fn 2 (for n 2):

Generating functions will now allow us to derive an astonishing closed formula for
fn .
Let F .x/ be the generating function for the sequence of Fibonacci numbers, that
is,
F .x/ WWD f0 C f1 x C f2 x 2 C fn x n C :
Reasoning as we did at the start of this chapter to derive the formula for a geometric
series, we have

F .x/ D f0 C f1 x C f2 x 2 C C fn x n C :
xF .x/ D f0 x f1 x 2 fn 1 x n C :
x 2 F .x/ D f0 x 2 fn 2 x n C :
F .x/.1 x x2/ D f0 C .f1 2
f0 /x C 0x C C 0x n C :
D 0 C 1x C 0x 2 D x;
so
x
F .x/ D :
1 x x2
But F .x/ is the same as the function we used to illustrate the partial fraction method
for finding coefficients in Section 16.3. So by equation (16.12), we arrive at what
is called Binet’s formula:
p !n p !n !
1 1C 5 1 5
fn D p (16.14)
5 2 2

Binet’s formula for Fibonacci numbers is astonishing and maybe scary. It’s not
even obvious that the expression on the right-hand side (16.14) is an integer. But
the formula is very useful. For example, it provides—via the repeated squaring
method—a much more efficient way to compute Fibonacci numbers than crunch-
ing through the recurrence. It also make explicit the exponential growth of these
numbers.
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16.4. Solving Linear Recurrences 695

Figure 16.1 The initial configuration of the disks in the Towers of Hanoi problem.

16.4.2 The Towers of Hanoi
According to legend, there is a temple in Hanoi with three posts and 64 gold disks
of different sizes. Each disk has a hole through the center so that it fits on a post.
In the misty past, all the disks were on the first post, with the largest on the bottom
and the smallest on top, as shown in Figure 16.1.
Monks in the temple have labored through the years since to move all the disks
to one of the other two posts according to the following rules:
The only permitted action is removing the top disk from one post and drop-
ping it onto another post.

A larger disk can never lie above a smaller disk on any post.
So, for example, picking up the whole stack of disks at once and dropping them on
another post is illegal. That’s good, because the legend says that when the monks
complete the puzzle, the world will end!
To clarify the problem, suppose there were only 3 gold disks instead of 64. Then
the puzzle could be solved in 7 steps as shown in Figure 16.2.
The questions we must answer are, “Given sufficient time, can the monks suc-
ceed?” If so, “How long until the world ends?” And, most importantly, “Will this
happen before the final exam?”

A Recursive Solution
The Towers of Hanoi problem can be solved recursively. As we describe the pro-
cedure, we’ll also analyze the minimum number tn of steps required to solve the
n-disk problem. For example, some experimentation shows that t1 D 1 and t2 D 3.
The procedure illustrated above uses 7 steps, which shows that t3 is at most 7.
The recursive solution has three stages, which are described below and illustrated
in Figure 16.3. For clarity, the largest disk is shaded in the figures.

Stage 1. Move the top n 1 disks from the first post to the second using the solution
for n 1 disks. This can be done in tn 1 steps.
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696 Chapter 16 Generating Functions

Figure 16.2 The 7-step solution to the Towers of Hanoi problem when there are
n D 3 disks.

Figure 16.3 A recursive solution to the Towers of Hanoi problem.
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16.4. Solving Linear Recurrences 697

Stage 2. Move the largest disk from the first post to the third post. This takes just
1 step.
Stage 3. Move the n 1 disks from the second post to the third post, again using
the solution for n 1 disks. This can also be done in tn 1 steps.
This algorithm shows that tn , the minimum number of steps required to move n
disks to a different post, is at most tn 1 C 1 C tn 1 D 2tn 1 C 1. We can use this
fact to upper bound the number of operations required to move towers of various
heights:
t3 2 t2 C 1 D 7
t4 2 t3 C 1 15
Continuing in this way, we could eventually compute an upper bound on t64 , the
number of steps required to move 64 disks. So this algorithm answers our first
question: given sufficient time, the monks can finish their task and end the world.
This is a shame. After all that effort, they’d probably want to smack a few high-fives
and go out for burgers and ice cream, but nope—world’s over.

Finding a Recurrence
We cannot yet compute the exact number of steps that the monks need to move the
64 disks, only an upper bound. Perhaps, having pondered the problem since the
beginning of time, the monks have devised a better algorithm.
Lucky for us, there is no better algorithm. Here’s why: at some step, the monks
must move the largest disk from the first post to a different post. For this to happen,
the n 1 smaller disks must all be stacked out of the way on the only remaining
post. Arranging the n 1 smaller disks this way requires at least tn 1 moves. After
the largest disk is moved, at least another tn 1 moves are required to pile the n 1
smaller disks on top.
This argument shows that the number of steps required is at least 2tn 1 C 1.
Since we gave an algorithm using exactly that number of steps, we can now write
an expression for tn , the number of moves required to complete the Towers of Hanoi
problem with n disks:
t0 D 0
tn D 2tn 1 C1 (for n 1):
Solving the Recurrence
We can now find a formula for tn using generating functions. Let T .x/ be the
generating function for the tn ’s, that is,
T .x/ WWD t0 C t1 x C t2 x 2 C tn x n C :
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698 Chapter 16 Generating Functions

Reasoning as we did for the Fibonacci recurrence, we have

T .x/ D t0 C t1 x C C tn x n C
2xT .x/ D 2t0 x 2tn 1 x n C
1=.1 x/ D 1 1x 1x n C
T .x/.1 2x/ 1=.1 x/ D t0 1 C 0x C C 0x n C
D 1;
so
1 x
T .x/.1 2x/ D 1D ;
1 x 1 x
and
x
T .x/ D :
.1 2x/.1 x/
Using partial fractions,
x c1 c2
D C
.1 2x/.1 x/ 1 2x 1 x
for some constants c1 ; c2 . Now multiplying both sides by the left hand denominator
gives
x D c1 .1 x/ C c2 .1 2x/:
Substituting 1=2 for x yields c1 D 1 and substituting 1 for x yields c2 D 1,
which gives
1 1
T .x/ D :
1 2x 1 x
Finally we can read off the simple formula for the numbers of steps needed to move
a stack of n disks:

n n 1 n 1
tn D Œx T .x/ D Œx Œx D 2n 1:
1 2x 1 x

16.4.3 Solving General Linear Recurrences
An equation of the form

f .n/ D c1 f .n 1/ C c2 f .n 2/ C C cd f .n d / C h.n/ (16.15)

for constants ci 2 C is called a degree d linear recurrence with inhomogeneous
term h.n/.
The methods above can be used to solve linear recurrences with a large class of
inhomogeneous terms. In particular, when the inhomogeneous term itself has a gen-
erating function that can be expressed as a quotient of polynomials, the approach
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16.5. Formal Power Series 699

used above to derive generating functions for the Fibonacci and Tower of Hanoi
examples carries over to yield a quotient of polynomials that defines the generating
function f .0/ C f .1/x C f .2/x 2 C . Then partial fractions can be used to find
a formula for f .n/ that is a linear combination of terms of the form nk ˛ n where k
is a nonnegative integer d and ˛ is the reciprocal of a root of the denominator
polynomial. For example, see Problems 16.14, 16.15, 16.18, and 16.19.

16.5 Formal Power Series
16.5.1 Divergent Generating Functions
Let F .x/ be the generating function for nŠ, that is,

F .x/ WWD 1 C 1x C 2x 2 C C nŠx n C :

Because x n D o.nŠ/ for all x ¤ 0, this generating function converges only at
x D 0.3
Next, let H.x/ be the generating function for n nŠ, that is,

H.x/ WWD 0 C 1x C 4x 2 C C n nŠx n C :

Again, H.x/ converges only for x D 0, so H.x/ and F .x/ describe the same,
trivial, partial function on the reals.
On the other hand, F .x/ and H.x/ have different coefficients for all powers of
x greater than 1, and we can usefully distinguish them as formal, symbolic objects.
To illustrate this, note than by subtracting 1 from F .x/ and then dividing each
of the remaining terms by x, we get a series where the coefficient if x n is .n C 1/Š.
That is
n F .x/ 1
Œx D .n C 1/Š : (16.16)
x
Now a little further formal reasoning about F .x/ and H.x/ will allow us to
deduce the following identity for nŠ:4
n
X
nŠ D 1 C .i 1/ .i 1/Š (16.17)
i D1
3 This
section is based on an example from “Use of everywhere divergent generating function,”
mathoverflow, response 8,147 by Aaron Meyerowitz, Nov. 12, 2010.
4 A combinatorial proof of (16.17) is given in Problem 15.70
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700 Chapter 16 Generating Functions

To prove this identity, note that from (16.16), we have

n n F .x/ 1
Œx H.x/ WWD n nŠ D .n C 1/Š nŠ D Œx Œx n F .x/:
x

In other words,
F .x/ 1
H.x/ D F .x/; (16.18)
x
Solving (16.18) for F .x/, we get

xH.x/ C 1
F .x/ D : (16.19)
1 x
But Œx n .xH.x/ C 1/ is .n 1/ .n 1/Š for n 1 and is 1 for n D 0, so by the
convolution formula,
n
n xH.x/ C 1 X
Œx D1C .i 1/ .i 1/Š :
1 x
i D1

The identity (16.17) now follows immediately from (16.19).

16.5.2 The Ring of Power Series
So why don’t we have to worry about series whose radius of convergence is zero,
and how do we justify the kind of manipulation in the previous section to derive
the formula (16.19)? The answer comes from thinking abstractly about infinite
sequences of numbers and operations that can be performed on them.
For example, one basic operation combining two infinite sequences is adding
them coordinatewise. That is, if we let

G WWD .g0 ; g1 ; g2 ; : : : /;
H WWD .h0 ; h1 ; h2 ; : : : /;

then we can define the sequence sum ˚ by the rule:

G ˚ H WWD .g0 C h0 ; g1 C h1 ; : : : ; gn C hn ; : : : /:

Another basic operation is sequence multiplication ˝ defined by the convolution
rule (not coordinatewise):
n
!
X
G ˝ H WWD g0 C h0 ; g0 h1 C g1 h0 ; : : : ; gi hn i ; : : : :
i D0
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16.5. Formal Power Series 701

These operations on infinite sequences have lots of nice properties. For example,
it’s easy to check that sequence addition and multiplication are commutative:

G ˚ H D H ˚ G;
G ˝ H D H ˝ G:

If we let

Z WWD .0; 0; 0; : : : /;
I WWD .1; 0; 0; : : : ; 0; : : : /;

then it’s equally easy to check that Z acts like a zero for sequences and I acts like
the number one:

Z ˚ G D G;
Z ˝ G D Z; (16.20)
I ˝ G D G:

Now if we define
G WWD . g0 ; g1 ; g2 ; : : : /
then
G ˚ . G/ D Z:
In fact, the operations ˚ and ˝ satisfy all the commutative ring axioms described
in Section 9.7.1. The set of infinite sequences of numbers together with these op-
erations is called the ring of formal power series over these numbers.5
A sequence H is the reciprocal of a sequence G when

G ˝ H D I:

A reciprocal of G is also called a multiplicative inverse or simply an “inverse”
of G. The ring axioms imply that if there is a reciprocal, it is unique (see Prob-
lem 9.32), so the suggestive notation 1=G can be used unambiguously to denote
this reciprocal, if it exists. For example, letting

J WWD .1; 1; 0; 0; : : : ; 0; : : : /
K WWD .1; 1; 1; 1; : : : ; 1; : : : /;

the definition of ˝ implies that J ˝ K D I , and so K D 1=J and J D 1=K.
5 Theelements in the sequences may be the real numbers, complex numbers, or, more generally,
may be the elements from any given commutative ring.
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702 Chapter 16 Generating Functions

In the ring of formal power series, equation (16.20) implies that the zero se-
quence Z has no inverse, so 1=Z is undefined—just as the expression 1/0 is unde-
fined over the real numbers or the ring Zn of Section 9.7.1. It’s not hard to verify
that a series has an inverse iff its initial element is nonzero (see Problem 16.25).
Now we can explain the proper way to understand a generating function defini-
tion
X1
G.x/ WWD gn x n :
nD0

It simply means that G.x/ really refers to its infinite sequence of coefficients .g0 ; g1 ; : : : /
in the ring of formal power series. The simple expression x can be understood as
referring to the sequence

X WWD .0; 1; 0; 0; : : : ; 0; : : : /:

Likewise, 1 x abbreviates the sequence J above, and the familiar equation
1
D 1 C x C x2 C x3 C (16.21)
1 x
can be understood as a way of restating the assertion that K is 1=J . In other words,
the powers of the variable x just serve as a place holders—and as reminders of the
definition of convolution. The equation (16.21) has nothing to do with the values
of x or the convergence of the series. Rather, it is stating a property that holds in
the ring of formal power series. The reasoning about the divergent series in the
previous section is completely justified as properties of formal power series.

16.6 References
[47], [23], [9] [18]

Problems for Section 16.1
Practice Problems
Problem 16.1.
The notation Œx n F .x/ refers to the coefficient of x n in the generating function
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16.6. References 703

F .x/. Indicate all the expressions below that equal Œx n 4xG.x/ (most of them do).

4Œx n xG.x/ 4xŒx n G.x/ Œx n 1 4G.x/

.Œx n 4x/ Œx n G.x/ .Œx4x/ Œx n xG.x/ Œx nC1 4x 2 G.x/

Problem 16.2.
What is the coefficient of x n in the generating function
1Cx
‹
.1 x/2

Problems for Section 16.2
Practice Problems
Problem 16.3.
You would like to buy a bouquet of flowers. You find an online service that will
make bouquets of lilies, roses and tulips, subject to the following constraints:

there must be at most 1 lily,

there must be an odd number of tulips,

there must be at least two roses.

Example: A bouquet of no lilies, 3 tulips, and 5 roses satisfies the constraints.
Express B.x/, the generating function for the number of ways to select a bouquet
of n flowers, as a quotient of polynomials (or products of polynomials). You do not
need to simplify this expression.

Problem 16.4.
Write a formula for the generating function whose successive coefficients are given
by the sequence:
(a) 0, 0, 1, 1, 1,. . .

(b) 1, 1, 0, 0, 0,. . .
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704 Chapter 16 Generating Functions

(d) 1, 4, 6, 4, 1, 0, 0, 0,. . .

(e) 1, 2, 3, 4, 5,. . .

(f) 1, 4, 9, 16, 25,. . .

(g) 1, 1, 1/2, 1/6, 1/24, 1/120,. . .

Class Problems
Problem 16.5.
Let A.x/ D 1 n
P
nD0 an x . Then it’s easy to check that

A.n/ .0/
an D ;
nŠ

where A.n/ is the nth derivative of A. Use this fact (which you may assume) instead
of the Convolution Counting Principle 16.2.3, to prove that
1
!
1 X nCk 1 n
D x :
.1 x/k k 1
nD0

So if we didn’t already know the Bookkeeper Rule from Section 15.6, we could
have proved it from this calculation and the Convolution Rule for generating func-
tions.

Problem 16.6. (a) Let
x2 C x
S.x/ WWD :
.1 x/3
What is the coefficient of x n in the generating function series for S.x/?

(b) Explain why S.x/=.1 x/ is the generating function for P the sums of squares.
That is, the coefficient of x n in the series for S.x/=.1 x/ is nkD1 k 2 .

16.6. References 705

Homework Problems
Problem 16.7.
We will use generating functions to determine how many ways there are to use
pennies, nickels, dimes, quarters, and half-dollars to give n cents change.
(a) Write the generating function P .x/ for for the number of ways to use only
pennies to make n cents.

(b) Write the generating function N.x/ for the number of ways to use only nickels
to make n cents.

(d) Write the generating function for the number of ways to use pennies, nickels,
dimes, quarters, and half-dollars to give n cents change.

(e) Explain how to use this function to find out how many ways are there to change
50 cents; you do not have to provide the answer or actually carry out the process.

Problem 16.8.
The answer derived by generating functions for the “absurd” counting problem
in Section 16.2.6 is not impossibly complicated at all. Describe a direct simple
counting argument to derive this answer without using generating functions.

Problems for Section 16.3
Class Problems
Problem 16.9.
We are interested in generating functions for the number of different ways to com-
pose a bag of n donuts subject to various restrictions. For each of the restrictions in
parts (a)-(e) below, find a closed form for the corresponding generating function.
(a) All the donuts are chocolate and there are at least 3.

(b) All the donuts are glazed and there are at most 2.

(d) All the donuts are plain and their number is a multiple of 4.
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706 Chapter 16 Generating Functions

(e) The donuts must be chocolate, glazed, coconut, or plain with the numbers of
each flavor subject to the constraints above.

(f) Now find a closed form for the number of ways to select n donuts subject to
the above constraints.

Homework Problems
Problem 16.10.
Miss McGillicuddy never goes outside without a collection of pets. In particular:

She brings a positive number of songbirds, which always come in pairs.

She may or may not bring her alligator, Freddy.

She brings at least 2 cats.

She brings two or more chihuahuas and labradors leashed together in a line.

Let Pn denote the number of different collections of n pets that can accompany
her, where we regard chihuahuas and labradors leashed in different orders as dif-
ferent collections.
For example, P6 D 4 since there are 4 possible collections of 6 pets:

2 songbirds, 2 cats, 2 chihuahuas leashed in line

2 songbirds, 2 cats, 2 labradors leashed in line

2 songbirds, 2 cats, a labrador leashed behind a chihuahua

2 songbirds, 2 cats, a chihuahua leashed behind a labrador

(a) Let
P .x/ WWD P0 C P1 x C P2 x 2 C P3 x 3 C
be the generating function for the number of Miss McGillicuddy’s pet collections.
Verify that
4x 6
P .x/ D :
.1 x/2 .1 2x/
(b) Find a closed form expression for Pn .
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16.6. References 707

Exam Problems
Problem 16.11.
T-Pain is planning an epic boat trip and he needs to decide what to bring with him.

He must bring some burgers, but they only come in packs of 6.

He and his two friends can’t decide whether they want to dress formally or
casually. He’ll either bring 0 pairs of flip flops or 3 pairs.

He doesn’t have very much room in his suitcase for towels, so he can bring
at most 2.

In order for the boat trip to be truly epic, he has to bring at least 1 nautical-
themed pashmina afghan.

(a) Let B.x/ be the generating function for the number of ways to bring n burgers,
F .x/ for the number of ways to bring n pairs of flip flops, T .x/ for towels, and
A.x/ for Afghans. Write simple formulas for each of these.

(b) Let gn be the the number of different ways for T-Pain to bring n items (burg-
P1towels, nand/or afghans) on his boat trip. Let G.x/ be the
ers, pairs of flip flops,
generating function nD0 gn x . Verify that

x7
G.x/ D :
.1 x/2

Problem 16.12.
Every day in the life of Dangerous Dan is a potential disaster:

Dan may or may not spill his breakfast cereal on his computer keyboard.

Dan may or may not fall down the front steps on his way out the door.

Dan stubs his toe zero or more times.

Dan blurts something foolish an even number of times.
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708 Chapter 16 Generating Functions

Let Tn be the number of different combinations of n mishaps Dan can suffer in one
day. For example, T3 D 7, because there are seven possible combinations of three
mishaps:
spills 0 1 0 1 1 0 0
falls 0 0 1 1 0 1 0
stubs 3 2 2 1 0 0 1
blurts 0 0 0 0 2 2 2
(a) Express the generating function

T .x/ WWD T0 C T1 x C T2 x 2 C

as a quotient of polynomials.

(b) Put integers in the boxes that make this equation true:

g.x/ D C
1 x .1 x/2

Problems for Section 16.4
Practice Problems
Problem 16.13.
Let b, c, a0 , a1 , a2 ,. . . be real numbers such that

an D b.an 1/ Cc
for n 1.
Let G.x/ be the generating function for this sequence.
(a) Express the coefficient of x n for n 1 in the series expansion of bxG.x/ in
terms of b and ai for suitable i .
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16.6. References 709

(b) What is the coefficient of x n for n 1 in the series expansion of cx=.1 x/?

(d) Using the method of partial fractions, we can find real numbers d and e such
that
G.x/ D d=L.x/ C e=M.x/:
What are L.x/ and M.x/?

Class Problems
Problem 16.14.
The famous mathematician, Fibonacci, has decided to start a rabbit farm to fill up
his time while he’s not making new sequences to torment future college students.
Fibonacci starts his farm on month zero (being a mathematician), and at the start of
month one he receives his first pair of rabbits. Each pair of rabbits takes a month
to mature, and after that breeds to produce one new pair of rabbits each month.
Fibonacci decides that in order never to run out of rabbits or money, every time a
batch of new rabbits is born, he’ll sell a number of newborn pairs equal to the total
number of pairs he had three months earlier. Fibonacci is convinced that this way
he’ll never run out of stock.
(a) Define the number rn of pairs of rabbits Fibonacci has in month n, using a
recurrence relation. That is, define rn in terms of various ri where i < n.

(b) Let R.x/ be the generating function for rabbit pairs,

R.x/ WWD r0 C r1 x C r2 x 2 C

Express R.x/ as a quotient of polynomials.

(d) Finally, use the partial fraction decomposition to come up with a closed form
expression for the number of pairs of rabbits Fibonacci has on his farm on month
n.

Problem 16.15.
Less well-known than the Towers of Hanoi—but no less fascinating—are the Tow-
ers of Sheboygan. As in Hanoi, the puzzle in Sheboygan involves 3 posts and n
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710 Chapter 16 Generating Functions

rings of different sizes. The rings are placed on post #1 in order of size with the
smallest ring on top and largest on bottom.
The objective is to transfer all n rings to post #2 via a sequence of moves. As
in the Hanoi version, a move consists of removing the top ring from one post and
dropping it onto another post with the restriction that a larger ring can never lie
above a smaller ring. But unlike Hanoi, a local ordinance requires that a ring can
only be moved from post #1 to post #2, from post #2 to post #3, or from post
#3 to post #1. Thus, for example, moving a ring directly from post #1 to post #3 is
not permitted.
(a) One procedure that solves the Sheboygan puzzle is defined recursively: to
move an initial stack of n rings to the next post, move the top stack of n 1 rings
to the furthest post by moving it to the next post two times, then move the big, nth
ring to the next post, and finally move the top stack another two times to land on
top of the big ring. Let sn be the number of moves that this procedure uses. Write
a simple linear recurrence for sn .

(b) Let S.x/ be the generating function for the sequence hs0 ; s1 ; s2 ; : : : i. Care-
fully show that
x
S.x/ D :
.1 x/.1 4x/
(c) Give a simple formula for sn .

(d) A better (indeed optimal, but we won’t prove this) procedure to solve the Tow-
ers of Sheboygan puzzle can be defined in terms of two mutually recursive proce-
dures, procedure P1 .n/ for moving a stack of n rings 1 pole forward, and P2 .n/
for moving a stack of n rings 2 poles forward. This is trivial for n D 0. For n > 0,
define:
P1 .n/: Apply P2 .n 1/ to move the top n 1 rings two poles forward to the third
pole. Then move the remaining big ring once to land on the second pole. Then
apply P2 .n 1/ again to move the stack of n 1 rings two poles forward from the
third pole to land on top of the big ring.
P2 .n/: Apply P2 .n 1/ to move the top n 1 rings two poles forward to land on
the third pole. Then move the remaining big ring to the second pole. Then apply
P1 .n 1/ to move the stack of n 1 rings one pole forward to land on the first
pole. Now move the big ring 1 pole forward again to land on the third pole. Finally,
apply P2 .n 1/ again to move the stack of n 1 rings two poles forward to land
on the big ring.
Let tn be the number of moves needed to solve the Sheboygan puzzle using proce-
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16.6. References 711

dure P1 .n/. Show that
tn D 2tn 1 C 2tn 2 C 3; (16.22)
for n > 1.
Hint: Let un be the number of moves used by procedure P2 .n/. Express each of tn
and un as linear combinations of tn 1 and un 1 and solve for tn .

(e) Derive values a; b; c; ˛; ˇ such that

tn D a˛ n C bˇ n C c:

Conclude that tn D o.sn /.

Homework Problems
Problem 16.16.
Taking derivatives of generating functions is another useful operation. This is done
termwise, that is, if

F .x/ D f0 C f1 x C f2 x 2 C f3 x 3 C ;

then
F 0 .x/ WWD f1 C 2f2 x C 3f3 x 2 C :
For example,
0
1 1
D D 1 C 2x C 3x 2 C
.1 x/2 .1 x/
so
x
H.x/ WWD D 0 C 1x C 2x 2 C 3x 3 C
.1 x/2
is the generating function for the sequence of nonnegative integers. Therefore
1Cx
D H 0 .x/ D 1 C 22 x C 32 x 2 C 42 x 3 C ;
.1 x/3
so
x2 C x
D xH 0 .x/ D 0 C 1x C 22 x 2 C 32 x 3 C C n2 x n C
.1 x/3
is the generating function for the nonnegative integer squares.
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712 Chapter 16 Generating Functions

(a) Prove that for all k 2 N, the generating function for the nonnegative integer
kth powers is a quotient of polynomials in x. That is, for all k 2 N there are
polynomials Rk .x/ and Sk .x/ such that

Rk .x/
Œx n D nk : (16.23)
Sk .x/

Hint: Observe that the derivative of a quotient of polynomials is also a quotient of
polynomials. It is not necessary work out explicit formulas for Rk and Sk to prove
this part.

(b) Conclude that if f .n/ is a function on the nonnegative integers defined recur-
sively in the form

f .n/ D af .n 1/ C bf .n 2/ C cf .n 3/ C p.n/˛ n

where the a; b; c; ˛ 2 C and p is a polynomial with complex coefficients, then
the generating function for the sequence f .0/; f .1/; f .2/; : : : will be a quotient of
polynomials in x, and hence there is a closed form expression for f .n/.
Hint: Consider
Rk .˛x/
Sk .˛x/

Problem 16.17.
Generating functions provide an interesting way to count the number of strings of
matched brackets. To do this, we’ll use a description of these strings as the set
GoodCount of strings of brackets with a good count.6
Namely, one precise way to determine if a string is matched is to start with 0
and read the string from left to right, adding 1 to the count for each left bracket
and subtracting 1 from the count for each right bracket. For example, here are the
counts for the two strings above

[ ] ] [ [ [ [ [ ] ] ] ]
0 1 0 1 0 1 2 3 4 3 2 1 0

[ [ [ ] ] [ ] ] [ ]
0 1 2 3 2 1 2 1 0 1 0
6 Problem 7.20 also examines these strings.
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16.6. References 713

A string has a good count if its running count never goes negative and ends with 0.
So the second string above has a good count, but the first one does not because its
count went negative at the third step.
Definition. Let

GoodCount WWD fs 2 f] ; [ g j s has a good countg:

The matched strings can now be characterized precisely as this set of strings with
good counts.
Let cn be the number of strings in GoodCount with exactly n left brackets, and
let C.x/ be the generating function for these numbers:

C.x/ WWD c0 C c1 x C c2 x 2 C :

(a) The wrap of a string s is the string, [ s ] , that starts with a left bracket fol-
lowed by the characters of s, and then ends with a right bracket. Explain why the
generating function for the wraps of strings with a good count is xC.x/.
Hint: The wrap of a string with good count also has a good count that starts and
ends with 0 and remains positive everywhere else.

(b) Explain why, for every string s with a good count, there is a unique sequence
of strings s1 ; : : : ; sk that are wraps of strings with good counts and s D s1 sk .
For example, the string r WWD [ [ ] ] [ ] [ [ ] [ ] ] 2 GoodCount equals s1 s2 s3 where
s1 WWD [ [ ] ] ; s2 WWD [ ] ; s3 WWD [ [ ] [ ] ] , and this is the only way to express r as a
sequence of wraps of strings with good counts.

C D 1 C xC C .xC /2 C C .xC /n C ; (i)

so
1
C D ; (ii)
1
xC
and hence p
1 ˙ 1 4x
C D : (iii)
2x
Let D.x/ WWD 2xC.x/. Expressing D as a power series

D.x/ D d0 C d1 x C d2 x 2 C ;

we have
dnC1
cn D : (iv)
2
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714 Chapter 16 Generating Functions

(d) Use (iii), (iv), and the value of c0 to conclude that
p
D.x/ D 1 1 4x:

(e) Prove that
.2n 3/ .2n 5/ 5 3 1 2n
dn D :
nŠ

Hint: dn D D .n/ .0/=nŠ

(f) Conclude that !
1 2n
cn D :
nC1 n

Exam Problems
Problem 16.18.
Define the sequence r0 ; r1 ; r2 ; : : : recursively by the rule that r0 WWD 1 and

rn WWD 7rn 1 C .n C 1/ for n > 0:

Let R.x/ WWD 1 n
P
0 rn x be the generating function of this sequence. Express R.x/
as a quotient of polynomials or products of polynomials. You do not have to find a
closed form for rn .

Problem 16.19.
Alyssa Hacker sends out a video that spreads like wildfire over the UToob network.
On the day of the release—call it day zero—and the day following—call it day
one—the video doesn’t receive any hits. However, starting with day two, the num-
ber of hits rn can be expressed as seven times the number of hits on the previous
day, four times the number of hits the day before that, and the number of days that
has passed since the release of the video plus one. So, for example on day 2, there
will be 7 0 C 4 0 C 3 D 3 hits.
(a) Give a linear a recurrence for rn .

(b) Express the generating function R.x/ WWD 1 n
P
0 rn x as a quotient of polyno-
mials or products of polynomials. You do not have to find a closed form for rn .
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16.6. References 715

Problem 16.20.
Consider the following sequence of predicates:

Q1 .x1 / WWD x1
Q2 .x1 ; x2 / WWD x1 IMPLIES x2
Q3 .x1 ; x2 ; x3 / WWD .x1 IMPLIES x2 / IMPLIES x3
Q4 .x1 ; x2 ; x3 ; x4 / WWD ..x1 IMPLIES x2 / IMPLIES x3 / IMPLIES x4
Q5 .x1 ; x2 ; x3 ; x4 ; x5 / WWD ...x1 IMPLIES x2 / IMPLIES x3 / IMPLIES x4 / IMPLIES x5
:: ::
: :

Let Tn be the number of different true/false settings of the variables x1 ; x2 ; : : : ; xn
for which Qn .x1 ; x2 ; : : : ; xn / is true. For example, T2 D 3 since Q2 .x1 ; x2 / is
true for 3 different settings of the variables x1 and x2 :

x1 x2 Q2 .x1 ; x2 /
T T T
T F F
F T T
F F T

We let T0 D 1 by convention.
(a) Express TnC1 in terms of Tn and n, assuming n 0.

(b) Use a generating function to prove that

2nC1 C . 1/n
Tn D
3
for n 1.

Problem 16.21.
Define the Triple Fibonacci numbers T0 ; T1 ; : : : recursively by the rules

T0 D T1 WWD 3;
Tn WWD Tn 1 C Tn 2 (for n 2): (16.24)

(a) Prove that all Triple Fibonacci numbers are divisible by 3.

(b) Prove that the gcd of every pair of consecutive Triple Fibonacci numbers is 3.

(c) Express the generating function T .x/ for the Triple Fibonacci as a quotient of
polynomials. (You do not have to find a formula for Œx n T .x/.)
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716 Chapter 16 Generating Functions

Problem 16.22.
Define the Double Fibonacci numbers D0 ; D1 ; : : : recursively by the rules

D0 D D1 WWD 1;
Dn WWD 2Dn 1 C Dn 2 (for n 2): (16.25)

(a) Prove that all Double Fibonacci numbers are odd.

(b) Prove that every two consecutive Double Fibonacci numbers are relatively
prime.

(c) Express the generating function D.x/ for the Double Fibonacci as a quotient
of polynomials. (You do not have to find a formula for Œx n D.x/.)

Problems for Section 16.5
Practice Problems
Problem 16.23.
In the context of formal series, a number r may be used to indicate the sequence

.r; 0; 0; : : : ; 0; : : : /:

For example the number 1 may be used to indicate the identity series I and 0
may indicate the zero series Z. Whether “r” means the number or the sequence is
supposed to be clear from context.
Verify that in the ring of formal power series,

r ˝ .g0 ; g1 ; g2 ; : : : / D .rg0 ; rg1 ; rg2 ; : : : /:

In particular,
.g0 ; g1 ; g2 ; : : : / D 1 ˝ .g0 ; g1 ; g2 ; : : : /:

Problem 16.24.
Define the formal power series

X WWD .0; 1; 0; 0; : : : ; 0; : : : /:

(a) Explain why X has no reciprocal.
Hint: What can you say about x .g0 C g1 x C g2 x 2 C /?
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16.6. References 717

(b) Use the definition of power series multiplication ˝ to prove carefully that

X ˝ .g0 ; g1 ; g2 ; : : : / D .0; g0 ; g1 ; g2 ; : : : /:

X 0 WWD I WWD .1; 0; 0; : : : ; 0; : : : /;
X nC1 WWD X ˝ X n :

Verify that the monomial x n refers to the same power series as X n .

Class Problems
Problem 16.25.
Show that a sequence G WWD .g0 ; g1 ; : : : / has a multiplicative inverse in the ring of
formal power series iff g0 ¤ 0.
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IV Probability
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Introduction
Probability is one of the most important disciplines in all of the sciences. It is also
one of the least well understood.
Probability is especially important in computer science—it arises in virtually
every branch of the field. In algorithm design and game theory, for example, al-
gorithms and strategies that make random choices at certain steps frequently out-
perform deterministic algorithms and strategies. In information theory and signal
processing, an understanding of randomness is critical for filtering out noise and
compressing data. In cryptography and digital rights management, probability is
crucial for achieving security. The list of examples is long.
Given the impact that probability has on computer science, it seems strange that
probability should be so misunderstood by so many. The trouble is that “common-
sense” intuition is demonstrably unreliable when it comes to problems involving
random events. As a consequence, many students develop a fear of probability.
We’ve witnessed many graduate oral exams where a student will solve the most
horrendous calculation, only to then be tripped up by the simplest probability ques-
tion. Even some faculty will start squirming if you ask them a question that starts
“What is the probability that. . . ?”
Our goal in the remaining chapters is to equip you with the tools that will enable
you to solve basic problems involving probability easily and confidently.
Chapter 17 introduces the basic definitions and an elementary 4-step process
that can be used to determine the probability that a specified event occurs. We il-
lustrate the method on two famous problems where your intuition will probably fail
you. The key concepts of conditional probability and independence are introduced,
along with examples of their use, and regrettable misuse, in practice: the probabil-
ity you have a disease given that a diagnostic test says you do, and the probability
that a suspect is guilty given that his blood type matches the blood found at the
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722 Part IV Probability

scene of the crime.
Random variables provide a more quantitative way to measure random events,
and we study them in Chapter 19. For example, instead of determining the proba-
bility that it will rain, we may want to determine how much or how long it is likely
to rain. The fundamental concept of the expected value of a random variable is
introduced and some of its key properties are developed.
Chapter 20 examines the probability that a random variable deviates significantly
from its expected value. Probability of deviation provides the theoretical basis for
estimation by sampling which is fundamental in science, engineering, and human
affairs. It is also especially important in engineering practice, where things are
generally fine if they are going as expected, and you would like to be assured that
the probability of an unexpected event is very low.
A final chapter applies the previous probabilistic tools to solve problems involv-
ing more complex random processes. You will see why you will probably never
get very far ahead at the casino and how two Stanford graduate students became
billionaires by combining graph theory and probability theory to design a better
search engine for the web.
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17 Events and Probability Spaces

17.1 Let’s Make a Deal
In the September 9, 1990 issue of Parade magazine, columnist Marilyn vos Savant
responded to this letter:

Suppose you’re on a game show, and you’re given the choice of three
doors. Behind one door is a car, behind the others, goats. You pick a
door, say number 1, and the host, who knows what’s behind the doors,
opens another door, say number 3, which has a goat. He says to you,
“Do you want to pick door number 2?” Is it to your advantage to
switch your choice of doors?

Craig. F. Whitaker
Columbia, MD

The letter describes a situation like one faced by contestants in the 1970’s game
show Let’s Make a Deal, hosted by Monty Hall and Carol Merrill. Marilyn replied
that the contestant should indeed switch. She explained that if the car was behind
either of the two unpicked doors—which is twice as likely as the the car being
behind the picked door—the contestant wins by switching. But she soon received
a torrent of letters, many from mathematicians, telling her that she was wrong. The
problem became known as the Monty Hall Problem and it generated thousands of
hours of heated debate.
This incident highlights a fact about probability: the subject uncovers lots of
examples where ordinary intuition leads to completely wrong conclusions. So until
you’ve studied probabilities enough to have refined your intuition, a way to avoid
errors is to fall back on a rigorous, systematic approach such as the Four Step
Method that we will describe shortly. First, let’s make sure we really understand
the setup for this problem. This is always a good thing to do when you are dealing
with probability.

17.1.1 Clarifying the Problem
Craig’s original letter to Marilyn vos Savant is a bit vague, so we must make some
assumptions in order to have any hope of modeling the game formally. For exam-
ple, we will assume that:
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724 Chapter 17 Events and Probability Spaces

1. The car is equally likely to be hidden behind each of the three doors.

2. The player is equally likely to pick each of the three doors, regardless of the
car’s location.

3. After the player picks a door, the host must open a different door with a goat
behind it and offer the player the choice of staying with the original door or
switching.

4. If the host has a choice of which door to open, then he is equally likely to
select each of them.

In making these assumptions, we’re reading a lot into Craig Whitaker’s letter. There
are other plausible interpretations that lead to different answers. But let’s accept
these assumptions for now and address the question, “What is the probability that
a player who switches wins the car?”

17.2 The Four Step Method
Every probability problem involves some sort of randomized experiment, process,
or game. And each such problem involves two distinct challenges:

1. How do we model the situation mathematically?

2. How do we solve the resulting mathematical problem?

In this section, we introduce a four step approach to questions of the form, “What
is the probability that. . . ?” In this approach, we build a probabilistic model step
by step, formalizing the original question in terms of that model. Remarkably, this
structured approach provides simple solutions to many famously confusing prob-
lems. For example, as you’ll see, the four step method cuts through the confusion
surrounding the Monty Hall problem like a Ginsu knife.

17.2.1 Step 1: Find the Sample Space
Our first objective is to identify all the possible outcomes of the experiment. A
typical experiment involves several randomly-determined quantities. For example,
the Monty Hall game involves three such quantities:

1. The door concealing the car.

2. The door initially chosen by the player.
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17.2. The Four Step Method 725

car location

Figure 17.1 The first level in a tree diagram for the Monty Hall Problem. The
branches correspond to the door behind which the car is located.

3. The door that the host opens to reveal a goat.

Every possible combination of these randomly-determined quantities is called an
outcome. The set of all possible outcomes is called the sample space for the exper-
iment.
A tree diagram is a graphical tool that can help us work through the four step
approach when the number of outcomes is not too large or the problem is nicely
structured. In particular, we can use a tree diagram to help understand the sample
space of an experiment. The first randomly-determined quantity in our experiment
is the door concealing the prize. We represent this as a tree with three branches, as
shown in Figure 17.1. In this diagram, the doors are called A, B and C instead of
1, 2, and 3, because we’ll be adding a lot of other numbers to the picture later.
For each possible location of the prize, the player could initially choose any of
the three doors. We represent this in a second layer added to the tree. Then a third
layer represents the possibilities of the final step when the host opens a door to
reveal a goat, as shown in Figure 17.2.
Notice that the third layer reflects the fact that the host has either one choice
or two, depending on the position of the car and the door initially selected by the
player. For example, if the prize is behind door A and the player picks door B, then
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726 Chapter 17 Events and Probability Spaces

car location player’s door
initial revealed
guess
B
A
C
B C

A C B

C
A
A
B B

C
C A

B
C A
A
B
A
C
B

Figure 17.2 The full tree diagram for the Monty Hall Problem. The second level
indicates the door initially chosen by the player. The third level indicates the door
revealed by Monty Hall.
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17.2. The Four Step Method 727

the host must open door C. However, if the prize is behind door A and the player
picks door A, then the host could open either door B or door C.
Now let’s relate this picture to the terms we introduced earlier: the leaves of the
tree represent outcomes of the experiment, and the set of all leaves represents the
sample space. Thus, for this experiment, the sample space consists of 12 outcomes.
For reference, we’ve labeled each outcome in Figure 17.3 with a triple of doors
indicating:
.door concealing prize; door initially chosen; door opened to reveal a goat/:
In these terms, the sample space is the set

.A; A; B/; .A; A; C /; .A; B; C /; .A; C; B/; .B; A; C /; .B; B; A/;
SD
.B; B; C /; .B; C; A/; .C; A; B/; .C; B; A/; .C; C; A/; .C; C; B/
The tree diagram has a broader interpretation as well: we can regard the whole
experiment as following a path from the root to a leaf, where the branch taken at
each stage is “randomly” determined. Keep this interpretation in mind; we’ll use it
again later.

17.2.2 Step 2: Define Events of Interest
Our objective is to answer questions of the form “What is the probability that . . . ?”,
where, for example, the missing phrase might be “the player wins by switching,”
“the player initially picked the door concealing the prize,” or “the prize is behind
door C.”
A set of outcomes is called an event. Each of the preceding phrases characterizes
an event. For example, the event Œprize is behind door C refers to the set:
f.C; A; B/; .C; B; A/; .C; C; A/; .C; C; B/g;
and the event Œprize is behind the door first picked by the player is:
f.A; A; B/; .A; A; C /; .B; B; A/; .B; B; C /; .C; C; A/; .C; C; B/g:
Here we’re using square brackets around a property of outcomes as a notation for
the event whose outcomes are the ones that satisfy the property.
What we’re really after is the event Œplayer wins by switching:
f.A; B; C /; .A; C; B/; .B; A; C /; .B; C; A/; .C; A; B/; .C; B; A/g: (17.1)
The outcomes in this event are marked with checks in Figure 17.4.
Notice that exactly half of the outcomes are checked, meaning that the player
wins by switching in half of all outcomes. You might be tempted to conclude that
a player who switches wins with probability 1=2. This is wrong. The reason is that
these outcomes are not all equally likely, as we’ll see shortly.
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728 Chapter 17 Events and Probability Spaces

car location player’s door outcome
intial revealed
guess
B .A;A;B/
A
C .A;A;C/
B C
.A;B;C/

A C B
.A;C;B/
C
.B;A;C/
A
A .B;B;A/
B B

C .B;B;C/
C A
.B;C;A/
B
C .C;A;B/
A
A
.C;B;A/
B
A .C;C;A/
C
B .C;C;B/

Figure 17.3 The tree diagram for the Monty Hall Problem with the outcomes la-
beled for each path from root to leaf. For example, outcome .A; A; B/ corresponds
to the car being behind door A, the player initially choosing door A, and Monty
Hall revealing the goat behind door B.
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17.2. The Four Step Method 729

car location player’s door outcome switch
intial revealed wins
guess
B .A;A;B/
A
C .A;A;C/
B C
.A;B;C/ 

A C B
.A;C;B/ 
C
.B;A;C/ 
A
A .B;B;A/
B B

C .B;B;C/
C A 
.B;C;A/
B
.C;A;B/ 
C A
A
.C;B;A/ 
B
A .C;C;A/
C
B .C;C;B/

Figure 17.4 The tree diagram for the Monty Hall Problem, where the outcomes
where the player wins by switching are denoted with a check mark.
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730 Chapter 17 Events and Probability Spaces

17.2.3 Step 3: Determine Outcome Probabilities
So far we’ve enumerated all the possible outcomes of the experiment. Now we
must start assessing the likelihood of those outcomes. In particular, the goal of this
step is to assign each outcome a probability, indicating the fraction of the time this
outcome is expected to occur. The sum of all the outcome probabilities must equal
one, reflecting the fact that there always must be an outcome.
Ultimately, outcome probabilities are determined by the phenomenon we’re mod-
eling and thus are not quantities that we can derive mathematically. However, math-
ematics can help us compute the probability of every outcome based on fewer and
more elementary modeling decisions. In particular, we’ll break the task of deter-
mining outcome probabilities into two stages.

Step 3a: Assign Edge Probabilities
First, we record a probability on each edge of the tree diagram. These edge-
probabilities are determined by the assumptions we made at the outset: that the
prize is equally likely to be behind each door, that the player is equally likely to
pick each door, and that the host is equally likely to reveal each goat, if he has a
choice. Notice that when the host has no choice regarding which door to open, the
single branch is assigned probability 1. For example, see Figure 17.5.

Step 3b: Compute Outcome Probabilities
Our next job is to convert edge probabilities into outcome probabilities. This is a
purely mechanical process:
calculate the probability of an outcome by multiplying the edge-probabilities
on the path from the root to that outcome.
For example, the probability of the topmost outcome in Figure 17.5, .A; A; B/, is
1 1 1 1
D : (17.2)
3 3 2 18
We’ll examine the official justification for this rule in Section 18.4, but here’s an
easy, intuitive justification: as the steps in an experiment progress randomly along
a path from the root of the tree to a leaf, the probabilities on the edges indicate
how likely the path is to proceed along each branch. For example, a path starting
at the root in our example is equally likely to go down each of the three top-level
branches.
How likely is such a path to arrive at the topmost outcome .A; A; B/? Well,
there is a 1-in-3 chance that a path would follow the A-branch at the top level, a
1-in-3 chance it would continue along the A-branch at the second level, and 1-in-2
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17.2. The Four Step Method 731

chance it would follow the B-branch at the third level. Thus, there is half of a one
third of a one third chance, of arriving at the .A; A; B/ leaf. That is, the chance is
1=3 1=3 1=2 D 1=18—the same product (in reverse order) we arrived at in (17.2).
We have illustrated all of the outcome probabilities in Figure 17.5.
Specifying the probability of each outcome amounts to defining a function that
maps each outcome to a probability. This function is usually called PrŒ. In these
terms, we’ve just determined that:
1
PrŒ.A; A; B/ D ;
18
1
PrŒ.A; A; C / D ;
18
1
PrŒ.A; B; C / D ;
9
etc.

17.2.4 Step 4: Compute Event Probabilities
We now have a probability for each outcome, but we want to determine the proba-
bility of an event. The probability of an event E is denoted by PrŒE, and it is the
sum of the probabilities of the outcomes in E. For example, the probability of the
[switching wins] event (17.1) is

PrŒswitching wins
D PrŒ.A; B; C / C PrŒ.A; C; B/ C PrŒ.B; A; C /C
PrŒ.B; C; A/ C PrŒ.C; A; B/ C PrŒ.C; B; A/
1 1 1 1 1 1
D C C C C C
9 9 9 9 9 9
2
D :
3
It seems Marilyn’s answer is correct! A player who switches doors wins the car
with probability 2=3. In contrast, a player who stays with his or her original door
wins with probability 1=3, since staying wins if and only if switching loses.
We’re done with the problem! We didn’t need any appeals to intuition or inge-
nious analogies. In fact, no mathematics more difficult than adding and multiplying
fractions was required. The only hard part was resisting the temptation to leap to
an “intuitively obvious” answer.
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732 Chapter 17 Events and Probability Spaces

car location player’s door outcome switch probability
intial revealed wins
guess
B 1=2 .A;A;B/ 1=18
A 1=3
C 1=2 .A;A;C/ 1=18
B 1=3 C 1
.A;B;C/  1=9

A 1=3 C 1=3 B 1 
.A;C;B/ 1=9
C 1 
.B;A;C/ 1=9
A 1=3
A 1=2 .B;B;A/ 1=18
B 1=3 B 1=3
C 1=2 .B;B;C/ 1=18
C 1=3 A 1 
.B;C;A/ 1=9
B 1
.C;A;B/  1=9
C 1=3 A 1=3
A 1
.C;B;A/  1=9
B 1=3
A 1=2 .C;C;A/ 1=18
C 1=3
B 1=2 .C;C;B/ 1=18

Figure 17.5 The tree diagram for the Monty Hall Problem where edge weights
denote the probability of that branch being taken given that we are at the parent of
that branch. For example, if the car is behind door A, then there is a 1/3 chance that
the player’s initial selection is door B. The rightmost column shows the outcome
probabilities for the Monty Hall Problem. Each outcome probability is simply the
product of the probabilities on the path from the root to the outcome leaf.
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17.3. Strange Dice 733

17.2.5 An Alternative Interpretation of the Monty Hall Problem
Was Marilyn really right? Our analysis indicates that she was. But a more accurate
conclusion is that her answer is correct provided we accept her interpretation of the
question. There is an equally plausible interpretation in which Marilyn’s answer
is wrong. Notice that Craig Whitaker’s original letter does not say that the host is
required to reveal a goat and offer the player the option to switch, merely that he
did these things. In fact, on the Let’s Make a Deal show, Monty Hall sometimes
simply opened the door that the contestant picked initially. Therefore, if he wanted
to, Monty could give the option of switching only to contestants who picked the
correct door initially. In this case, switching never works!

17.3 Strange Dice
The four-step method is surprisingly powerful. Let’s get some more practice with
it. Imagine, if you will, the following scenario.
It’s a typical Saturday night. You’re at your favorite pub, contemplating the true
meaning of infinite cardinalities, when a burly-looking biker plops down on the
stool next to you. Just as you are about to get your mind around pow.pow.R//,
biker dude slaps three strange-looking dice on the bar and challenges you to a $100
wager. His rules are simple. Each player selects one die and rolls it once. The
player with the lower value pays the other player $100.
Naturally, you are skeptical, especially after you see that these are not ordinary
dice. Each die has the usual six sides, but opposite sides have the same number on
them, and the numbers on the dice are different, as shown in Figure 17.6.
Biker dude notices your hesitation, so he sweetens his offer: he will pay you
$105 if you roll the higher number, but you only need pay him $100 if he rolls
higher, and he will let you pick a die first, after which he will pick one of the other
two. The sweetened deal sounds persuasive since it gives you a chance to pick what
you think is the best die, so you decide you will play. But which of the dice should
you choose? Die B is appealing because it has a 9, which is a sure winner if it
comes up. Then again, die A has two fairly large numbers, and die C has an 8 and
no really small values.
In the end, you choose die B because it has a 9, and then biker dude selects
die A. Let’s see what the probability is that you will win. (Of course, you probably
should have done this before picking die B in the first place.) Not surprisingly, we
will use the four-step method to compute this probability.
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734 Chapter 17 Events and Probability Spaces

A B C

Figure 17.6 The strange dice. The number of pips on each concealed face is the
same as the number on the opposite face. For example, when you roll die A, the
probabilities of getting a 2, 6, or 7 are each 1=3.

17.3.1 Die A versus Die B
Step 1: Find the sample space.
The tree diagram for this scenario is shown in Figure 17.7. In particular, the sample
space for this experiment are the nine pairs of values that might be rolled with Die A
and Die B:
For this experiment, the sample space is a set of nine outcomes:

S D f .2; 1/; .2; 5/; .2; 9/; .6; 1/; .6; 5/; .6; 9/; .7; 1/; .7; 5/; .7; 9/ g:

Step 2: Define events of interest.
We are interested in the event that the number on die A is greater than the number
on die B. This event is a set of five outcomes:

f .2; 1/; .6; 1/; .6; 5/; .7; 1/; .7; 5/ g:

These outcomes are marked A in the tree diagram in Figure 17.7.
Step 3: Determine outcome probabilities.
To find outcome probabilities, we first assign probabilities to edges in the tree di-
agram. Each number on each die comes up with probability 1=3, regardless of
the value of the other die. Therefore, we assign all edges probability 1=3. The
probability of an outcome is the product of the probabilities on the correspond-
ing root-to-leaf path, which means that every outcome has probability 1=9. These
probabilities are recorded on the right side of the tree diagram in Figure 17.7.
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17.3. Strange Dice 735

die A die B winner probability
of outcome
1=3 A 1=9
1
5 1=3
B 1=9
9
1=3
2 1=3 B 1=9

1=3 A 1=9
1
6 1=3 5 1=3
A 1=9
9
1=3
B 1=9

1=3 A 1=9
7 1=3 1
5 1=3
A 1=9
9
1=3
B 1=9

Figure 17.7 The tree diagram for one roll of die A versus die B. Die A wins with
probability 5=9.
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736 Chapter 17 Events and Probability Spaces

Step 4: Compute event probabilities.
The probability of an event is the sum of the probabilities of the outcomes in that
event. In this case, all the outcome probabilities are the same, so we say that the
sample space is uniform. Computing event probabilities for uniform sample spaces
is particularly easy since you just have to compute the number of outcomes in the
event. In particular, for any event E in a uniform sample space S,

jEj
PrŒE D : (17.3)
jSj

In this case, E is the event that die A beats die B, so jEj D 5, jSj D 9, and

PrŒE D 5=9:

This is bad news for you. Die A beats die B more than half the time and, not
surprisingly, you just lost $100.
Biker dude consoles you on your “bad luck” and, given that he’s a sensitive guy
beneath all that leather, he offers to go double or nothing.1 Given that your wallet
only has $25 in it, this sounds like a good plan. Plus, you figure that choosing die A
will give you the advantage.
So you choose A, and then biker dude chooses C . Can you guess who is more
likely to win? (Hint: it is generally not a good idea to gamble with someone you
don’t know in a bar, especially when you are gambling with strange dice.)

17.3.2 Die A versus Die C
We can construct the tree diagram and outcome probabilities as before. The result
is shown in Figure 17.8, and there is bad news again. Die C will beat die A with
probability 5=9, and you lose once again.
You now owe the biker dude $200 and he asks for his money. You reply that you
need to go to the bathroom.

17.3.3 Die B versus Die C
Being a sensitive guy, biker dude nods understandingly and offers yet another wa-
ger. This time, he’ll let you have die C . He’ll even let you raise the wager to $200
so you can win your money back.
This is too good a deal to pass up. You know that die C is likely to beat die A
and that die A is likely to beat die B, and so die C is surely the best. Whether biker
1 Double or nothing is slang for doing another wager after you have lost the first. If you lose again,

you will owe biker dude double what you owed him before. If you win, you will owe him nothing;
in fact, since he should pay you $210 if he loses, you would come out $10 ahead.
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17.3. Strange Dice 737

die C die A winner probability
of outcome
1=3 C 1=9
2
6 1=3
A 1=9
7
1=3
3 1=3 A 1=9

1=3 C 1=9
2
4 1=3 6 1=3
A 1=9
7
1=3
A 1=9

1=3 C 1=9
8 1=3 2
6 1=3
C 1=9
7
1=3
C 1=9

Figure 17.8 The tree diagram for one roll of die C versus die A. Die C wins with
probability 5=9.
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738 Chapter 17 Events and Probability Spaces

dude picks A or B, the odds would be in your favor this time. Biker dude must
really be a nice guy.
So you pick C , and then biker dude picks B. Wait—how come you haven’t
caught on yet and worked out the tree diagram before you took this bet? If you do
it now, you’ll see by the same reasoning as before that B beats C with probabil-
ity 5=9. But surely there is a mistake! How is it possible that

C beats A with probability 5=9,
A beats B with probability 5=9,
B beats C with probability 5=9?

The problem is not with the math, but with your intuition. Since A will beat B
more often than not, and B will beat C more often than not, it seems like A ought
to beat C more often than not, that is, the “beats more often” relation ought to
be transitive. But this intuitive idea is simply false: whatever die you pick, biker
dude can pick one of the others and be likely to win. So picking first is actually a
disadvantage, and as a result, you now owe biker dude $400.
Just when you think matters can’t get worse, biker dude offers you one final
wager for $1,000. This time, instead of rolling each die once, you will each roll
your die twice, and your score is the sum of your rolls, and he will even let you
pick your die second, that is, after he picks his. Biker dude chooses die B. Now
you know that die A will beat die B with probability 5=9 on one roll, so, jumping
at this chance to get ahead, you agree to play, and you pick die A. After all, you
figure that since a roll of die A beats a roll of die B more often that not, two rolls
of die A are even more likely to beat two rolls of die B, right?
Wrong! (Did we mention that playing strange gambling games with strangers in
a bar is a bad idea?)

17.3.4 Rolling Twice
If each player rolls twice, the tree diagram will have four levels and 34 D 81
outcomes. This means that it will take a while to write down the entire tree dia-
gram. But it’s easy to write down the first two levels as in Figure 17.9(a) and then
notice that the remaining two levels consist of nine identical copies of the tree in
Figure 17.9(b).
The probability of each outcome is .1=3/4 D 1=81 and so, once again, we have a
uniform probability space. By equation (17.3), this means that the probability that
A wins is the number of outcomes where A beats B divided by 81.
To compute the number of outcomes where A beats B, we observe that the two
rolls of die A result in nine equally likely outcomes in a sample space SA in which
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17.3. Strange Dice 739

1st A 2nd A sum of 1st B 2nd B sum of
roll roll A rolls roll roll B rolls

2
6
4
8
‹ 1
5
2
6
2 7 9 1 9 10
2 8 1 6
6 6 12 5 5 10
7 13 9 14
7 2 9 9 1 10
6 13 5 14
7 14 9 18

Figure 17.9 Parts of the tree diagram for die B versus die A where each die is
rolled twice. The first two levels are shown in (a). The last two levels consist of
nine copies of the tree in (b).

the two-roll sums take the values

.4; 8; 8; 9; 9; 12; 13; 13; 14/:

Likewise, two rolls of die B result in nine equally likely outcomes in a sample
space SB in which the two-roll sums take the values

.2; 6; 6; 10; 10; 10; 14; 14; 18/:

We can treat the outcome of rolling both dice twice as a pair .x; y/ 2 SA SB ,
where A wins iff the sum of the two A-rolls of outcome x is larger the sum of the
two B-rolls of outcome y. If the A-sum is 4, there is only one y with a smaller
B-sum, namely, when the B-sum is 2. If the A-sum is 8, there are three y’s with
a smaller B-sum, namely, when the B-sum is 2 or 6. Continuing the count in this
way, the number of pairs .x; y/ for which the A-sum is larger than the B-sum is

1 C 3 C 3 C 3 C 3 C 6 C 6 C 6 C 6 D 37:

A similar count shows that there are 42 pairs for which B-sum is larger than the
A-sum, and there are two pairs where the sums are equal, namely, when they both
equal 14. This means that A loses to B with probability 42=81 > 1=2 and ties with
probability 2=81. Die A wins with probability only 37=81.
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740 Chapter 17 Events and Probability Spaces

How can it be that A is more likely than B to win with one roll, but B is more
likely to win with two rolls? Well, why not? The only reason we’d think otherwise
is our unreliable, untrained intuition. (Even the authors were surprised when they
first learned about this, but at least they didn’t lose $1400 to biker dude.) In fact,
the die strength reverses no matter which two die we picked. So for one roll,

A B C A;

but for two rolls,
A B C A;
where we have used the symbols and to denote which die is more likely to
result in the larger value.
The weird behavior of the three strange dice above generalizes in a remarkable
way: there are arbitrarily large sets of dice which will beat each other in any desired
pattern according to how many times the dice are rolled.2

17.4 The Birthday Principle
There are 95 students in a class. What is the probability that some birthday is
shared by two people? Comparing 95 students to the 365 possible birthdays, you
might guess the probability lies somewhere around 1=4—but you’d be wrong: the
probability that there will be two people in the class with matching birthdays is
actually more than 0:9999.
To work this out, we’ll assume that the probability that a randomly chosen stu-
dent has a given birthday is 1=d . We’ll also assume that a class is composed of n
randomly and independently selected students. Of course d D 365 and n D 95
in this case, but we’re interested in working things out in general. These random-
ness assumptions are not really true, since more babies are born at certain times
of year, and students’ class selections are typically not independent of each other,
but simplifying in this way gives us a start on analyzing the problem. More impor-
tantly, these assumptions are justifiable in important computer science applications
of birthday matching. For example, birthday matching is a good model for colli-
sions between items randomly inserted into a hash table. So we won’t worry about
things like spring procreation preferences that make January birthdays more com-
mon, or about twins’ preferences to take classes together (or not).
2 TBA - Reference Ron Graham paper.
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17.4. The Birthday Principle 741

17.4.1 Exact Formula for Match Probability
There are d n sequences of n birthdays, and under our assumptions, these are
equally likely. There are d.d 1/.d 2/ .d .n 1// length n sequences of
distinct birthdays. That means the probability that everyone has a different birthday
is:
d.d 2/ .d .n
1/.d 1//
dn
d d 1 d 2 d .n 1/
D (17.4)
d
d d d
0 1 2 n 1
D 1 1 1 1 (17.5)
d d d d

Now we simplify (17.5) using the fact that 1 x < e x for all x > 0. This
follows by truncating the Taylor series e x D 1 x C x 2 =2Š x 3 =3Š C . The
approximation e x 1 x is pretty accurate when x is small.

0 1 2 n 1
1 1 1 1
d d d d
0 1=d 2=d .n 1/=d
<e e e e (17.6)
P
n 1
iD1 i=d
De
.n.n 1/=2d /
De : (17.7)

For n D 95 and d D 365, the value of (17.7) is less than 1=200; 000, which
means the probability of having some pair of matching birthdays actually is more
than 1 1=200; 000 > 0:99999. So it would be pretty astonishing if there were no
pair of students in the class with matching birthdays.
For d n2 =2, the probability of no match turns out to be asymptotically equal
to the upper bound (17.7). For d D n2 =2 in particular, the probability of no match
is asymptotically equal to 1=e. This leads to a rule of thumb which is useful in
many contexts in computer science:

The Birthday Principle
p
If there are d days in a year and 2d people in a room, then the probability that
two share a birthday is about 1 1=e 0:632.
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742 Chapter 17 Events and Probability Spaces

p
For example, the Birthday Principle says that if you have 2 365 27 people
in a room, then the probability that two share a birthday is about 0:632. The actual
probability is about 0:626, so the approximation is quite good.
Among other applications, it implies that to use a hash function that maps n
items into a hash table of size d , you can expect many collisions if n2 is more than
a small fraction of d . The Birthday Principle also famously comes into play as the
basis of “birthday attacks” that crack certain cryptographic systems.

17.5 Set Theory and Probability
Let’s abstract what we’ve just done into a general mathematical definition of sample
spaces and probability.

17.5.1 Probability Spaces
Definition 17.5.1. A countable sample space S is a nonempty countable set.3 An
element ! 2 S is called an outcome. A subset of S is called an event.
Definition 17.5.2. A probability function on a sample space S is a total function
Pr W S ! R such that
PrŒ! 0 for all ! 2 S, and
P
!2S PrŒ! D 1.

A sample space together with a probability function is called a probability space.
For any event E S, the probability of E is defined to be the sum of the probabil-
ities of the outcomes in E:
X
PrŒE WWD PrŒ!:
!2E

In the previous examples there were only finitely many possible outcomes, but
we’ll quickly come to examples that have a countably infinite number of outcomes.
The study of probability is closely tied to set theory because any set can be a
sample space and any subset can be an event. General probability theory deals
with uncountable sets like the set of real numbers, but we won’t need these, and
sticking to countable sets lets us define the probability of events using sums instead
of integrals. It also lets us avoid some distracting technical problems in set theory
like the Banach-Tarski “paradox” mentioned in Chapter 8.
3 Yes,sample spaces can be infinite. If you did not read Chapter 8, don’t worry—countable just
means that you can list the elements of the sample space as !0 , !1 , !2 , . . . .
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17.5. Set Theory and Probability 743

17.5.2 Probability Rules from Set Theory
Most of the rules and identities that we have developed for finite sets extend very
naturally to probability.
An immediate consequence of the definition of event probability is that for dis-
joint events E and F ,

PrŒE [ F D PrŒE C PrŒF :

This generalizes to a countable number of events:
Rule 17.5.3 (Sum Rule). If E0 ; E1 ; : : : ; En ; : : : are pairwise disjoint events, then
" #
[ X
Pr En D PrŒEn :
n2N n2N

The Sum Rule lets us analyze a complicated event by breaking it down into
simpler cases. For example, if the probability that a randomly chosen MIT student
is native to the United States is 60%, to Canada is 5%, and to Mexico is 5%, then
the probability that a random MIT student is native to one of these three countries
is 70%.
Another consequence of the Sum Rule is that PrŒA C PrŒA D 1, which follows
because PrŒS D 1 and S is the union of the disjoint sets A and A. This equation
often comes up in the form:

PrŒA D 1 PrŒA: (Complement Rule)

Sometimes the easiest way to compute the probability of an event is to compute the
probability of its complement and then apply this formula.
Some further basic facts about probability parallel facts about cardinalities of
finite sets. In particular:
PrŒB A D PrŒB PrŒA \ B, (Difference Rule)
PrŒA [ B D PrŒA C PrŒB PrŒA \ B, (Inclusion-Exclusion)
PrŒA [ B PrŒA C PrŒB, (Boole’s Inequality)
If A B, then PrŒA PrŒB. (Monotonicity Rule)
The Difference Rule follows from the Sum Rule because B is the union of the
disjoint sets B A and A \ B. Inclusion-Exclusion then follows from the Sum
and Difference Rules, because A [ B is the union of the disjoint sets A and B
A. Boole’s inequality is an immediate consequence of Inclusion-Exclusion since
probabilities are nonnegative. Monotonicity follows from the definition of event
probability and the fact that outcome probabilities are nonnegative.
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744 Chapter 17 Events and Probability Spaces

The two-event Inclusion-Exclusion equation above generalizes to any finite set
of events in the same way as the corresponding Inclusion-Exclusion rule for n
sets. Boole’s inequality also generalizes to both finite and countably infinite sets of
events:

Rule 17.5.4 (Union Bound).

PrŒE1 [ [ En [ PrŒE1 C C PrŒEn C : (17.8)

The Union Bound is useful in many calculations. For example, suppose that Ei is
the event that the i -th critical component among n components in a spacecraft fails.
Then E1 [ [ En is the event that some critical component fails. If niD1 PrŒEi
P
is small, then the Union Bound can provide a reassuringly small upper bound on
this overall probability of critical failure.

17.5.3 Uniform Probability Spaces
Definition 17.5.5. A finite probability space S is said to be uniform if PrŒ! is the
same for every outcome ! 2 S.

As we saw in the strange dice problem, uniform sample spaces are particularly
easy to work with. That’s because for any event E S,

jEj
PrŒE D : (17.9)
jSj
This means that once we know the cardinality of E and S, we can immediately
obtain PrŒE. That’s great news because we developed lots of tools for computing
the cardinality of a set in Part III.
For example, suppose that you select five cards at random from a standard deck
of 52 cards. What is the probability of having a full house? Normally, this question
would take some effort to answer. But from the analysis in Section 15.7.2, we know
that !
52
jSj D
5
and ! !
4 4
jEj D 13 12
3 2
where E is the event that we have a full house. Since every five-card hand is equally
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17.5. Set Theory and Probability 745

2nd
1st player 1=2
player 1=2 T
2nd
player 1=2 T
1st H
player 1=2 T 1=2
H
T 1=2
H 1=16
1=2
H 1=8
1=2
1=4
1=2

Figure 17.10 The tree diagram for the game where players take turns flipping a
fair coin. The first player to flip heads wins.

likely, we can apply equation (17.9) to find that

13 12 43 42

PrŒE D 52
5
13 12 4 6 5 4 3 2 18
D D
52 51 50 49 48 12495
1
:
694
17.5.4 Infinite Probability Spaces
Infinite probability spaces are fairly common. For example, two players take turns
flipping a fair coin. Whoever flips heads first is declared the winner. What is the
probability that the first player wins? A tree diagram for this problem is shown in
Figure 17.10.
The event that the first player wins contains an infinite number of outcomes, but
we can still sum their probabilities:
1 1 1 1
PrŒfirst player wins D C C C C
2 8 32 128
1
1X 1 n
D
2 4
nD0

1 1 2
D D :
2 1 1=4 3
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746 Chapter 17 Events and Probability Spaces

Similarly, we can compute the probability that the second player wins:
1 1 1 1 1
PrŒsecond player wins D C C C C D :
4 16 64 256 3
In this case, the sample space is the infinite set

S WWD f Tn H j n 2 N g;

where Tn stands for a length n string of T’s. The probability function is
1
PrŒTn H WWD :
2nC1
To verify that this is a probability space, we just have to check that all the prob-
abilities are nonnegative and that they sum to 1. The given probabilities are all
nonnegative, and applying the formula for the sum of a geometric series, we find
that
X X 1
PrŒTn H D D 1:
2nC1
n2N n2N

Notice that this model does not have an outcome corresponding to the possi-
bility that both players keep flipping tails forever. (In the diagram, flipping for-
ever corresponds to following the infinite path in the tree without ever reaching
a leaf/outcome.) If leaving this possibility out of the model bothers you, you’re
welcome to fix it by adding another outcome !forever to indicate that that’s what
happened. Of course since the probabililities of the other outcomes already sum to
1, you have to define the probability of !forever to be 0. Now outcomes with prob-
ability zero will have no impact on our calculations, so there’s no harm in adding
it in if it makes you happier. On the other hand, in countable probability spaces
it isn’t necessary to have outcomes with probability zero, and we will generally
ignore them.

17.6 References
[17], [24], [28], [31], [35], [36] [40], [39], [48]
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17.6. References 747

Problems for Section 17.2
Practice Problems
Problem 17.1.
Let B be the number of heads that come up on 2n independent tosses of a fair coin.
(a) PrŒB D n is asymptotically equal to one of the expressions given below.
Explain which one.

1. p1
2 n
2. p2
n

3. p1
n
q
2
4. n

Exam Problems
Problem 17.2. (a) What’s the probability that 0 doesn’t appear among k digits
chosen independently and uniformly at random?

(b) A box contains 90 good and 10 defective screws. What’s the probability that
if we pick 10 screws from the box, none will be defective?

(c) First one digit is chosen uniformly at random from f1; 2; 3; 4; 5g and is re-
moved from the set; then a second digit is chosen uniformly at random from the
remaining digits. What is the probability that an odd digit is picked the second
time?

(d) Suppose that you randomly permute the digits 1; 2; ; n, that is, you select
a permutation uniformly at random. What is the probability the digit k ends up in
the i th position after the permutation?

(e) A fair coin is flipped n times. What’s the probability that all the heads occur
at the end of the sequence? (If no heads occur, then “all the heads are at the end of
the sequence” is vacuously true.)

Class Problems
Problem 17.3.
The New York Yankees and the Boston Red Sox are playing a two-out-of-three
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748 Chapter 17 Events and Probability Spaces

series. In other words, they play until one team has won two games. Then that
team is declared the overall winner and the series ends. Assume that the Red Sox
win each game with probability 3=5, regardless of the outcomes of previous games.
Answer the questions below using the four step method. You can use the same
tree diagram for all three problems.
(a) What is the probability that a total of 3 games are played?

(b) What is the probability that the winner of the series loses the first game?

Problem 17.4.
To determine which of two people gets a prize, a coin is flipped twice. If the flips
are a Head and then a Tail, the first player wins. If the flips are a Tail and then a
Head, the second player wins. However, if both coins land the same way, the flips
don’t count and the whole process starts over.
Assume that on each flip, a Head comes up with probability p, regardless of
what happened on other flips. Use the four step method to find a simple formula
for the probability that the first player wins. What is the probability that neither
player wins?
Hint: The tree diagram and sample space are infinite, so you’re not going to
finish drawing the tree. Try drawing only enough to see a pattern. Summing all
the winning outcome probabilities directly is cumbersome. However, a neat trick
solves this problem—and many others. Let s be the sum of all winning outcome
probabilities in the whole tree. Notice that you can write the sum of all the winning
probabilities in certain subtrees as a function of s. Use this observation to write an
equation in s and then solve.

Homework Problems
Problem 17.5.
Let’s see what happens when Let’s Make a Deal is played with four doors. A prize
is hidden behind one of the four doors. Then the contestant picks a door. Next, the
host opens an unpicked door that has no prize behind it. The contestant is allowed
to stick with their original door or to switch to one of the two unopened, unpicked
doors. The contestant wins if their final choice is the door hiding the prize.
Let’s make the same assumptions as in the original problem:

1. The prize is equally likely to be behind each door.
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17.6. References 749

2. The contestant is equally likely to pick each door initially, regardless of the
prize’s location.

3. The host is equally likely to reveal each door that does not conceal the prize
and was not selected by the player.

Use The Four Step Method to find the following probabilities. The tree diagram
may become awkwardly large, in which case just draw enough of it to make its
structure clear.
(a) Contestant Stu, a sanitation engineer from Trenton, New Jersey, stays with his
original door. What is the probability that Stu wins the prize?

(b) Contestant Zelda, an alien abduction researcher from Helena, Montana, switches
to one of the remaining two doors with equal probability. What is the probability
that Zelda wins the prize?
Now let’s revise our assumptions about how contestants choose doors. Say the
doors are labeled A, B, C, and D. Suppose that Carol always opens the earliest door
possible (the door whose label is earliest in the alphabet) with the restriction that
she can neither reveal the prize nor open the door that the player picked.
This gives contestant Mergatroid—an engineering student from Cambridge, MA—
just a little more information about the location of the prize. Suppose that Merga-
troid always switches to the earliest door, excluding his initial pick and the one
Carol opened.
(c) What is the probability that Mergatroid wins the prize?

Problem 17.6.
There were n Immortal Warriors born into our world, but in the end there can be
only one. The Immortals’ original plan was to stalk the world for centuries, dueling
one another with ancient swords in dramatic landscapes until only one survivor
remained. However, after a thought-provoking discussion probability, they opt to
give the following protocol a try:

(i) The Immortals forge a coin that comes up heads with probability p.

(ii) Each Immortal flips the coin once.

(iii) If exactly one Immortal flips heads, then they are declared The One. Other-
wise, the protocol is declared a failure, and they all go back to hacking each
other up with swords.
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750 Chapter 17 Events and Probability Spaces

One of the Immortals (Kurgan from the Russian steppe) argues that as n grows
large, the probability that this protocol succeeds must tend to zero. Another (McLeod
from the Scottish highlands) argues that this need not be the case, provided p is
chosen carefully.
(a) A natural sample space to use to model this problem is fH; T gn of length-n
sequences of H and T’s, where the successive H’s and T’s in an outcome correspond
to the Head or Tail flipped on each one of the n successive flips. Explain how a tree
diagram approach leads to assigning a probability to each outcome that depends
only on p; n and the number h of H’s in the outcome.

(b) What is the probability that the experiment succeeds as a function of p and n?

(d) What is the probability of success if p is chosen in this way? What quantity
does this approach when n, the number of Immortal Warriors, grows large?

Problem 17.7.
We play a game with a deck of 52 regular playing cards, of which 26 are red and
26 are black. I randomly shuffle the cards and place the deck face down on a table.
You have the option of “taking” or “skipping” the top card. If you skip the top card,
then that card is revealed and we continue playing with the remaining deck. If you
take the top card, then the game ends; you win if the card you took was revealed
to be black, and you lose if it was red. If we get to a point where there is only one
card left in the deck, you must take it. Prove that you have no better strategy than
to take the top card—which means your probability of winning is 1/2.
Hint: Prove by induction the more general claim that for a randomly shuffled
deck of n cards that are red or black—not necessarily with the same number of red
cards and black cards—there is no better strategy than taking the top card.

Problems for Section 17.5
Class Problems
Problem 17.8.
Suppose there is a system, built by Caltech graduates, with n components. We
know from past experience that any particular component will fail in a given year
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17.6. References 751

with probability p. That is, letting Fi be the event that the i th component fails
within one year, we have
PrŒFi D p
for 1 i n. The system will fail if any one of its components fails. What can we
say about the probability that the system will fail within one year?
Let F be the event that the system fails within one year. Without any additional
assumptions, we can’t get an exact answer for PrŒF . However, we can give useful
upper and lower bounds, namely,

p PrŒF np: (17.10)

We may as well assume p < 1=n, since the upper bound is trivial otherwise. For
example, if n D 100 and p D 10 5 , we conclude that there is at most one chance
in 1000 of system failure within a year and at least one chance in 100,000.
Let’s model this situation with the sample space S WWD pow.Œ1; n/ whose out-
comes are subsets of positive integers n, where s 2 S corresponds to the indices
of exactly those components that fail within one year. For example, f2; 5g is the
outcome that the second and fifth components failed within a year and none of the
other components failed. So the outcome that the system did not fail corresponds
to the empty set ;.
(a) Show that the probability that the system fails could be as small as p by de-
scribing appropriate probabilities for the outcomes. Make sure to verify that the
sum of your outcome probabilities is 1.

(b) Show that the probability that the system fails could actually be as large as np
by describing appropriate probabilities for the outcomes. Make sure to verify that
the sum of your outcome probabilities is 1.

Problem 17.9.
Here are some handy rules for reasoning about probabilities that all follow directly
from the Disjoint Sum Rule. Prove them.
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752 Chapter 17 Events and Probability Spaces

PrŒA B D PrŒA PrŒA \ B (Difference Rule)
PrŒA D 1 PrŒA (Complement Rule)
PrŒA [ B D PrŒA C PrŒB PrŒA \ B (Inclusion-Exclusion)
PrŒA [ B PrŒA C PrŒB (2-event Union Bound)
A B IMPLIES PrŒA PrŒB (Monotonicity)

Homework Problems
Problem 17.10.
Prove the following probabilistic inequality, referred to as the Union Bound.
Let A1 ; A2 ; : : : ; An ; : : : be events. Then
" #
[ X
Pr An PrŒAn :
n2N n2N

Hint: Replace the An ’s by pairwise disjoint events and use the Sum Rule.

Problem 17.11.
The results of a round robin tournament in which every two people play each other
and one of them wins can be modelled a tournament digraph—a digraph with ex-
actly one edge between each pair of distinct vertices, but we’ll continue to use the
language of players beating each other.
An n-player tournament is k-neutral for some k 2 Œ0; n/, when, for every set of
k players, there is another player who beats them all. For example, being 1-neutral
is the same as not having a “best” player who beats everyone else.
This problem shows that for any fixed k, if n is large enough, there will be a
k-neutral tournament of n players. We will do this by reformulating the question in
terms of probabilities. In particular, for any fixed n, we assign probabilities to each
n-vertex tournament digraph by choosing a direction for the edge between any two
vertices, independently and with equal probability for each edge.
(a) For any set S of k players, let BS be the event that no contestant beats every-
one in S . Express PrŒBS in terms of n and k.

(b) Let Qk be the event equal to the set of n-vertex tournament digraphs that are
not k-neutral. Prove that !
n n k
PrŒQk ˛ ;
k
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17.6. References 753

where ˛ WWD 1 .1=2/k .
Hint: Let S range over the size-k subsets of players, so
[
Qk D BS :
S

Use Boole’s inequality.

(d) Explain why the previous result implies that for every integer k, there is a
k-neutral tournament.

Homework Problems
Problem 17.12.
Suppose you repeatedly flip a fair coin until three consecutive flips match the pat-
tern HHT or the pattern TTH occurs. What is the probability you will see HHT
first? Define a suitable probability space that models the coin flipping and use it to
explain your answer.
Hint: Symmetry between Heads and Tails.
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18 Conditional Probability

18.1 Monty Hall Confusion
Remember how we said that the Monty Hall problem confused even professional
mathematicians? Based on the work we did with tree diagrams, this may seem
surprising—the conclusion we reached followed routinely and logically. How could
this problem be so confusing to so many people?
Well, one flawed argument goes as follows: let’s say the contestant picks door
A. And suppose that Carol, Monty’s assistant, opens door B and shows us a goat.
Let’s use the tree diagram 17.3 from Chapter 17 to capture this situation. There are
exactly three outcomes where contestant chooses door A, and there is a goat behind
door B:
.A; A; B/; .A; A; C /; .C; A; B/: (18.1)
These outcomes have respective probabilities 1/18, 1/18, 1/9.
Among those outcomes, switching doors wins only on the last outcome .C; A; B/.
The other two outcomes together have the same 1/9 probability as the last one So
in this situation, the probability that we win by switching is the same as the proba-
bility that we lose. In other words, in this situation, switching isn’t any better than
sticking!
Something has gone wrong here, since we know that the actual probability of
winning by switching in 2/3. The mistaken conclusion that sticking or switching
are equally good strategies comes from a common blunder in reasoning about how
probabilities change given some information about what happened. We have asked
for the probability that one event, [win by switching], happens, given that another
event, [pick A AND goat at B], happens. We use the notation

Pr [win by switching] j [pick A AND goat at B]

for this probability which, by the reasoning above, equals 1/2.

18.1.1 Behind the Curtain
A “given” condition is essentially an instruction to focus on only some of the possi-
ble outcomes. Formally, we’re defining a new sample space consisting only of some
of the outcomes. In this particular example, we’re given that the player chooses
door A and that there is a goat behind B. Our new sample space therefore consists
solely of the three outcomes listed in (18.1). In the opening of Section 18.1, we
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756 Chapter 18 Conditional Probability

calculated the conditional probability of winning by switching given that one of
these outcome happened, by weighing the 1/9 probability of the win-by-switching
outcome .C; A; B/ against the 1=18C1=18C1=9 probability of the three outcomes
in the new sample space.

Pr [win by switching] j [pick A AND goat at B]

D Pr .C; A; B/ j f.C; A; B/; .A; A; B/; .A; A; C /g C
PrŒ.C; A; B/
PrŒf.C; A; B/; .A; A; B/; .A; A; C /g
1=9 1
D D :
1=18 C 1=18 C 1=9 2
There is nothing wrong with this calculation. So how come it leads to an incorrect
conclusion about whether to stick or switch? The answer is that this was the wrong
thing to calculate, as we’ll explain in the next section.

18.2 Definition and Notation

The expression Pr X j Y denotes the probability of event X, given that event
Y happens. In the example above, event X is the event of winning on a switch,
and event Y is the event that a goat
is behind door B and the contestant chose
door A. We calculated Pr X j Y using a formula which serves as the definition
of conditional probability:

Definition 18.2.1. Let X and Y be events where Y has nonzero probability. Then
PrŒX \ Y
Pr X j Y WWD :
PrŒY

The conditional probability Pr X j Y is undefined when the probability of
event Y is zero. To avoid cluttering up statements with uninteresting hypotheses
that conditioning events like Y have nonzero probability, we will make an implicit
assumption from now on that all such events have nonzero probability.
Pure probability is often counterintuitive, but conditional probability can be even
worse. Conditioning can subtly alter probabilities and produce unexpected results
in randomized algorithms and computer systems as well as in betting games. But
Definition 18.2.1 is very simple and causes no trouble—provided it is properly
applied.
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18.2. Definition and Notation 757

18.2.1 What went wrong
So if everything in the opening Section 18.1 is mathematically sound, why does it
seem to contradict the results that we established in Chapter 17? The problem is a
common one: we chose the wrong condition. In our initial description of the sce-
nario, we learned the location of the goat when Carol opened door B. But when we
defined our condition as “the contestant opens A and the goat is behind B,” we in-
cluded the outcome .A; A; C / in which Carol opens door C! The correct conditional
probability should have been “what are the odds of winning by switching given the
contestant chooses door A and Carol opens door B.” By choosing a condition that
did not reflect everything known. we inadvertently included an extraneous outcome
in our calculation. With the correct conditioning, we still win by switching 1/9 of
the time, but the smaller set of known outcomes has smaller total probability:
1 1 3
PrŒf.A; A; B/; .C; A; B/g D C D :
18 9 18
The conditional probability would then be:

Pr [win by switching] j [pick A AND Carol opens B]
PrŒ.C; A; B/
D Pr .C; A; B/ j f.C; A; B/; .A; A; B/g C
PrŒf.C; A; B/; .A; A; B/g
1=9 2
D D ;
1=9 C 1=18 3
which is exactly what we already deduced from the tree diagram 17.2 in Sec-
tion 17.2.
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758 Chapter 18 Conditional Probability

The O. J. Simpson Trial
In an opinion article in the New York Times, Steven Strogatz points to the O. J.
Simpson trial as an example of poor choice of conditions. O. J. Simpson was
a retired football player who was accused, and later acquitted, of the murder of
his wife, Nicole Brown Simpson. The trial was widely publicized and called
the “trial of the century.” Racial tensions, allegations of police misconduct, and
new-at-the-time DNA evidence captured the public’s attention. But Strogatz, cit-
ing mathematician and author I.J. Good, focuses on a less well-known aspect of
the case: whether O. J.’s history of abuse towards his wife was admissible into
evidence.
The prosecution argued that abuse is often a precursor to murder, pointing to
statistics indicating that an abuser was as much as ten times more likely to commit
murder than was a random individual. The defense, however, countered with
statistics indicating that the odds of an abusive husband murdering his wife were
“infinitesimal,” roughly 1 in 2500. Based on those numbers, the actual relevance
of a history of abuse to a murder case would appear limited at best. According
to the defense, introducing that history would prejudice the jury against Simpson
but would lack any probitive value, so the discussion should be barred.
In other words, both the defense and the prosecution were arguing conditional
probability, specifically the likelihood that a woman will be murdered by her
husband, given that her husband abuses her. But both defense and prosecution
omitted a vital piece of data from their calculations: Nicole Brown Simpson was
murdered. Strogatz points out that based on the defense’s numbers and the crime
statistics of the time, the probability that a woman was murdered by her abuser,
given that she was abused and murdered, is around 80%.
Strogatz’s article goes into more detail about the calculations behind that 80%
figure. But the issue we want to illustrate is that conditional probability is used
and misused all the time, and even experts under public scrutiny make mistakes.

18.3 The Four-Step Method for Conditional Probability
In a best-of-three tournament, the local C-league hockey team wins the first game
with probability 1=2. In subsequent games, their probability of winning is deter-
mined by the outcome of the previous game. If the local team won the previous
game, then they are invigorated by victory and win the current game with proba-
bility 2=3. If they lost the previous game, then they are demoralized by defeat and
win the current game with probability only 1=3. What is the probability that the
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18.3. The Four-Step Method for Conditional Probability 759

local team wins the tournament, given that they win the first game?
This is a question about a conditional probability. Let A be the event that the
local team wins the tournament, and let B be the event that they
win the first game.
Our goal is then to determine the conditional probability Pr A j B .
We can tackle conditional probability questions just like ordinary probability
problems: using a tree diagram and the four step method. A complete tree diagram
is shown in Figure 18.1.
game 1 game 2 game 3 outcome event A: event B: outcome
win the win probability
series game 1
W WW   1=3
2=3
W WLW   1=18
1=3 1=3
W L
1=2 2=3
L WLL  1=9

W LWW  1=9
1=2 2=3
L W
1=3 L 1=3
LWL 1=18
2=3
L LL 1=3

Figure 18.1 The tree diagram for computing the probability that the local team
wins two out of three games given that they won the first game.

Step 1: Find the Sample Space
Each internal vertex in the tree diagram has two children, one corresponding to a
win for the local team (labeled W ) and one corresponding to a loss (labeled L).
The complete sample space is:
S D fW W; W LW; W LL; LW W; LW L; LLg:
Step 2: Define Events of Interest
The event that the local team wins the whole tournament is:
T D fW W; W LW; LW W g:
And the event that the local team wins the first game is:
F D fW W; W LW; W LLg:
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760 Chapter 18 Conditional Probability

The outcomes in these events are indicated with check marks in the tree diagram in
Figure 18.1.
Step 3: Determine Outcome Probabilities
Next, we must assign a probability to each outcome. We begin by labeling edges as
specified in the problem statement. Specifically, the local team has a 1=2 chance of
winning the first game, so the two edges leaving the root are each assigned probabil-
ity 1=2. Other edges are labeled 1=3 or 2=3 based on the outcome of the preceding
game. We then find the probability of each outcome by multiplying all probabilities
along the corresponding root-to-leaf path. For example, the probability of outcome
W LL is:
1 1 2 1
D :
2 3 3 9
Step 4: Compute Event Probabilities
We can now compute the probability that the local team wins the tournament, given
that they win the first game:
PrŒA \ B
Pr A j B D
PrŒB
PrŒfW W; W LW g
D
PrŒfW W; W LW; W LLg
1=3 C 1=18
D
1=3 C 1=18 C 1=9
7
D :
9
We’re done! If the local team wins the first game, then they win the whole tourna-
ment with probability 7=9.

18.4 Why Tree Diagrams Work
We’ve now settled into a routine of solving probability problems using tree dia-
grams. But we’ve left a big question unaddressed: mathematical justification be-
hind those funny little pictures. Why do they work?
The answer involves conditional probabilities. In fact, the probabilities that
we’ve been recording on the edges of tree diagrams are conditional probabilities.
For example, consider the uppermost path in the tree diagram for the hockey team
problem, which corresponds to the outcome W W . The first edge is labeled 1=2,
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18.4. Why Tree Diagrams Work 761

which is the probability that the local team wins the first game. The second edge
is labeled 2=3, which is the probability that the local team wins the second game,
given that they won the first—a conditional probability! More generally, on each
edge of a tree diagram, we record the probability that the experiment proceeds
along that path, given that it reaches the parent vertex.
So we’ve been using conditional probabilities all along. For example, we con-
cluded that:
1 2 1
PrŒW W D D :
2 3 3
Why is this correct?
The answer goes back to Definition 18.2.1 of conditional probability which could
be written in a form called the Product Rule for conditional probabilities:
Rule (Conditional Probability Product Rule: 2 Events).

PrŒE1 \ E2 D PrŒE1 Pr E2 j E1 :

Multiplying edge probabilities in a tree diagram amounts to evaluating the right
side of this equation. For example:

PrŒwin first game \ win second game

D PrŒwin first game Pr win second game j win first game
1 2
D :
2 3
So the Conditional Probability Product Rule is the formal justification for multiply-
ing edge probabilities to get outcome probabilities.
To justify multiplying edge probabilities along a path of length three, we need a
rule for three events:
Rule (Conditional Probability Product Rule: 3 Events).

PrŒE1 \ E2 \ E3 D PrŒE1 Pr E2 j E1 Pr E3 j E1 \ E2 :

An n-event version of the Rule is given in Problem 18.1, but its form should be
clear from the three event version.

18.4.1 Probability of Size-k Subsets
As a simple application of the product rule for conditional probabilities, we can use
the rule to calculate the number ofsize-k subsets of the integers Œ1::n. Of course
we already know this number is kn , but now the rule will give us a new derivation
of the formula for kn .
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762 Chapter 18 Conditional Probability

Let’s pick some size-k subset S Œ1::n as a target. Suppose we choose a size-k
subset at random, with all subsets of Œ1::n equally likely to be chosen, and let p be
the probability that our randomly chosen equals this target. That is, the probability
of picking S is p, and since all sets are equally likely to be chosen, the number of
size-k subsets equals 1=p.
So what’s p? Well, the probability that the smallest number in the random set
is one of the k numbers in S is k=n. Then, given that the smallest number in the
random set is in S , the probability that the second smallest number in the random
set is one of the remaining k 1 elements in S is .k 1/=.n 1/. So by the product
rule, the probability that the two smallest numbers in the random set are both in S
is
k k 1
:
n n 1
Next, given that the two smallest numbers in the random set are in S, the probability
that the third smallest number is one of the k 2 remaining elements in S is .k
2/=.n 2/. So by the product rule, the probability that the three smallest numbers
in the random set are all in S is
k k 1 k 2
:
n n 1 n 2
Continuing in this way, it follows that the probability that all k elements in the
randomly chosen set are in S , that is, the probabilty that the randomly chosen set
equals the target, is
k k 1 k 2 k .k 1/
pD
n n 1 n 2 n .k 1/
k .k 1/ .k 1/ 1
D
n .n 1/ .n 2/ .n .k 1//
kŠ
D
nŠ=.n k/Š
kŠ.n k/Š
D :
nŠ
So we have again shown the number of size-k subsets of Œ1::n, namely 1=p, is
nŠ
:
kŠ.n k/Š

18.4.2 Medical Testing
Breast cancer is a deadly disease that claims thousands of lives every year. Early
detection and accurate diagnosis are high priorities, and routine mammograms are
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18.4. Why Tree Diagrams Work 763

one of the first lines of defense. They’re not very accurate as far as medical tests go,
but they are correct between 90% and 95% of the time, which seems pretty good
for a relatively inexpensive non-invasive test.1 However, mammogram results are
also an example of conditional probabilities having counterintuitive consequences.
If the test was positive for breast cancer in you or a loved one, and the test is better
than 90% accurate, you’d naturally expect that to mean there is better than 90%
chance that the disease was present. But a mathematical analysis belies that naive
intuitive expectation. Let’s start by precisely defining how accurate a mammogram
is:

If you have the condition, there is a 10% chance that the test will say you do
not have it. This is called a “false negative.”

If you do not have the condition, there is a 5% chance that the test will say
you do. This is a “false positive.”

18.4.3 Four Steps Again
Now suppose that we are testing middle-aged women with no family history of
cancer. Among this cohort, incidence of breast cancer rounds up to about 1%.
Step 2: Define Events of Interest
Let A be the event that the person has breast cancer. Let B be the event that the
test was positive.
The outcomes
in each event are marked in the tree diagram. We
want to find Pr A j B , the probability that a person has breast cancer, given that
the test was positive.
Step 3: Find Outcome Probabilities
First, we assign probabilities to edges. These probabilities are drawn directly from
the problem statement. By the Product Rule, the probability of an outcome is the
product of the probabilities on the corresponding root-to-leaf path. All probabilities
are shown in Figure 18.2.
Step 4: Compute Event Probabilities
From Definition 18.2.1, we have
PrŒA \ B 0:009
Pr A j B D D 15:4%:
PrŒB 0:009 C 0:0495
So, if the test is positive, then there is an 84.6% chance that the result is incorrect,
even though the test is nearly 95% accurate! So this seemingly pretty accurate
1 The statistics in this example are roughly based on actual medical data, but have been rounded

or simplified for illustrative purposes.
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764 Chapter 18 Conditional Probability

Step 1: Find the Sample Space
The sample space is found with the tree diagram in Figure 18.2.

Figure 18.2 The tree diagram for a breast cancer test.
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18.4. Why Tree Diagrams Work 765

test doesn’t tell us much. To see why percent accuracy is no guarantee of value,
notice that there is a simple way to make a test that is 99% accurate: always return
a negative result! This test gives the right answer for all healthy people and the
wrong answer only for the 1% that actually have cancer. This 99% accurate test
tells us nothing; the “less accurate” mammogram is still a lot more useful.

18.4.4 Natural Frequencies
That there is only about a 15% chance that the patient actually has the condition
when the test say so may seem surprising at first, but it makes sense with a little
thought. There are two ways the patient could test positive: first, the patient could
have the condition and the test could be correct; second, the patient could be healthy
and the test incorrect. But almost everyone is healthy! The number of healthy
individuals is so large that even the mere 5% with false positive results overwhelm
the number of genuinely positive results from the truly ill.
Thinking like this in terms of these “natural frequencies” can be a useful tool for
interpreting some of the strange seeming results coming from those formulas. For
example, let’s take a closer look at the mammogram example.
Imagine 10,000 women in our demographic. Based on the frequency of the
disease, we’d expect 100 of them to have breast cancer. Of those, 90 would have
a positve result. The remaining 9,900 woman are healthy, but 5% of them—500,
give or take—will show a false positive on the mammogram. That gives us 90
real positives out of a little fewer than 600 positives. An 85% error rate isn’t so
surprising after all.

18.4.5 A Posteriori Probabilities
If you think about it much, the medical testing problem we just considered could
start to trouble you. You may wonder if a statement like “If someone tested positive,
then that person has the condition with probability 18%” makes sense, since a given
person being tested either has the disease or they don’t.
One way to understand such a statement is that it just means that 15% of the
people who test positive will actually have the condition. Any particular person has
it or they don’t, but a randomly selected person among those who test positive will
have the condition with probability 15%.
But what does this 15% probability tell you if you personally got a positive
result? Should you be relieved that there is less than one chance in five that you
have the disease? Should you worry that there is nearly one chance in five that you
do have the disease? Should you start treatment just in case? Should you get more
tests?
These are crucial practical questions, but it is important to understand that they
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are not mathematical questions. Rather, these are questions about statistical judge-
ments and the philosophical meaning of probability. We’ll say a bit more about this
after looking at one more example of after-the-fact probabilities.

The Hockey Team in Reverse
Suppose that we turn the hockey question around: what is the probability that the
local C-league hockey team won their first game, given that they won the series?
As we discussed earlier, some people find this question absurd. If the team has
already won the tournament, then the first game is long since over. Who won the
first game is a question of fact, not of probability. However, our mathematical
theory of probability contains no notion of one event preceding another. There
is no notion of time at all. Therefore, from a mathematical perspective, this is a
perfectly valid question. And this is also a meaningful question from a practical
perspective. Suppose that you’re told that the local team won the series, but not
told the results of individual games. Then, from your perspective, it makes perfect
sense to wonder how likely it isthat local
team won the first game.
A conditional probability Pr B j A is called a posteriori if event B precedes
event A in time. Here are some other examples of a posteriori probabilities:
The probability it was cloudy this morning, given that it rained in the after-
noon.
The probability that I was initially dealt two queens in Texas No Limit Hold
’Em poker, given that I eventually got four-of-a-kind.
from ordinary probabilities; the distinction comes from our view of causality, which
is a philosophical question rather than a mathematical one.
Let’s return to the original problem. The probability
that the local team won their
first game, given that they won the series is Pr B j A . We can compute this using
the definition of conditional probability and the tree diagram in Figure 18.1:
PrŒB \ A 1=3 C 1=18 7
Pr B j A D D D :
PrŒA 1=3 C 1=18 C 1=9 9
In general, such pairs of probabilities are related by Bayes’ Rule:
Theorem 18.4.1 (Bayes’ Rule).

Pr A j B PrŒB
Pr B j A D (18.2)
PrŒA
Proof. We have

Pr B j A PrŒA D PrŒA \ B D Pr A j B PrŒB
by definition of conditional probability. Dividing by PrŒA gives (18.2).
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18.4. Why Tree Diagrams Work 767

18.4.6 Philosphy of Probability
Let’s try to assign a probability to the event

Œ26972607 1 is a prime number

It’s not obvious how to check whether such a large number is prime, so you might
try an estimation based on the density of primes. The Prime Number Theorem
implies that only about 1 in 5 million numbers in this range are prime, so you might
say that the probability is about 2 10 8 . On the other hand, given that we chose
this example to make some philosophical point, you might guess that we probably
purposely chose an obscure looking prime number, and you might be willing to
make an even money bet that the number is prime. In other words, you might think
the probability is 1/2. Finally, we can take the position that assigning a probability
to this statement is nonsense because there is no randomness involved; the number
is either prime or it isn’t. This is the view we take in this text.
An alternate view is the Bayesian approach, in which a probability is interpreted
as a degree of belief in a proposition. A Bayesian would agree that the number
above is either prime or composite, but they would be perfectly willing to assign a
probability to each possibility. The Bayesian approach is very broad in its willing-
ness to assign probabilities to any event, but the problem is that there is no single
“right” probability for an event, since the probability depends on one’s initial be-
liefs. On the other hand, if you have confidence in some set of initial beliefs, then
Bayesianism provides a convincing framework for updating your beliefs as further
information emerges.
As an aside, it is not clear whether Bayes himself was Bayesian in this sense.
However, a Bayesian would be willing to talk about the probability that Bayes was
Bayesian.
Another school of thought says that probabilities can only be meaningfully ap-
plied to repeatable processes like rolling dice or flipping coins. In this frequen-
tist view, the probability of an event represents the fraction of trials in which the
event occurred. So we can make sense of the a posteriori probabilities of the C-
league hockey example of Section 18.4.5 by imagining that many hockey series
were played, and the probability that the local team won their first game, given that
they won the series, is simply the fraction of series where they won the first game
among all the series they won.
Getting back to prime numbers, we mentioned in Section 9.5.1 that there is a
probabilistic primality test. If a number N is composite, there is at least a 3=4
chance that the test will discover this. In the remaining 1=4 of the time, the test is
inconclusive. But as long as the result is inconclusive, the test can be run indepen-
dently again and again up to, say, 100 times. So if N actually is composite, then
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the probability that 000 repetitions of the probabilistic test do not discover this is at
most: 100
1
:
4
If the test remained inconclusive after 100 repetitions, it is still logically possible
that N is composite, but betting that N is prime would be the best bet you’ll ever get
to make! If you’re comfortable using probability to describe your personal belief
about primality after such an experiment, you are being a Bayesian. A frequentist
would not assign a probability to N ’s primality, but they would also be happy to
bet on primality with tremendous confidence. We’ll examine this issue again when
we discuss polling and confidence levels in Section 18.9.
Despite the philosophical divide, the real world conclusions Bayesians and Fre-
quentists reach from probabilities are pretty much the same, and even where their
interpretations differ, they use the same theory of probability.

18.5 The Law of Total Probability
Breaking a probability calculation into cases simplifies many problems. The idea
is to calculate the probability of an event A by splitting into two cases based on
whether or not another event E occurs. That is, calculate the probability of A \ E
and A \ E. By the Sum Rule, the sum of these probabilities equals PrŒA. Express-
ing the intersection probabilities as conditional probabilities yields:

Rule 18.5.1 (Law of Total Probability: single event).
ˇ
PrŒA D Pr A j E PrŒE C Pr A ˇ E PrŒE:

For example, suppose we conduct the following experiment. First, we flip a fair
coin. If heads comes up, then we roll one die and take the result. If tails comes up,
then we roll two dice and take the sum of the two results. What is the probability
that this process yields a 2? Let E be the event that the coin comes up heads,
and let A be the event that we get a 2 overall. Assuming that the coin is fair,
PrŒE D PrŒE D 1=2. There are now two cases. If we flip heads, then we roll
a 2 on a single die with probability Pr A j E D 1=6. On the other hand, if we
ˇ
flip tails, then we get a sum of 2 on two dice with probability Pr A ˇ E D 1=36.
Therefore, the probability that the whole process yields a 2 is
1 1 1 1 7
PrŒA D C D :
2 6 2 36 72
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This rule extends to any set of disjoint events that make up the entire sample
space. For example,
Rule (Law of Total Probability: 3-events). If E1 ; E2 and E3 are disjoint, and
PrŒE1 [ E2 [ E3 D 1, then

PrŒA D Pr A j E1 PrŒE1 C Pr A j E2 PrŒE2 C Pr A j E3 PrŒE3 :
This in turn leads to a three-event version of Bayes’ Rule in which the probability
of event E1 given A is calculated from the “inverse” conditional probabilities of A
given E1 , E2 , and E3 :
Rule (Bayes’ Rule: 3-events).

Pr A j E1 PrŒE1
Pr E1 j A D
Pr A j E1 PrŒE1 C Pr A j E2 PrŒE2 C Pr A j E3 PrŒE3
The generalization of these rules to n disjoint events is a routine exercise (Prob-
lems 18.3 and 18.4).

18.5.1 Conditioning on a Single Event
The probability rules that we derived in Section 17.5.2 extend to probabilities con-
ditioned on the same event. For example, the Inclusion-Exclusion formula for two
sets holds when all probabilities are conditioned on an event C :

Pr A [ B j C D Pr A j C C Pr B j C Pr A \ B j C :
This is easy to verify by plugging in the Definition 18.2.1 of conditional probabil-
ity.2
It is important not to mix up events before and after the conditioning bar. For
example, the following is not a valid identity:
False Claim.

Pr A j B [ C D Pr A j B C Pr A j C Pr A j B \ C : (18.3)
A simple counter-example is to let B and C be events over a uniform spacewith
most
of their
outcomes in A, but not overlapping. This ensures that Pr A j B and
Pr A j C are both close to 1. For example,
B WWD Œ0::9;
C WWD Œ10::18 [ f0g;
A WWD Œ1::18;
2 Problem 18.13 explains why this and similar conditional identities follow on general principles
from the corresponding unconditional identities.
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770 Chapter 18 Conditional Probability

so
9
Pr A j B D D Pr A j C :
10
Also, since 0 is the only outcome in B \ C and 0 … A, we have

Pr A j B \ C D 0

So the right-hand side of (18.3) is 1.8, while the left-hand side is a probability
which can be at most 1—actually, it is 18/19.

18.6 Simpson’s Paradox
In 1973, a famous university was investigated for gender discrimination [6]. The
investigation was prompted by evidence that, at first glance, appeared definitive: in
1973, 44% of male applicants to the school’s graduate programs were accepted, but
only 35% of female applicants were admitted.
However, this data turned out to be completely misleading. Analysis of the in-
dividual departments, showed not only that few showed significant evidence of
bias, but also that among the few departments that did show statistical irregulari-
ties, most were slanted in favor of women. This suggests that if there was any sex
discrimination, then it was against men!
Given the discrepancy in these findings, it feels like someone must be doing bad
math—intentionally or otherwise. But the numbers are not actually inconsistent.
In fact, this statistical hiccup is common enough to merit its own name: Simpson’s
Paradox occurs when multiple small groups of data all exhibit a similar trend, but
that trend reverses when those groups are aggregated. To explain how this is pos-
sible, let’s first clarify the problem by expressing both arguments in terms of con-
ditional probabilities. For simplicity, suppose that there are only two departments
EE and CS. Consider the experiment where we pick a random candidate. Define
the following events:

AWWD the candidate is admitted to his or her program of choice,

FEE WWD the candidate is a woman applying to the EE department,

FCS WWD the candidate is a woman applying to the CS department,

MEE WWD the candidate is a man applying to the EE department,

MCS WWD the candidate is a man applying to the CS department.
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18.6. Simpson’s Paradox 771

CS 2 men admitted out of 5 candidates 40%
50 women admitted out of 100 candidates 50%
EE 70 men admitted out of 100 candidates 70%
4 women admitted out of 5 candidates 80%
Overall 72 men admitted, 105 candidates 69%
54 women admitted, 105 candidates 51%

Table 18.1 A scenario in which men are overall more likely than women to be
admitted to a school, despite being less likely to be admitted into any given pro-
gram.

Assume that all candidates are either men or women, and that no candidate belongs
to both departments. That is, the events FEE , FCS , MEE and MCS are all disjoint.
In these terms, the plaintiff’s assertion—that a male candidate is more likely to
be admitted to the university than a female—can be expressed by the following
inequality:
Pr A j MEE [ MCS > Pr A j FEE [ FCS :
The university’s retort that in any given department, a male applicant is less
likely to be admitted than a female can be expressed by a pair of inequalities:

Pr A j MEE < Pr A j FEE and

Pr A j MCS < Pr A j FCS :

We can explain how there could be such a discrepancy between university-wide
and department-by-department admission statistics by supposing that the CS de-
partment is more selective than the EE department, but CS attracts a far larger
number of woman applicants than EE.3 . Table 18.1 shows some admission statis-
tics for which the inequalities asserted by both the plaintiff and the university hold.
Initially, we and the plaintiffs both assumed that the overall admissions statistics
for the university could only be explained by gender discrimination. The depart-
ment by department statistics seems to belie the accusation of discrimination. But
do they really?
Suppose we replaced “the candidate is a man/woman applying to the EE depart-
ment,” by “the candidate is a man/woman for whom an admissions decision was
made during an odd-numbered day of the month,” and likewise with CS and an
even-numbered day of the month. Since we don’t think the parity of a date is a
3 Atthe actual university in the lawsuit, the “exclusive” departments more popular among women
were those that did not require a mathematical foundation, such as English and education. Women’s
disproportionate choice of these careers reflects gender bias, but one which predates the university’s
involvement.
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772 Chapter 18 Conditional Probability

cause for the outcome of an admission decision, we would most likely dismiss the
“coincidence” that on both odd and even dates, women are more frequently admit-
ted. Instead we would judge, based on the overall data showing women less likely
to be admitted, that gender bias against women was an issue in the university.
Bear in mind that it would be the same numerical data that we would be using
to justify our different conclusions in the department-by-department case and the
even-day-odd-day case. We interpreted the same numbers differently based on our
implicit causal beliefs, specifically that departments matter and date parity does
not. It is circular to claim that the data corroborated our beliefs that there is or is not
discrimination. Rather, our interpretation of the data correlation depended on our
beliefs about the causes of admission in the first place.4 This example highlights
a basic principle in statistics which people constantly ignore: never assume that
correlation implies causation.

18.7 Independence
Suppose that we flip two fair coins simultaneously on opposite sides of a room.
Intuitively, the way one coin lands does not affect the way the other coin lands.
The mathematical concept that captures this intuition is called independence.

Definition 18.7.1. An event with probability 0 is defined to be independent of every
event (including itself). If PrŒB ¤ 0, then event A is independent of event B iff

Pr A j B D PrŒA: (18.4)

In other words, A and B are independent if knowing that B happens does not al-
ter the probability that A happens, as is the case with flipping two coins on opposite
sides of a room.

Potential Pitfall
Students sometimes get the idea that disjoint events are independent. The opposite
is true: if A \ B D ;, then knowing that A happens means you know that B
does not happen. Disjoint events are never independent—unless one of them has
probability zero.
4 These issues are thoughtfully examined in Causality: Models, Reasoning and Inference, Judea

Pearl, Cambridge U. Press, 2001.
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18.7. Independence 773

18.7.1 Alternative Formulation
Sometimes it is useful to express independence in an alternate form which follows
immediately from Definition 18.7.1:

Theorem 18.7.2. A is independent of B if and only if

PrŒA \ B D PrŒA PrŒB: (18.5)

Notice that Theorem 18.7.2 makes apparent the symmetry between A being in-
dependent of B and B being independent of A:

Corollary 18.7.3. A is independent of B iff B is independent of A.

18.7.2 Independence Is an Assumption
Generally, independence is something that you assume in modeling a phenomenon.
For example, consider the experiment of flipping two fair coins. Let A be the event
that the first coin comes up heads, and let B be the event that the second coin is
heads. If we assume that A and B are independent, then the probability that both
coins come up heads is:
1 1 1
PrŒA \ B D PrŒA PrŒB D D :
2 2 4
In this example, the assumption of independence is reasonable. The result of one
coin toss should have negligible impact on the outcome of the other coin toss. And
if we were to repeat the experiment many times, we would be likely to have A \ B
about 1/4 of the time.
On the other hand, there are many examples of events where assuming indepen-
dence isn’t justified. For example, an hourly weather forecast for a clear day might
list a 10% chance of rain every hour from noon to midnight, meaning each hour has
a 90% chance of being dry. But that does not imply that the odds of a rainless day
are a mere 0:912 0:28. In reality, if it doesn’t rain as of 5pm, the odds are higher
than 90% that it will stay dry at 6pm as well—and if it starts pouring at 5pm, the
chances are much higher than 10% that it will still be rainy an hour later.
Deciding when to assume that events are independent is a tricky business. In
practice, there are strong motivations to assume independence since many useful
formulas (such as equation (18.5)) only hold if the events are independent. But you
need to be careful: we’ll describe several famous examples where (false) assump-
tions of independence led to trouble. This problem gets even trickier when there
are more than two events in play.
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18.8 Mutual Independence
We have defined what it means for two events to be independent. What if there are
more than two events? For example, how can we say that the flips of n coins are
all independent of one another? A set of events is said to be mutually independent
if the probability of each event in the set is the same no matter which of the other
events has occurred. This is equivalent to saying that for any selection of two or
more of the events, the probability that all the selected events occur equals the
product of the probabilities of the selected events.
For example, four events E1 ; E2 ; E3 ; E4 are mutually independent if and only if
all of the following equations hold:
PrŒE1 \ E2 D PrŒE1 PrŒE2
PrŒE1 \ E3 D PrŒE1 PrŒE3
PrŒE1 \ E4 D PrŒE1 PrŒE4
PrŒE2 \ E3 D PrŒE2 PrŒE3
PrŒE2 \ E4 D PrŒE2 PrŒE4
PrŒE3 \ E4 D PrŒE3 PrŒE4
PrŒE1 \ E2 \ E3 D PrŒE1 PrŒE2 PrŒE3
PrŒE1 \ E2 \ E4 D PrŒE1 PrŒE2 PrŒE4
PrŒE1 \ E3 \ E4 D PrŒE1 PrŒE3 PrŒE4
PrŒE2 \ E3 \ E4 D PrŒE2 PrŒE3 PrŒE4
PrŒE1 \ E2 \ E3 \ E4 D PrŒE1 PrŒE2 PrŒE3 PrŒE4
The generalization to mutual independence of n events should now be clear.

18.8.1 DNA Testing
Assumptions about independence are routinely made in practice. Frequently, such
assumptions are quite reasonable. Sometimes, however, the reasonableness of an
independence assumption is not so clear, and the consequences of a faulty assump-
tion can be severe.
Let’s return to the O. J. Simpson murder trial. The following expert testimony
was given on May 15, 1995:
Mr. Clarke: When you make these estimations of frequency—and I believe you
touched a little bit on a concept called independence?
Dr. Cotton: Yes, I did.
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18.8. Mutual Independence 775

Mr. Clarke: And what is that again?

Dr. Cotton: It means whether or not you inherit one allele that you have is not—
does not affect the second allele that you might get. That is, if you inherit
a band at 5,000 base pairs, that doesn’t mean you’ll automatically or with
some probability inherit one at 6,000. What you inherit from one parent is
[independent of] what you inherit from the other.

Mr. Clarke: Why is that important?

Dr. Cotton: Mathematically that’s important because if that were not the case, it
would be improper to multiply the frequencies between the different genetic
locations.

Mr. Clarke: How do you—well, first of all, are these markers independent that
you’ve described in your testing in this case?
Presumably, this dialogue was as confusing to you as it was for the jury. Es-
sentially, the jury was told that genetic markers in blood found at the crime scene
matched Simpson’s. Furthermore, they were told that the probability that the mark-
ers would be found in a randomly-selected person was at most 1 in 170 million.
This astronomical figure was derived from statistics such as:
1 person in 100 has marker A.

1 person in 50 marker B.

1 person in 40 has marker C .

1 person in 5 has marker D.

1 person in 170 has marker E.
Then these numbers were multiplied to give the probability that a randomly-selected
person would have all five markers:

PrŒA \ B \ C \ D \ E D PrŒA PrŒB PrŒC PrŒD PrŒE
1 1 1 1 1 1
D D :
100 50 40 5 170 170;000;000
The defense pointed out that this assumes that the markers appear mutually in-
dependently. Furthermore, all the statistics were based on just a few hundred blood
samples.
After the trial, the jury was widely mocked for failing to “understand” the DNA
evidence. If you were a juror, would you accept the 1 in 170 million calculation?
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776 Chapter 18 Conditional Probability

18.8.2 Pairwise Independence
The definition of mutual independence seems awfully complicated—there are so
many selections of events to consider! Here’s an example that illustrates the sub-
tlety of independence when more than two events are involved. Suppose that we
flip three fair, mutually-independent coins. Define the following events:
A1 is the event that coin 1 matches coin 2.
A2 is the event that coin 2 matches coin 3.
A3 is the event that coin 3 matches coin 1.
Are A1 , A2 , A3 mutually independent?
The sample space for this experiment is:
fHHH; HH T; H TH; H T T; THH; TH T; T TH; T T T g:
Every outcome has probability .1=2/3 D 1=8 by our assumption that the coins are
mutually independent.
To see if events A1 , A2 and A3 are mutually independent, we must check a
sequence of equalities. It will be helpful first to compute the probability of each
event Ai :
PrŒA1 D PrŒHHH C PrŒHH T C PrŒT TH C PrŒT T T
1 1 1 1 1
D C C C D :
8 8 8 8 2
By symmetry, PrŒA2 D PrŒA3 D 1=2 as well. Now we can begin checking all the
equalities required for mutual independence:
1 1 1 1 1
PrŒA1 \ A2 D PrŒHHH C PrŒT T T D C D D
8 8 4 2 2
D PrŒA1 PrŒA2 :
By symmetry, PrŒA1 \ A3 D PrŒA1 PrŒA3 and PrŒA2 \ A3 D PrŒA2 PrŒA3
must hold also. Finally, we must check one last condition:

1 1 1
PrŒA1 \ A2 \ A3 D PrŒHHH C PrŒT T T D C D
8 8 4
1
¤ D PrŒA1 PrŒA2 PrŒA3 :
8
The three events A1 , A2 and A3 are not mutually independent even though any
two of them are independent! This not-quite mutual independence seems weird at
first, but it happens. It even generalizes:
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18.8. Mutual Independence 777

Definition 18.8.1. A set A1 , A2 , . . . , of events is k-way independent iff every set
of k of these events is mutually independent. The set is pairwise independent iff it
is 2-way independent.

So the events A1 , A2 , A3 above are pairwise independent, but not mutually inde-
pendent. Pairwise independence is a much weaker property than mutual indepen-
dence.
For example, suppose that the prosecutors in the O. J. Simpson trial were wrong
and markers A, B, C , D and E are only pairwise independently. Then the proba-
bility that a randomly-selected person has all five markers is no more than:

PrŒA \ B \ C \ D \ E PrŒA \ E D PrŒA PrŒE
1 1 1
D D :
100 170 17;000
The first line uses the fact that A \ B \ C \ D \ E is a subset of A \ E. (We picked
out the A and E markers because they’re the rarest.) We use pairwise independence
on the second line. Now the probability of a random match is 1 in 17,000—a far cry
from 1 in 170 million! And this is the strongest conclusion we can reach assuming
only pairwise independence.
On the other hand, the 1 in 17,000 bound that we get by assuming pairwise
independence is a lot better than the bound that we would have if there were no
independence at all. For example, if the markers are dependent, then it is possible
that

everyone with marker E has marker A,
everyone with marker A has marker B,
everyone with marker B has marker C , and
everyone with marker C has marker D.

In such a scenario, the probability of a match is
1
PrŒE D :
170
So a stronger independence assumption leads to a smaller bound on the prob-
ability of a match. The trick is to figure out what independence assumption is
reasonable. Assuming that the markers are mutually independent may well not be
reasonable unless you have examined hundreds of millions of blood samples. Oth-
erwise, how would you know that marker D does not show up more frequently
whenever the other four markers are simultaneously present?
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778 Chapter 18 Conditional Probability

18.9 Probability versus Confidence
Let’s look at some other problems like the breast cancer test of Section 18.4.2, but
this time we’ll use more extreme numbers to highlight some key issues.

18.9.1 Testing for Tuberculosis
Let’s suppose we have a really terrific diagnostic test for tuberculosis (TB): if you
have TB, the test is guaranteed to detect it, and if you don’t have TB, then the test
will report that correctly 99% of the time!
In other words, let “TB” be the event that a person has TB, “pos” be the event
that the person tests positive for TB, so “pos” is the event that they test negative.
Now we can restate these guarantees in terms of conditional probabilities:

Pr pos j TB D 1; (18.6)
ˇ
Pr pos ˇ TB D 0:99: (18.7)

This means that the test produces the correct result at least 99% of the time,
regardless of whether or not the person has TB. A careful statistician would assert:5

Lemma. You can be 99% confident that the test result is correct.

Corollary 18.9.1. If you test positive, then

either you have TB or something very unlikely (probability 1/100) hap-
pened.

Lemma 18.9.1 and Corollary 18.9.1 may seem to be saying that

False Claim. If you test positive, then the probability that you have TB is 0:99.

But this would be a mistake.
To highlight the difference between confidence in the test diagnosis versus the
probability of TB, let’s think about what to do if you test positive. Corollary 18.9.1
5 Confidence is usually used to describe the probability that a statistical estimations of some quan-

tity is correct (Section 20.5). We are trying to simplify the discussion by using this one concept to
illustrate standard approaches to both hypothesis testing and estimation.
In the context of hypothesis testing, statisticians would normally distinguish the “false positive”
probability, in this case the probability 0.01 that a healthy person is incorrectly diagnosed as having
TB, and call this the significance of the test. The “false negative” probability would be the probability
that person with TB is incorrectly diagnosed as healthy; it is zero. The power of the test is one minus
the false negative probability, so in this case the power is the highest possible, namely, one.
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18.9. Probability versus Confidence 779

seems to suggest that it’s worth betting with high odds that you have TB, because
it makes sense to bet against something unlikely happening—like the test being
wrong. But having TB actually turns out to be a lot less likely than the test being
wrong. So the either-or of Corollary 18.9.1 is really an either-or between some-
thing happening that is extremely unlikely—having TB—and something that is
only very unlikely—the diagnosis being wrong. You’re better off betting against
the extremely unlikely event, that is, it is better to bet the diagnosis is wrong.
So some knowledge of the probability of having TB is needed in order to figure
out how seriously to take a positive diagnosis, even when the diagnosis is given with
what seems like a high level of confidence. We can see exactly how the frequency
of TB in a population influences the importance of a positive diagnosis by actually
calculating the probability that someone who tests positive has TB. That is, we want
to calculate Pr TB j pos , which we do next.

18.9.2 Updating the Odds
Bayesian Updating
A standard way to convert the test probabilities into outcome probabilities is to use
Bayes Theorem (18.2). It will be helpful to rephrase Bayes Theorem in terms of
“odds” instead of probabilities.
If H is an event, we define the odds of H to be
PrŒH PrŒH
Odds.H / WWD D :
PrŒH 1 PrŒH

For example, if H is the event of rolling a four using a fair, six-sided die, then

PrŒroll four D 1=6; so
1=6 1
Odds.roll four/ D D :
5=6 5
A gambler would say the odds of rolling a four were “one to five,” or equivalently,
“five to one against” rolling a four.
Odds are just another way to talk about probabilities. For example, saying the
odds that a horse will win a race are “three to one” means that the horse will win
with probability 1=4. In general,
Odds.H /
PrŒH D :
1 C Odds.H /
Now suppose an event E offers some evidence about H . We now want to find
the conditional probability of H given E. We can just as well find the odds of H
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780 Chapter 18 Conditional Probability

given E,

Pr H j E
Odds.H j E/ WWD ˇ
Pr H ˇ E

Pr E j H PrŒH = PrŒE
D ˇ (Bayes Theorem)
Pr E ˇ H PrŒH = PrŒE

Pr E j H PrŒH
D ˇ
Pr E ˇ H PrŒH
D Bayes-factor.E; H / Odds.H /;

where
Pr E j H
Bayes-factor.E; H / WWD ˇ :
Pr E ˇ H
So to update the odds of H given the evidence E, we just multiply by Bayes Factor:

Lemma 18.9.2.

Odds.H j E/ D Bayes-factor.E; H / Odds.H /:

Odds for the TB test
The probabilities of test outcomes given in (18.6) and (18.7) are exactly what we
need to find Bayes factor for the TB test:

Pr pos j TB
Bayes-factor.TB; pos/ D ˇ
Pr pos ˇ TB
1
D ˇ
1 Pr pos ˇ TB
1
D D 100:
1 0:99
So testing positive for TB increases the odds you have TB by a factor of 100, which
means a positive test is significant evidence supporting a diagnosis of TB. That
seems good to know. But Lemma 18.9.2 also makes it clear that when a random
person tests positive, we still can’t determine the odds they have TB unless we
know what are the odds of their having TB in the first place, so let’s examine that.
In 2011, the United States Center for Disease Control got reports of 11,000 cases
of TB in US. We can estimate that there were actually about 30,000 cases of TB
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18.9. Probability versus Confidence 781

that year, since it seems that only about one third of actual cases of TB get reported.
The US population is a little over 300 million, which means
30; 000 1
PrŒTB D :
300; 000; 000 10; 000
So the odds of TB are 1=9999. Therefore,
1 1
Odds.TB j pos/ D 100 :
9; 999 100
In other words, even if someone tests positive for TB at the 99% confidence level,
the odds remain about 100 to one against their having TB. The 99% confidence
level is not nearly high enough to overcome the relatively tiny probability of having
TB.

18.9.3 Facts that are Probably True
We have figured out that if a random person tests positive for TB, the probability
they have TB is about 1/100. Now if you personally happened to test positive for
TB, a competent doctor typically would tell you that the probability that you have
TB has risen from 1/10,000 to 1/100. But has it? Not really.
Your doctor should have not have been talking in this way about your particular
situation. He should just have stuck to the statement that for randomly chosen
people, the positive test would be right only one percent of the time. But you are
not a random person, and whether or not you have TB is a fact about reality. The
truth about your having TB may be unknown to your doctor and you, but that does
not mean it has some probability of being true. It is either true or false, we just
don’t know which.
In fact, if you were worried about a 1/100 probability of having this serious
disease, you could use additional information about yourself to change this proba-
bility. For example, native born residents of the US are about half as likely to have
TB as foreign born residents. So if you are native born, “your” probability of hav-
ing TB halves. Conversely, TB is twenty-five times more frequent among native
born Asian/Pacific Islanders than native born Caucasions. So your probability of
TB would increase dramatically if your family was from an Asian/Pacific Island.
The point is that the probability of having TB that your doctor reports to you
depends on the probability of TB for a random person whom the doctor thinks
is like you. The doctor has made a judgment about you based, for example, on
what personal factors he considers relevant to getting TB, or how serious he thinks
the consequences of a mistaken diagnosis would be. These are important medical
judgments, but they are not mathematical. Different doctors will make different
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782 Chapter 18 Conditional Probability

judgments about who is like you, and they will report differing probabilities. There
is no “true” model of who you are, and there is no true individual probability of
your having TB.

18.9.4 Extreme events
The definition of a fair coin is one where the probability of flipping a Head is 1/2
and likewise for flipping a Tail. Now suppose you flip the coin one hundred times
and get a Head every time. What do you think the odds are that the next flip will
also be a Head?
The official answer is that, by definition of “fair coin,” the probability of Heads
on the next flip is still 1/2. But this reasoning completely contradicts what any
sensible person would do, which is to bet heavily on the next flip being another
Head.
How to make sense of this? To begin, let’s recognize how absurd it is to wonder
about what happens after one hundred heads, because the probability that a hundred
flips of a fair coin will all come up Heads is unimaginably tiny. For example, the
probability that just the first fifty out of the hundred fair flips come up Heads is 2 50 .
We can try to make some sense of how small this number is with the observation
that, using a reasonable estimation of the number of people worldwide who are
killed by lightning in a given year, 2 50 is about equal to the probability that a
random person would be struck by lightning during the time it takes to read this
paragraph. Ain’t gonna happen.
The negligible probability that one hundred flips of a fair coin will all be Heads
simply undermines the credibility of the assumption that the coin is fair. Despite
being told the coin is fair, we can’t help but acknowledge at least some remote
possibility that the coin being flipped was one that rarely produced heads. So let’s
assume that there are two coins, a fair one and a biased one that comes up Heads
with probability 99/100. One of these coins is randomly chosen with the fair coin
hugely favored: the biased coin will be chosen only with extremely small probabil-
ity 2 50 . The chosen coin is then flipped one hundred times. Let E be the event of
flipping one hundred heads and H be the event that the biased coin was chosen.
Now
2 50
Odds.H / D 2 50 ;
1 2 50
Pr E j H .99=100/100
Bayes-factor.E; H / D ˇ D > 0:36 2100 ;
Pr E ˇ H 2 100
Odds.H j E/ D Bayes-factor.E; H / Odds.H /
> 0:36 2100 2 50
D 0:36 250 :
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18.9. Probability versus Confidence 783

This shows that after flipping one hundred heads, the odds that the biased coin was
chosen are overwhelming, and so with high probability the next flip will be a Head.
Thus, by assuming some tiny probability for the coin being heavily biased toward
Heads, we can justify our intuition that after one hundred consecutive Heads, the
next flip is very likely to be a Head.
Making an assumption about the probability that some unverified fact is true is
known as the Bayesian approach to a hypthesis testing problem. By granting a tiny
probability that the biased coin was being flipped, this Bayesian approach provided
a reasonable justification for estimating that the odds of a Head on the next flip are
ninety-nine to one in favor.

18.9.5 Confidence in the Next Flip
If we stick to confidence rather than probability, we don’t need to make any Bayesian
assumptions about the probability of a fair coin. We know that if one hundred
Heads are flipped, then either the coin is biased, or something that virtually never
happens (probability 2 100 ) has occurred. That means we can assert that the coin
is biased at the 1 2 100 confidence level. In short, when one hundred Heads are
flipped, we can be essentially 100% confident that the coin is biased.

Problems for Section 18.4
Homework Problems
Problem 18.1.
The Conditional Probability Product Rule for n Events is

Rule.

PrŒE1 \ E2 \ : : : \ En D PrŒE1 Pr E2 j E1 Pr E3 j E1 \ E2

Pr En j E1 \ E2 \ : : : \ En 1 :

(a) Restate the Rule without using elipses (. . . ).

(b) Prove it by induction.
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784 Chapter 18 Conditional Probability

Problems for Section 18.5
Practice Problems
Problem 18.2.
Dirty Harry places two bullets in random chambers of the six-bullet cylinder of his
revolver. He gives the cylinder a random spin and says “Feeling lucky?” as he
holds the gun against your heart.
(a) What is the probability that you will get shot if he pulls the trigger?

(b) Suppose he pulls the trigger and you don’t get shot. What is the probability
that you will get shot if he pulls the trigger a second time?

(c) Suppose you noticed that he placed the two shells next to each other in the
cylinder. How does this change the answers to the previous two questions?

Problem 18.3.
State and prove a version of the Law of Total Probability that applies to disjoint
events E1 ; : : : ; En whose union is the whole sample space.

Problem 18.4.
State and prove a version of Bayes Rule that applies to disjoint events E1 ; : : : ; En
whose union is the whole sample space. You may assume the n-event Law of Total
Probability, Problem 18.3.

Class Problems
Problem 18.5.
There are two decks of cards. One is complete, but the other is missing the ace
of spades. Suppose you pick one of the two decks with equal probability and then
select a card from that deck uniformly at random. What is the probability that you
picked the complete deck, given that you selected the eight of hearts? Use the
four-step method and a tree diagram.

Problem 18.6.
Suppose you have three cards: A~, A and a jack. From these, you choose a
random hand (that is, each card is equally likely to be chosen) of two cards, and let
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18.9. Probability versus Confidence 785

n be the number of aces in your hand. You then randomly pick one of the cards in
the hand and reveal it.
(a) Describe a simple probability space (that is, outcomes and their probabilities)
for this scenario, and list the outcomes in each of the following events:

1. Œn 1, (that is, your hand has an ace in it),
2. A~ is in your hand,
3. the revealed card is an A~,
4. the revealed card is an ace.

(b) Then calculate Pr n D 2 j E for E equal to each of the four events in part (a).
Notice that most, but not all, of these probabilities are equal.
Now suppose you have a deck with d distinct cards, a different kinds of aces
(including an A~), you draw a random hand with h cards, and then reveal a random
card from your hand.
(c) Prove that PrŒA~ is in your hand D h=d .

(d) Prove that
2d
Pr n D 2 j A~ is in your hand D PrŒn D 2 : (18.8)
ah

(e) Conclude that

Pr n D 2 j the revealed card is an ace D Pr n D 2 j A~ is in your hand :

Problem 18.7.
There are three prisoners in a maximum-security prison for fictional villains: the
Evil Wizard Voldemort, the Dark Lord Sauron, and Little Bunny Foo-Foo. The
parole board has declared that it will release two of the three, chosen uniformly at
random, but has not yet released their names. Naturally, Sauron figures that he will
be released to his home in Mordor, where the shadows lie, with probability 2=3.
A guard offers to tell Sauron the name of one of the other prisoners who will be
released (either Voldemort or Foo-Foo). If the guard has a choice of naming either
Voldemort or Foo-Foo (because both are to be released), he names one of the two
with equal probability.
Sauron knows the guard to be a truthful fellow. However, Sauron declines this
offer. He reasons that knowing what the guards says will reduce his chances, so he
is better off not knowing. For example, if the guard says, “Little Bunny Foo-Foo
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786 Chapter 18 Conditional Probability

will be released”, then his own probability of release will drop to 1=2 because he
will then know that either he or Voldemort will also be released, and these two
events are equally likely.
Dark Lord Sauron has made a typical mistake when reasoning about conditional
probability. Using a tree diagram and the four-step method, explain his mistake.
What is the probability that Sauron is released given that the guard says Foo-Foo is
released?
Hint: Define the events S, F and “F ” as follows:

“F ” D Guard says Foo-Foo is released
F D Foo-Foo is released
S D Sauron is released

Problem 18.8.
Every Skywalker serves either the light side or the dark side.

The first Skywalker serves the dark side.

For n 2, the n-th Skywalker serves the same side as the .n 1/-st Sky-
walker with probability 1=4, and the opposite side with probability 3=4.

Let dn be the probability that the n-th Skywalker serves the dark side.
(a) Express dn with a recurrence equation and sufficient base cases.

P1
(b) Derive a simple expression for the generating function D.x/ WWD 1 dn x n .

Problem 18.9. (a) For the directed acyclic graph (DAG) G0 in Figure 18.3, a
minimum-edge DAG with the same walk relation can be obtained by removing
some edges. List these edges (use notation hu ! vi for an edge from u to v):

(b) List the vertices in a maximal chain in G0 .
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18.9. Probability versus Confidence 787

Figure 18.3 The DAG G0
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788 Chapter 18 Conditional Probability

Figure 18.4 Simple graph G

Let G be the simple graph shown in Figure 18.4.
!
A directed graph G can be randomly constructed from G by assigning a direction
to each edge independently with equal likelihood.
!
(c) What is the probability that G D G0 ?

!
Define the following events with respect to the random graph G :

T1 WWD vertices 2; 3; 4 are on a length three directed cycle;
T2 WWD vertices 1; 3; 4 are on a length three directed cycle;
T3 WWD vertices 1; 2; 4 are on a length three directed cycle;
T4 WWD vertices 1; 2; 3 are on a length three directed cycle:
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18.9. Probability versus Confidence 789

(d) What is

PrŒT1 ‹

PrŒT1 \ T2 ‹

PrŒT1 \ T2 \ T3 ‹

!
(e) G has the property that if it has a directed cycle, then it has a length three
!
directed cycle. Use this fact to find the probability that G is a DAG.

Homework Problems
Problem 18.10.
There is a subject—naturally not Math for Computer Science—in which 10% of the
assigned problems contain errors. If you ask a Teaching Assistant (TA) whether a
problem has an error, then they will answer correctly 80% of the time, regardless
of whether or not a problem has an error. If you ask a lecturer, he will identify
whether or not there is an error with only 75% accuracy.
We formulate this as an experiment of choosing one problem randomly and ask-
ing a particular TA and Lecturer about it. Define the following events:

E WWD Œthe problem has an error;
T WWD Œthe TA says the problem has an error;
L WWD Œthe lecturer says the problem has an error:

(a) Translate the description above into a precise set of equations involving con-
ditional probabilities among the events E, T and L.

(b) Suppose you have doubts about a problem and ask a TA about it, and they tell
you that the problem is correct. To double-check, you ask a lecturer, who says that
the problem has an error. Assuming that the correctness of the lecturer’s answer
and the TA’s answer are independent of each other, regardless of whether there is
an error, what is the probability that there is an error in the problem?

(c) Is event T independent of event L (that is, Pr T j L D PrŒT )? First, give
an argument based on intuition, and then calculate both probabilities to verify your
intuition.
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790 Chapter 18 Conditional Probability

Problem 18.11.
Suppose you repeatedly flip a fair coin until you see the sequence HTT or HHT.
What is the probability you see the sequence HTT first?
Hint: Try to find the probability that HHT comes before HTT conditioning on
whether you first toss an H or a T. The answer is not 1=2.

Problem 18.12.
A 52-card deck is thoroughly shuffled and you are dealt a hand of 13 cards.
(a) If you have one ace, what is the probability that you have a second ace?

(b) If you have the ace of spades, what is the probability that you have a second
ace? Remarkably, the answer is different from part (a).

Problem 18.13.
Suppose PrŒ W S ! Œ0; 1 is a probability function on a sample space S and let B
be an event such that PrŒB > 0. Define a function PrB Œ on outcomes ! 2 S by
the rule: (
PrŒ!= PrŒB if ! 2 B;
PrB Œ! WWD (18.9)
0 if ! … B:
(a) Prove that PrB Œ is also a probability function on S according to Defini-
tion 17.5.2.

(b) Prove that
PrŒA \ B
PrB ŒA D
PrŒB
for all A S.

Pr C [ D j B D Pr C j B C Pr D j B .C; D disjoint/:

Give examples of several further such rules.

Problem 18.14.
Professor Meyer has a deck of 52 randomly shuffled playing cards, 26 red, 26 black.
He proposes the following game: he will repeatedly draw a card off the top of the
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18.9. Probability versus Confidence 791

deck and turn it face up so that you can see it. At any point while there are still
cards left in the deck, you may choose to stop, and he will turn over the next card.
If the turned up card is black you win, and otherwise you lose. Either way, the
game ends.
Suppose that after drawing off some top cards without stopping, the deck is left
with r red cards and b black cards.
(a) Show that if you choose to stop at this point, the probability of winning is
b=.r C b/.

(b) Prove if you choose not to stop at this point, the probability of winning is still
b=.r C b/, regardless of your stopping strategy for the rest of the game.
Hint: Induction on r C b.

Exam Problems
Problem 18.15.
Sally Smart just graduated from high school. She was accepted to three reputable
colleges.

With probability 4=12, she attends Yale.
With probability 5=12, she attends MIT.
With probability 3=12, she attends Little Hoop Community College.

Sally is either happy or unhappy in college.

If she attends Yale, she is happy with probability 4=12.
If she attends MIT, she is happy with probability 7=12.
If she attends Little Hoop, she is happy with probability 11=12.

(a) A tree diagram to help Sally project her chance at happiness is shown below.
On the diagram, fill in the edge probabilities, and at each leaf write the probability
of the corresponding outcome.

(b) What is the probability that Sally is happy in college?

(d) Show that the event that Sally attends Yale is not independent of the event that
she is happy.
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792 Chapter 18 Conditional Probability

happy

unhappy
Yale

happy
MIT

unhappy

Little Hoop
happy

unhappy

(e) Show that the event that Sally attends MIT is independent of the event that she
is happy.

Problem 18.16.
Here’s a variation of Monty Hall’s game: the contestant still picks one of three
doors, with a prize randomly placed behind one door and goats behind the other
two. But now, instead of always opening a door to reveal a goat, Monty instructs
Carol to randomly open one of the two doors that the contestant hasn’t picked. This
means she may reveal a goat, or she may reveal the prize. If she reveals the prize,
then the entire game is restarted, that is, the prize is again randomly placed behind
some door, the contestant again picks a door, and so on until Carol finally picks a
door with a goat behind it. Then the contestant can choose to stick with his original
choice of door or switch to the other unopened door. He wins if the prize is behind
the door he finally chooses.
To analyze this setup, we define two events:

GP : The event that the contestant guesses the door with the prize behind it on his
first guess.

OP : The event that the game is restarted at least once. Another way to describe
this is as the event that the door Carol first opens has a prize behind it.

Give the values of the following probabilities:
(a) PrŒGP
ˇ
(b) Pr OP ˇ GP

18.9. Probability versus Confidence 793

(d) the probability that the game will continue forever

(e) When Carol finally picks the goat, the contestant has the choice of sticking or
switching. Let’s say that the contestant adopts the strategy of sticking. Let W be
the event that the contestant wins with this strategy, and let w WWD PrŒW . Express
the following conditional probabilities as simple closed forms in terms of w.

i) Pr W j GP
ˇ
ii) Pr W ˇ GP \ OP
ˇ
iii) Pr W ˇ GP \ OP

(f) What is the value of PrŒW ?

(g) For any final outcome where the contestant wins with a “stick” strategy, he
would lose if he had used a “switch” strategy, and vice versa. In the original Monty
Hall game, we concluded immediately that the probability that he would win with
a “switch” strategy was 1 PrŒW . Why isn’t this conclusion quite as obvious for
this new, restartable game? Is this conclusion still sound? Briefly explain.

Problem 18.17.
There are two decks of cards, the red deck and the blue deck. They differ slightly
in a way that makes drawing the eight of hearts slightly more likely from the red
deck than from the blue deck.
One of the decks is randomly chosen and hidden in a box. You reach in the
box and randomly pick a card that turns out to be the eight of hearts. You believe
intuitively that this makes the red deck more likely to be in the box than the blue
deck.
Your intuitive judgment about the red deck can be formalized and verified using
some inequalities between probabilities and conditional probabilities involving the
events

R WWD Red deck is in the box;
B WWD Blue deck is in the box;
E WWD Eight of hearts is picked from the deck in the box:

(a) State an inequality between probabilities and/or conditional probabilities that
formalizes the assertion, “picking the eight of hearts from the red deck is more
likely than from the blue deck.”
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794 Chapter 18 Conditional Probability

(b) State a similar inequality that formalizes the assertion “picking the eight of
hearts from the deck in the box makes the red deck more likely to be in the box
than the blue deck.”

(c) Assuming the each deck is equally likely to be the one in the box, prove that
the inequality of part (a) implies the inequality of part (b).

(d) Suppose you couldn’t be sure that the red deck and blue deck were equally
likely to be in the box. Could you still conclude that picking the eight of hearts
from the deck in the box makes the red deck more likely to be in the box than the
blue deck? Briefly explain.

Problem 18.18.
A flip of Coin 1 is x times as likely to come up Heads as a flip of Coin 2. A
biased random choice of one of these coins will be made, where the probability of
choosing Coin 1 is w times that of Coin 2.
(a) Restate the information above as equations between conditional probabilities
involving the events

C1 WWD Coin 1 was chosen;
C 2 WWD Coin 2 was chosen;
H WWD the chosen coin came up Heads:

(b) State an inequality involving conditional probabilities of the above events that
formalizes the assertion “Given that the chosen coin came up Heads, the chosen
coin is more likely to have been Coin 1 than Coin 2.”

(c) Prove that, given that the chosen coin came up Heads, the chosen coin is more
likely to have been Coin 1 than Coin 2 iff

wx > 1:

Problem 18.19.
There is an unpleasant, degenerative disease called Beaver Fever which causes peo-
ple to tell math jokes unrelentingly in social settings, believing other people will
think they’re funny. Fortunately, Beaver Fever is rare, afflicting only about 1 in
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18.9. Probability versus Confidence 795

1000 people. Doctor Meyer has a fairly reliable diagnostic test to determine who is
going to suffer from this disease:

If a person will suffer from Beaver Fever, the probability that Dr. Meyer
diagnoses this is 0.99.

If a person will not suffer from Beaver Fever, the probability that Dr. Meyer
diagnoses this is 0.97.

Let B be the event that a randomly chosen person will suffer Beaver Fever, and Y
be the event that Dr. Meyer’s diagnosis is “Yes, this person will suffer from Beaver
Fever,” with B and Y being the complements of these events.
(a) The description above explicitly gives the values of the following quantities.
What are their values?
ˇ
PrŒB Pr Y j B Pr Y ˇ B
ˇ
(b) Write formulas for PrŒB and Pr Y ˇ B solely in terms of the explicitly given
quantities in part (a)—literally use their expressions, not their numeric values.

from Beaver Fever solely in terms of PrŒB, PrŒB, Pr Y j B and Pr Y ˇ B .

(d) Write a formula solely in terms of the expressions given in part (a) for the
probability that a person will suffer Beaver Fever given that Doctor Meyer says
they will. Then calculate the numerical value of the formula.

Suppose there was a vaccine to prevent Beaver Fever, but the vaccine was expen-
sive or slightly risky itself. If you were sure you were going to suffer from Beaver
Fever, getting vaccinated would be worthwhile, but even if Dr. Meyer diagnosed
you as a future sufferer of Beaver Fever, the probability you actually will suffer
Beaver Fever remains low (about 1/32 by part (d)).
In this case, you might sensibly decide not to be vaccinated—after all, Beaver
Fever is not that bad an affliction. So the diagnostic test serves no purpose in your
case. You may as well not have bothered to get diagnosed. Even so, the test may
be useful:
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796 Chapter 18 Conditional Probability

(e) Suppose Dr. Meyer had enough vaccine to treat 2% of the population. If he
randomly chose people to vaccinate, he could expect to vaccinate only 2% of the
people who needed it. But by testing everyone and only vaccinating those diag-
nosed as future sufferers, he can expect to vaccinate a much larger fraction people
who were going to suffer from Beaver Fever. Estimate this fraction.

Problem 18.20.
Suppose that Let’s Make a Deal is played according to slightly different rules and
with a red goat and a blue goat. There are three doors, with a prize hidden behind
one of them and the goats behind the others. No doors are opened until the con-
testant makes a final choice to stick or switch. The contestant is allowed to pick a
door and ask a certain question that the host then answers honestly. The contestant
may then stick with their chosen door, or switch to either of the other doors.
(a) If the contestant asks “is there is a goat behind one of the unchosen doors?”
and the host answers “yes,” is the contestant more likely to win the prize if they
stick, switch, or does it not matter? Clearly identify the probability space of out-
comes and their probabilities you use to model this situation. What is the contes-
tant’s probability of winning if he uses the best strategy?

(b) If the contestant asks “is the red goat behind one of the unchosen doors?” and
the host answers “yes,” is the contestant more likely to win the prize if they stick,
switch, or does it not matter? Clearly identify the probability space of outcomes
and their probabilities you use to model this situation. What is the contestant’s
probability of winning if he uses the best strategy?

Problem 18.21.
You are organizing a neighborhood census and instruct your census takers to knock
on doors and note the sex of any child that answers the knock. Assume that there
are two children in every household, that a random child is equally likely to be a
girl or a boy, and that the two children in a household are equally likely to be the
one that opens the door.
A sample space for this experiment has outcomes that are triples whose first
element is either B or G for the sex of the elder child, whose second element is
either B or G for the sex of the younger child, and whose third element is E or Y
indicating whether the elder child or younger child opened the door. For example,
.B; G; Y/ is the outcome that the elder child is a boy, the younger child is a girl, and
the girl opened the door.
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18.9. Probability versus Confidence 797

(a) Let O be the event that a girl opened the door, and let T be the event that the
household has two girls. List the outcomes in O and T.

(b) What is the probability Pr T j O , that both children are girls, given that a
girl opened the door?

(c) What mistake is made in the following argument? (Note: merely stating the
correct probability is not an explanation of the mistake.)

If a girl opens the door, then we know that there is at least one girl in the
household. The probability that there is at least one girl is

1 PrŒboth children are boys D 1 .1=2 1=2/ D 3=4:

So,

Pr T j there is at least one girl in the household
PrŒT \ there is at least one girl in the household
D
PrŒthere is at least one girl in the household
PrŒT
D
PrŒthere is at least one girl in the household
D .1=4/=.3=4/ D 1=3:

Therefore, given that a girl opened the door, the probability that there
are two girls in the household is 1/3.

Problem 18.22.
A guard is going to release exactly two of the three prisoners, Sauron, Voldemort,
and Bunny Foo Foo, and he’s equally likely to release any set of two prisoners.
(a) What is the probability that Voldemort will be released?

The guard will truthfully tell Voldemort the name of one of the prisoners to be
released. We’re interested in the following events:

V : Voldemort is released.

“F ”: The guard tells Voldemort that Foo Foo will be released.

“S”: The guard tells Voldemort that Sauron will be released.

The guard has two rules for choosing whom he names:
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798 Chapter 18 Conditional Probability

never say that Voldemort will be released,

if both Foo Foo and Sauron are getting released, say “Foo Foo.”

(b) What is Pr V j “F ” ?

(d) Show how to use the Law of Total Probability to combine your answers to
parts (b) and (c) to verify that the result matches the answer to part (a).

Problem 18.23.
We are interested in paths in the plane starting at .0; 0/ that go one unit right or one
unit up at each step. To model this, we use a state machine whose states are N N,
whose start state is .0; 0/, and whose transitions are

.x; y/ ! .x C 1; y/;
.x; y/ ! .x; y C 1/:

(a) How many length n paths are there starting from the origin?

(b) How many states are reachable in exactly n steps?

(d) If transitions occur independently at random, going right with probability p
and up with probability q WWD 1 p at each step, what is the probability of reaching
position .x; y/?

(e) What is the probability of reaching state .x; y/ given that the path to .x; y/
reached .m; n/ before getting to .x; y/?

(f) Show that the probability that a path ending at .x; y/ went through .m; n/ is
the same for all p.
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18.9. Probability versus Confidence 799

Problems for Section 18.6
Practice Problems
Problem 18.24.
Define the events A; FEE ; FCS ; MEE , and MCS as in Section 18.6.
In these terms, the plaintiff in a discrimination suit against a university makes
the argument that in both departments, the probability that a female is admitted is
less than the probability for a male. That is,

Pr A j FEE < Pr A j MEE and (18.10)

Pr A j FCS < Pr A j MCS : (18.11)
The university’s defence attorneys retort that overall, a female applicant is more
likely to be admitted than a male, namely, that

Pr A j FEE [ FCS > Pr A j MEE [ MCS : (18.12)
The judge then interrupts the trial and calls the plaintiff and defence attorneys to
a conference in his office to resolve what he thinks are contradictory statements of
facts about the admission data. The judge points out that:

Pr A j FEE [ FCS

D Pr A j FEE C Pr A j FCS (because FEE and FCS are disjoint)

< Pr A j MEE C Pr A j MCS (by (18.10) and (18.11))

D Pr A j MEE [ MCS (because MEE and MCS are disjoint)
so
Pr A j FEE [ FCS < Pr A j MEE [ MCS ;
which directly contradicts the university’s position (18.12)!
Of course the judge is mistaken; an example where the plaintiff and defence
assertions are all true appears in Section 18.6. What is the mistake in the judge’s
proof?

Problems for Section 18.7
Practice Problems
Problem 18.25.
Outside of their hum-drum duties as Math for Computer Science Teaching Assis-
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800 Chapter 18 Conditional Probability

tants, Oscar is trying to learn to levitate using only intense concentration and Liz is
trying to become the world champion flaming torch juggler. Suppose that Oscar’s
probability of success is 1=6, Liz’s chance of success is 1=4, and these two events
are independent.
(a) If at least one of them succeeds, what is the probability that Oscar learns to
levitate?

(b) If at most one of them succeeds, what is the probability that Liz becomes the
world flaming torch juggler champion?

Problem 18.26.
What is the smallest size sample space in which there are two independent events,
neither of which has probability zero or probability one? Explain.

Problem 18.27.
Give examples of event A; B; E such that
(a) A and B are independent, and are also conditionally independent given E, but
are not conditionally independent given E. That is,

PrŒA \ B D PrŒA PrŒB;

Pr A \ B j E D Pr A j E Pr B j E ;
ˇ ˇ ˇ
Pr A \ B ˇ E ¤ Pr A ˇ E Pr B ˇ E :

Hint: Let S D f1; 2; 3; 4g.

(b) A and B are conditionally independent given E, or given E, but are not inde-
pendent. That is,

Pr A \ B j E D Pr A j E Pr B j E ;
ˇ ˇ ˇ
Pr A \ B ˇ E D Pr A ˇ E Pr B ˇ E ;
PrŒA \ B ¤ PrŒA PrŒB:

Hint: Let S D f1; 2; 3; 4; 5g.
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18.9. Probability versus Confidence 801

An alternative example is

A WWD f1g
B WWD f1; 2g
E WWD f3; 4; 5g:

Class Problems
Problem 18.28.
H when Pr H j E > PrŒH , and it is
Event E is evidence in favor of event
evidence against H when Pr H j E < PrŒH .
(a) Give an example of events A; B; H such that A and B are independent, both
are evidence for H , but A [ B is evidence against H .
Hint: Let S D Œ1::8

(b) Prove E is evidence in favor of H iff E is evidence against H .

Problem 18.29.
Let G be a simple graph with n vertices. Let “A.u; v/” mean that vertices u and v
are adjacent, and let “W .u; v/” mean that there is a length-two walk between u and
v.
(a) Explain why W .u; u/ holds iff 9v: A.u; v/.

(b) Write a predicate-logic formula defining W .u; v/ in terms of the predicate
A.:; :/ when u ¤ v.
There are e WWD n2 possible edges between the n vertices of G. Suppose the

actual edges of E.G/ are chosen with randomly from this set of e possible edges.
Each edge is chosen with probability p, and the choices are mutually independent.
(c) Write a simple formula in terms of p; e, and k for PrŒjE.G/j D k.

(d) Write a simple formula in terms of p and n for PrŒW .u; u/.

Let w, x, y and z be four distinct vertices.
Because edges are chosen mutually independently, events that depend on disjoint
sets of edges will be mutually independent. For example, the events

A.w; y/ AND A.y; x/
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802 Chapter 18 Conditional Probability

and
A.w; z/ AND A.z; x/
are independent since hw—yi ; hy—xi ; hw—zi ; hz—xi are four distinct edges.
(e) Let
r WWD PrŒNOT.W .w; x//; (18.13)
where w and x are distinct vertices. Write a simple formula for r in terms of n and
p.
Hint: Different length-two paths between x and y don’t share any edges.

(f) Vertices x and y being on a three-cycle can be expressed simply as

A.x; y/ AND W .x; y/:

Write a simple expression in terms of p and r for the probability that x and y lie
on a three-cycle in G.

(g) Are W .w; x/ and W .y; z/ independent events? Briefly comment (proof not
required).

Problems for Section 18.8
Practice Problems
Problem 18.30.
Suppose A, B and C are mutually independent events, what about A \ B and
B [ C?

Class Problems
Problem 18.31.
Suppose you flip three fair, mutually independent coins. Define the following
events:

Let A be the event that the first coin is heads.

Let B be the event that the second coin is heads.
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18.9. Probability versus Confidence 803

Let C be the event that the third coin is heads.
Let D be the event that an even number of coins are heads.
(a) Use the four step method to determine the probability space for this experiment
and the probability of each of A; B; C; D.

(b) Show that these events are not mutually independent.

Problem 18.32.
Let A; B; C be events. For each of the following statements, prove it or give a
counterexample.
(a) If A is independent of B, then A is also independent of B.

(b) If A is independent of B, and A is independent of C , then A is independent of
B \ C.
Hint: Choose A; B; C pairwise but not 3-way independent.

(d) If A is independent of B, and A is independent of C , and A is independent of
B \ C , then A is independent of B [ C .

Problem 18.33.
Let A; B; C; D be events. Describe counterexamples showing that the following
claims are false.
(a)
False Claim. If A and B are independent given C , and are also independent given
D, then A and B are independent given C [ D.

(b)
False Claim. If A and B are independent given C , and are also independent given
D, then A and B are independent given C \ D.
Hint: Choose A; B; C; D 3-way but not 4-way independent.
so A and B are not independent given C \ D.
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804 Chapter 18 Conditional Probability

Homework Problems
Problem 18.34.
Describe events A, B and C that:

satisfy the “product rule,” namely,

PrŒA \ B \ C D PrŒA PrŒB PrŒC ;

no two out of the three events are independent.

Hint: Choose A; B; C events over the uniform probability space on Œ1::6.

Exam Problems
Problem 18.35.
A classroom has sixteen desks in a 4 4 arrangement as shown below.

If two desks are next to each other, vertically or horizontally, they are called an
adjacent pair. So there are three horizontally adjacent pairs in each row, for a total
of twelve horizontally adjacent pairs. Likewise, there are twelve vertically adjacent
pairs.
Boys and girls are assigned to desks mutually independently, with probability
p > 0 of a desk being occupied by a boy and probability q WWD 1 p > 0 of being
occupied by a girl. An adjacent pair D of desks is said to have a flirtation when
there is a boy at one desk and a girl at the other desk. Let FD be the event that D
has a flirtation.
(a) Different pairs D and E of adjacent desks are said to overlap when they share
a desk. For example, the first and second pairs in each row overlap, and so do the
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18.9. Probability versus Confidence 805

second and third pairs, but the first and third pairs do not overlap. Prove that if D
and E overlap, then FD and FE are independent events iff p D q.

(b) Find four pairs of desks D1 ; D2 ; D3 ; D4 and explain why FD1 ; FD2 ; FD3 ; FD4
are not mutually independent (even if p D q D 1=2).

Problems for Section 18.9
Problem 18.36.
An International Journal of Pharmacological Testing has a policy of publishing
drug trial results only if the conclusion holds at the 95% confidence level. The ed-
itors and reviewers always carefully check that any results they publish came from
a drug trial that genuinely deserved this level of confidence. They are also careful
to check that trials whose results they publish have been conducted independently
of each other.
The editors of the Journal reason that under this policy, their readership can be
confident that at most 5% of the published studies will be mistaken. Later, the
editors are embarrassed—and astonished—to learn that every one of the 20 drug
trial results they published during the year was wrong. The editors thought that
because the trials were conducted independently, the probability of publishing 20
wrong results was negligible, namely, .1=20/20 < 10 25 .
Write a brief explanation to these befuddled editors explaining what’s wrong
with their reasoning and how it could be that all 20 published studies were wrong.
Hint: xkcd comic: “significant” xkcd.com/882/

Practice Problems
Problem 18.37.
A somewhat reliable allergy test has the following properties:
If you are allergic, there is a 10% chance that the test will say you are not.
If you are not allergic, there is a 5% chance that the test will say you are.
(a) The test results are correct at what confidence level?

(b) What is the Bayes factor for being allergic when the test diagnoses a person as
allergic?

(c) What can you conclude about the odds of a random person being allergic given
that the test diagnoses them as allergic?
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806 Chapter 18 Conditional Probability

Suppose that your doctor tells you that because the test diagnosed you as allergic,
and about 25% of people are allergic, the odds are six to one that you are allergic.
(d) How would your doctor calculate these odds of being allergic based on what’s
known about the allergy test?

(e) Another doctor reviews your test results and medical record and says your
odds of being allergic are really much higher, namely thirty-six to one. Briefly
explain how two conscientious doctors could disagree so much. Is there a way you
could determine your actual odds of being allergic?
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19 Random Variables
Thus far, we have focused on probabilities of events. For example, we computed
the probability that you win the Monty Hall game or that you have a rare medical
condition given that you tested positive. But, in many cases we would like to know
more. For example, how many contestants must play the Monty Hall game until
one of them finally wins? How long will this condition last? How much will I lose
gambling with strange dice all night? To answer such questions, we need to work
with random variables.

19.1 Random Variable Examples
Definition 19.1.1. A random variable R on a probability space is a total function
whose domain is the sample space.

The codomain of R can be anything, but will usually be a subset of the real
numbers. Notice that the name “random variable” is a misnomer; random variables
are actually functions.
For example, suppose we toss three independent, unbiased coins. Let C be the
number of heads that appear. Let M D 1 if the three coins come up all heads or all
tails, and let M D 0 otherwise. Now every outcome of the three coin flips uniquely
determines the values of C and M . For example, if we flip heads, tails, heads, then
C D 2 and M D 0. If we flip tails, tails, tails, then C D 0 and M D 1. In effect,
C counts the number of heads, and M indicates whether all the coins match.
Since each outcome uniquely determines C and M , we can regard them as func-
tions mapping outcomes to numbers. For this experiment, the sample space is:

S D fHHH; HH T; H TH; H T T; THH; TH T; T TH; T T T g:

Now C is a function that maps each outcome in the sample space to a number as
follows:
C.HHH / D 3 C.THH / D 2
C.HH T / D 2 C.TH T / D 1
C.H TH / D 2 C.T TH / D 1
C.H T T / D 1 C.T T T / D 0:
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808 Chapter 19 Random Variables

Similarly, M is a function mapping each outcome another way:
M.HHH / D 1 M.THH / D 0
M.HH T / D 0 M.TH T / D 0
M.H TH / D 0 M.T TH / D 0
M.H T T / D 0 M.T T T / D 1:
So C and M are random variables.

19.1.1 Indicator Random Variables
An indicator random variable is a random variable that maps every outcome to
either 0 or 1. Indicator random variables are also called Bernoulli variables. The
random variable M is an example. If all three coins match, then M D 1; otherwise,
M D 0.
Indicator random variables are closely related to events. In particular, an in-
dicator random variable partitions the sample space into those outcomes mapped
to 1 and those outcomes mapped to 0. For example, the indicator M partitions the
sample space into two blocks as follows:
HHH
„ ƒ‚ T T T… HH
„ T H TH H T Tƒ‚ THH TH T T TH
…:
M D1 M D0

In the same way, an event E partitions the sample space into those outcomes
in E and those not in E. So E is naturally associated with an indicator random
variable, IE , where IE .!/ D 1 for outcomes ! 2 E and IE .!/ D 0 for outcomes
! … E. Thus, M D IE where E is the event that all three coins match.

19.1.2 Random Variables and Events
There is a strong relationship between events and more general random variables
as well. A random variable that takes on several values partitions the sample space
into several blocks. For example, C partitions the sample space as follows:
TTT
„ƒ‚… T
„ TH TH
ƒ‚T H T T… THH
„ Hƒ‚
TH HH T… HHH
„ ƒ‚ … :
C D0 C D1 C D2 C D3

Each block is a subset of the sample space and is therefore an event. So the assertion
that C D 2 defines the event
ŒC D 2 D fTHH; H TH; HH T g;
and this event has probability
1 1 1
PrŒC D 2 D PrŒTHH C PrŒH TH C PrŒHH T D C C D 3=8:
8 8 8
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19.2. Independence 809

Likewise ŒM D 1 is the event fT T T; HHH g and has probability 1=4.
More generally, any assertion about the values of random variables defines an
event. For example, the assertion that C 1 defines

ŒC 1 D fT T T; T TH; TH T; H T T g;

and so PrŒC 1 D 1=2.
Another example is the assertion that C M is an odd number. If you think about
it for a minute, you’ll realize that this is an obscure way of saying that all three
coins came up heads, namely,

ŒC M is odd D fHHH g:

19.2 Independence
The notion of independence carries over from events to random variables as well.
Random variables R1 and R2 are independent iff for all x1 ; x2 , the two events

ŒR1 D x1 and ŒR2 D x2

are independent.
For example, are C and M independent? Intuitively, the answer should be “no.”
The number of heads C completely determines whether all three coins match; that
is, whether M D 1. But, to verify this intuition, we must find some x1 ; x2 2 R
such that:

PrŒC D x1 AND M D x2 ¤ PrŒC D x1 PrŒM D x2 :

One appropriate choice of values is x1 D 2 and x2 D 1. In this case, we have:
1 3
PrŒC D 2 AND M D 1 D 0 ¤ D PrŒM D 1 PrŒC D 2:
4 8
The first probability is zero because we never have exactly two heads (C D 2)
when all three coins match (M D 1). The other two probabilities were computed
earlier.
On the other hand, let H1 be the indicator variable for the event that the first flip
is a Head, so
ŒH1 D 1 D fHHH; H TH; HH T; H T T g:
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810 Chapter 19 Random Variables

Then H1 is independent of M , since

PrŒM D 1 D 1=4 D Pr M D 1 j H1 D 1 D Pr M D 1 j H1 D 0

PrŒM D 0 D 3=4 D Pr M D 0 j H1 D 1 D Pr M D 0 j H1 D 0

This example is an instance of:

Lemma 19.2.1. Two events are independent iff their indicator variables are inde-
pendent.

The simple proof is left to Problem 19.1.
Intuitively, the independence of two random variables means that knowing some
information about one variable doesn’t provide any information about the other
one. We can formalize what “some information” about a variable R is by defining
it to be the value of some quantity that depends on R. This intuitive property of
independence then simply means that functions of independent variables are also
independent:

Lemma 19.2.2. Let R and S be independent random variables, and f and g be
functions such that domain.f / D codomain.R/ and domain.g/ D codomain.S /.
Then f .R/ and g.S / are independent random variables.

The proof is another simple exercise left to Problem 19.32.
As with events, the notion of independence generalizes to more than two random
variables.

Definition 19.2.3. Random variables R1 ; R2 ; : : : ; Rn are mutually independent iff
for all x1 ; x2 ; : : : ; xn , the n events

ŒR1 D x1 ; ŒR2 D x2 ; : : : ; ŒRn D xn

are mutually independent. They are k-way independent iff every subset of k of
them are mutually independent.

Lemmas 19.2.1 and 19.2.2 both extend straightforwardly to k-way independent
variables.

19.3 Distribution Functions
A random variable maps outcomes to values. The probability density function,
PDFR .x/, of a random variable R measures the probability that R takes the value
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19.3. Distribution Functions 811

x, and the closely related cumulative distribution function CDFR .x/ measures the
probability that R x. Random variables that show up for different spaces of
outcomes often wind up behaving in much the same way because they have the
same probability of taking different values, that is, because they have the same
pdf/cdf.

Definition 19.3.1. Let R be a random variable with codomain V . The probability
density function of R is a function PDFR W V ! Œ0; 1 defined by:
(
PrŒR D x if x 2 range.R/;
PDFR .x/ WWD
0 if x … range.R/:

If the codomain is a subset of the real numbers, then the cumulative distribution
function is the function CDFR W R ! Œ0; 1 defined by:

CDFR .x/ WWD PrŒR x:

A consequence of this definition is that
X
PDFR .x/ D 1:
x2range.R/

This is because R has a value for each outcome, so summing the probabilities over
all outcomes is the same as summing over the probabilities of each value in the
range of R.
As an example, suppose that you roll two unbiased, independent, 6-sided dice.
Let T be the random variable that equals the sum of the two rolls. This random
variable takes on values in the set V D f2; 3; : : : ; 12g. A plot of the probability
density function for T is shown in Figure 19.1. The lump in the middle indicates
that sums close to 7 are the most likely. The total area of all the rectangles is 1
since the dice must take on exactly one of the sums in V D f2; 3; : : : ; 12g.
The cumulative distribution function for T is shown in Figure 19.2: The height
of the i th bar in the cumulative distribution function is equal to the sum of the
heights of the leftmost i bars in the probability density function. This follows from
the definitions of pdf and cdf:
X X
CDFR .x/ D PrŒR x D PrŒR D y D PDFR .y/:
yx yx

It also follows from the definition that

lim CDFR .x/ D 1 and lim CDFR .x/ D 0:
x!1 x! 1
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812 Chapter 19 Random Variables

6=36
PDFT.x/
3=36

2 3 4 5 6 7 8 9 10 11 12
x2V

Figure 19.1 The probability density function for the sum of two 6-sided dice.

1 :::
CDFT.x/
1=2

0
0 1 2 3 4 5 6 7 8 9 10 11 12
x2V

Figure 19.2 The cumulative distribution function for the sum of two 6-sided dice.
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19.3. Distribution Functions 813

Both PDFR and CDFR capture the same information about R, so take your choice.
The key point here is that neither the probability density function nor the cumulative
distribution function involves the sample space of an experiment.
One of the really interesting things about density functions and distribution func-
tions is that many random variables turn out to have the same pdf and cdf. In other
words, even though R and S are different random variables on different probability
spaces, it is often the case that

PDFR D PDFS :

In fact, some pdf’s are so common that they are given special names. For exam-
ple, the three most important distributions in computer science are the Bernoulli
distribution, the uniform distribution, and the binomial distribution. We look more
closely at these common distributions in the next several sections.

19.3.1 Bernoulli Distributions
A Bernoulli distribution is the distribution function for a Bernoulli variable. Specif-
ically, the Bernoulli distribution has a probability density function of the form
fp W f0; 1g ! Œ0; 1 where

fp .0/ D p; and
fp .1/ D 1 p;

for some p 2 Œ0; 1. The corresponding cumulative distribution function is Fp W
R ! Œ0; 1 where 8
<0
ˆ if x < 0
Fp .x/ WWD p if 0 x < 1
ˆ
1 if 1 x:
:

19.3.2 Uniform Distributions
A random variable that takes on each possible value in its codomain with the same
probability is said to be uniform. If the codomain V has n elements, then the
uniform distribution has a pdf of the form

f W V ! Œ0; 1

where
1
f .v/ D
n
for all v 2 V .
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814 Chapter 19 Random Variables

If the elements of V in increasing order are a1 ; a2 ; : : : ; an , then the cumulative
distribution function would be F W R ! Œ0; 1 where
8
<0
ˆ if x < a1
F .x/ WWD k=n if ak x < akC1 for 1 k < n
ˆ
1 if an x:
:

Uniform distributions come up all the time. For example, the number rolled on
a fair die is uniform on the set f1; 2; : : : ; 6g. An indicator variable is uniform when
its pdf is f1=2 .

19.3.3 The Numbers Game
Enough definitions—let’s play a game! We have two envelopes. Each contains
an integer in the range 0; 1; : : : ; 100, and the numbers are distinct. To win the
game, you must determine which envelope contains the larger number. To give
you a fighting chance, we’ll let you peek at the number in one envelope selected
at random. Can you devise a strategy that gives you a better than 50% chance of
winning?
For example, you could just pick an envelope at random and guess that it contains
the larger number. But this strategy wins only 50% of the time. Your challenge is
to do better.
So you might try to be more clever. Suppose you peek in one envelope and see
the number 12. Since 12 is a small number, you might guess that the number in the
other envelope is larger. But perhaps we’ve been tricky and put small numbers in
both envelopes. Then your guess might not be so good!
An important point here is that the numbers in the envelopes may not be random.
We’re picking the numbers and we’re choosing them in a way that we think will
defeat your guessing strategy. We’ll only use randomization to choose the numbers
if that serves our purpose: making you lose!

Intuition Behind the Winning Strategy
People are surprised when they first learn that there is a strategy that wins more
than 50% of the time, regardless of what numbers we put in the envelopes.
Suppose that you somehow knew a number x that was in between the numbers
in the envelopes. Now you peek in one envelope and see a number. If it is bigger
than x, then you know you’re peeking at the higher number. If it is smaller than x,
then you’re peeking at the lower number. In other words, if you know a number x
between the numbers in the envelopes, then you are certain to win the game.
The only flaw with this brilliant strategy is that you do not know such an x. This
sounds like a dead end, but there’s a cool way to salvage things: try to guess x!
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19.3. Distribution Functions 815

There is some probability that you guess correctly. In this case, you win 100%
of the time. On the other hand, if you guess incorrectly, then you’re no worse off
than before; your chance of winning is still 50%. Combining these two cases, your
overall chance of winning is better than 50%.
Many intuitive arguments about probability are wrong despite sounding persua-
sive. But this one goes the other way: it may not convince you, but it’s actually
correct. To justify this, we’ll go over the argument in a more rigorous way—and
while we’re at it, work out the optimal way to play.

Analysis of the Winning Strategy
For generality, suppose that we can choose numbers from the integer interval Œ0::n.
Call the lower number L and the higher number H .
Your goal is to guess a number x between L and H . It’s simplest if x does not
equal L or H , so you should select x at random from among the half-integers:
1 3 5 2n 1
; ; ; :::;
2 2 2 2
But what probability distribution should you use?
The uniform distribution—selecting each of these half-integers with equal probability—
turns out to be your best bet. An informal justification is that if we figured out that
you were unlikely to pick some number—say 50 21 —then we’d always put 50 and 51
in the envelopes. Then you’d be unlikely to pick an x between L and H and would
have less chance of winning.
After you’ve selected the number x, you peek into an envelope and see some
number T . If T > x, then you guess that you’re looking at the larger number.
If T < x, then you guess that the other number is larger.
All that remains is to determine the probability that this strategy succeeds. We
can do this with the usual four step method and a tree diagram.
Step 1: Find the sample space.
You either choose x too low (< L), too high (> H ), or just right (L < x < H ).
Then you either peek at the lower number (T D L) or the higher number (T D H ).
This gives a total of six possible outcomes, as show in Figure 19.3.
Step 2: Define events of interest.
The four outcomes in the event that you win are marked in the tree diagram.
Step 3: Assign outcome probabilities.
First, we assign edge probabilities. Your guess x is too low with probability L=n,
too high with probability .n H /=n, and just right with probability .H L/=n.
Next, you peek at either the lower or higher number with equal probability. Multi-
plying along root-to-leaf paths gives the outcome probabilities.
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816 Chapter 19 Random Variables

choices number result probability
of x peeked at
TDL 1=2 lose L=2n

x too low
TDH 1=2 win L=2n
L=n
TDL 1=2 win .H�L/=2n
x just right
.H�L/=n
TDH 1=2 win .H�L/=2n

.n�H/=n TDL 1=2 win .n�H/=2n
x too high

lose .n�H/=2n
TDH 1=2

Figure 19.3 The tree diagram for the numbers game.

Step 4: Compute event probabilities.
The probability of the event that you win is the sum of the probabilities of the four
outcomes in that event:
L H L H L n H
PrŒwin D C C C
2n 2n 2n 2n
1 H L
D C
2 2n
1 1
C
2 2n
The final inequality relies on the fact that the higher number H is at least 1 greater
than the lower number L since they are required to be distinct.
Sure enough, you win with this strategy more than half the time, regardless of the
numbers in the envelopes! So with numbers chosen from the range 0; 1; : : : ; 100,
you win with probability at least 1=2 C 1=200 D 50:5%. If instead we agree to
stick to numbers 0; : : : ; 10, then your probability of winning rises to 55%. By Las
Vegas standards, those are great odds.
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19.3. Distribution Functions 817

Randomized Algorithms
The best strategy to win the numbers game is an example of a randomized algo-
rithm—it uses random numbers to influence decisions. Protocols and algorithms
that make use of random numbers are very important in computer science. There
are many problems for which the best known solutions are based on a random num-
ber generator.
For example, the most commonly-used protocol for deciding when to send a
broadcast on a shared bus or Ethernet is a randomized algorithm known as expo-
nential backoff. One of the most commonly-used sorting algorithms used in prac-
tice, called quicksort, uses random numbers. You’ll see many more examples if
you take an algorithms course. In each case, randomness is used to improve the
probability that the algorithm runs quickly or otherwise performs well.

19.3.4 Binomial Distributions
The third commonly-used distribution in computer science is the binomial distri-
bution. The standard example of a random variable with a binomial distribution is
the number of heads that come up in n independent flips of a coin. If the coin is
fair, then the number of heads has an unbiased binomial distribution, specified by
the pdf fn W Œ0::n ! Œ0; 1:
!
n
fn .k/ WWD 2 n:
k

This is because there are kn sequences of n coin tosses with exactly k heads, and

each such sequence has probability 2 n .
A plot of f20 .k/ is shown in Figure 19.4. The most likely outcome is k D 10
heads, and the probability falls off rapidly for larger and smaller values of k. The
falloff regions to the left and right of the main hump are called the tails of the
distribution.
In many fields, including Computer Science, probability analyses come down to
getting small bounds on the tails of the binomial distribution. In the context of a
problem, this typically means that there is very small probability that something
bad happens, which could be a server or communication link overloading or a ran-
domized algorithm running for an exceptionally long time or producing the wrong
result.
The tails do get small very fast. For example, the probability of flipping at most
25 heads in 100 tosses is less than 1 in 3,000,000. In fact, the tail of the distribution
falls off so rapidly that the probability of flipping exactly 25 heads is nearly twice
the probability of flipping exactly 24 heads plus the probability of flipping exactly
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818 Chapter 19 Random Variables

0:18
0:16
0:14
0:12
f20.k/ 0:10
0:08
0:06
0:04
0:02
0
0 5 10 15 20
k

Figure 19.4 The pdf for the unbiased binomial distribution for n D 20, f20 .k/.

23 heads plus . . . the probability of flipping no heads.

The General Binomial Distribution
If the coins are biased so that each coin is heads with probability p, then the
number of heads has a general binomial density function specified by the pdf
fn;p W Œ0::n ! Œ0; 1 where
!
n k
fn;p .k/ D p .1 p/n k : (19.1)
k

for some n 2 NC and p 2 Œ0; 1. This is because there are kn sequences with

k heads and n k tails, but now p k .1 p/n k is the probability of each such
sequence.
For example, the plot in Figure 19.5 shows the probability density function
fn;p .k/ corresponding to flipping n D 20 independent coins that are heads with
probability p D 0:75. The graph shows that we are most likely to get k D 15
heads, as you might expect. Once again, the probability falls off quickly for larger
and smaller values of k.
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19.4. Great Expectations 819

0:25

0:2

0:15

f20;:75.k/
0:1

0:05

0
0 5 10 15 20
k

Figure 19.5 The pdf for the general binomial distribution fn;p .k/ for n D 20
and p D :75.

19.4 Great Expectations
The expectation or expected value of a random variable is a single number that re-
veals a lot about the behavior of the variable. The expectation of a random variable
is also known as its mean or average. For example, the first thing you typically
want to know when you see your grade on an exam is the average score of the
class. This average score turns out to be precisely the expectation of the random
variable equal to the score of a random student.
More precisely, the expectation of a random variable is its “average” value when
each value is weighted according to its probability. Formally, the expected value of
a random variable is defined as follows:

Definition 19.4.1. If R is a random variable defined on a sample space S, then the
expectation of R is X
ExŒR WWD R.!/ PrŒ!: (19.2)
!2S

Let’s work through some examples.
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820 Chapter 19 Random Variables

19.4.1 The Expected Value of a Uniform Random Variable
Rolling a 6-sided die provides an example of a uniform random variable. Let R be
the value that comes up when you roll a fair 6-sided die. Then by (19.2), the
expected value of R is
1 1 1 1 1 1 7
ExŒR D 1 C2 C3 C4 C5 C6 D :
6 6 6 6 6 6 2
This calculation shows that the name “expected” value is a little misleading; the
random variable might never actually take on that value. No one expects to roll a
3 12 on an ordinary die!
In general, if Rn is a random variable with a uniform distribution on fa1 ; a2 ; : : : ; an g,
then the expectation of Rn is simply the average of the ai ’s:
a1 C a2 C C an
ExŒRn D :
n
19.4.2 The Expected Value of a Reciprocal Random Variable
Define a random variable S to be the reciprocal of the value that comes up when
you roll a fair 6-sided die. That is, S D 1=R where R is the value that you roll.
Now,

1 1 1 1 1 1 1 1 1 1 1 1 1 49
ExŒS D Ex D C C C C C D :
R 1 6 2 6 3 6 4 6 5 6 6 6 120
Notice that
Ex 1=R ¤ 1= ExŒR:
Assuming that these two quantities are equal is a common mistake.

19.4.3 The Expected Value of an Indicator Random Variable
The expected value of an indicator random variable for an event is just the proba-
bility of that event.
Lemma 19.4.2. If IA is the indicator random variable for event A, then
ExŒIA D PrŒA:
Proof.
ExŒIA D 1 PrŒIA D 1 C 0 PrŒIA D 0 D PrŒIA D 1
D PrŒA: (def of IA )
For example, if A is the event that a coin with bias p comes up heads, then
ExŒIA D PrŒIA D 1 D p.
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19.4. Great Expectations 821

19.4.4 Alternate Definition of Expectation
There is another standard way to define expectation.

Theorem 19.4.3. For any random variable R,
X
ExŒR D x PrŒR D x: (19.3)
x2range.R/

The proof of Theorem 19.4.3, like many of the elementary proofs about expec-
tation in this chapter, follows by regrouping of terms in equation (19.2):

Proof. Suppose R is defined on a sample space S. Then,
X
ExŒR WWD R.!/ PrŒ!
!2S
X X
D R.!/ PrŒ!
x2range.R/ !2ŒRDx
X X
D x PrŒ! (def of the event ŒR D x)
x2range.R/ !2ŒRDx
0 1
X X
D x@ PrŒ!A (factoring x from the inner sum)
x2range.R/ !2ŒRDx
X
D x PrŒR D x: (def of PrŒR D x)
x2range.R/

The first equality follows because the events ŒR D x for x 2 range.R/ partition
the sample space S, so summing over the outcomes in ŒR D x for x 2 range.R/
is the same as summing over S.

In general, equation (19.3) is more useful than the defining equation (19.2) for
calculating expected values. It also has the advantage that it does not depend on
the sample space, but only on the density function of the random variable. On
the other hand, summing over all outcomes as in equation (19.2) sometimes yields
easier proofs about general properties of expectation.

19.4.5 Conditional Expectation
Just like event probabilities, expectations can be conditioned on some event. Given
a random variable R, the expected value of R conditioned on an event A is the
probability-weighted average value of R over outcomes in A. More formally:
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822 Chapter 19 Random Variables

Definition 19.4.4. The conditional expectation ExŒR j A of a random variable R
given event A is:
X
ExŒR j A WWD r Pr R D r j A : (19.4)
r2range.R/

For example, we can compute the expected value of a roll of a fair die, given that
the number rolled is at least 4. We do this by letting R be the outcome of a roll of
the die. Then by equation (19.4),
6
X
i Pr R D i j R 4 D 10C20C30C4 31 C5 31 C6 13 D 5:

ExŒR j R 4 D
i D1

Conditional expectation is useful in dividing complicated expectation calcula-
tions into simpler cases. We can find a desired expectation by calculating the con-
ditional expectation in each simple case and averaging them, weighing each case
by its probability.
For example, suppose that 49.6% of the people in the world are male and the
rest female—which is more or less true. Also suppose the expected height of a
randomly chosen male is 50 1100 , while the expected height of a randomly chosen
female is 50 5:00 What is the expected height of a randomly chosen person? We can
calculate this by averaging the heights of men and women. Namely, let H be the
height (in feet) of a randomly chosen person, and let M be the event that the person
is male and F the event that the person is female. Then

ExŒH D ExŒH j M PrŒM C ExŒH j F PrŒF
D .5 C 11=12/ 0:496 C .5 C 5=12/ .1 0:496/
D 5:6646 : : : :

which is a little less than 5’ 8.”
This method is justified by:

Theorem 19.4.5 (Law of Total Expectation). Let R be a random variable on a
sample space S, and suppose that A1 , A2 , . . . , is a partition of S. Then
X
ExŒR D ExŒR j Ai PrŒAi :
i
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19.4. Great Expectations 823

Proof.
X
ExŒR D r PrŒR D r (by 19.3)
r2range.R/
X X
D r Pr R D r j Ai PrŒAi (Law of Total Probability)
r i
XX
D r Pr R D r j Ai PrŒAi (distribute constant r)
r i
XX
D r Pr R D r j Ai PrŒAi (exchange order of summation)
i r
X X
D PrŒAi r Pr R D r j Ai (factor constant PrŒAi )
i r
X
D PrŒAi ExŒR j Ai : (Def 19.4.4 of cond. expectation)
i

19.4.6 Mean Time to Failure
A computer program crashes at the end of each hour of use with probability p, if
it has not crashed already. What is the expected time until the program crashes?
This will be easy to figure out using the Law of Total Expectation, Theorem 19.4.5.
Specifically, we want to find ExŒC where C is the number of hours until the first
crash. We’ll do this by conditioning on whether or not the crash occurs in the first
hour.
So define A to be the event that the system fails on the first step and A to be the
complementary event that the system does not fail on the first step. Then the mean
time to failure ExŒC is

ExŒC D ExŒC j A PrŒA C ExŒC j A PrŒA: (19.5)

Since A is the condition that the system crashes on the first step, we know that

ExŒC j A D 1: (19.6)

Since A is the condition that the system does not crash on the first step, conditioning
on A is equivalent to taking a first step without failure and then starting over without
conditioning. Hence,
ExŒC j A D 1 C ExŒC : (19.7)
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824 Chapter 19 Random Variables

Plugging (19.6) and (19.7) into (19.5):

ExŒC D 1 p C .1 C ExŒC /.1 p/
DpC1 p C .1 p/ ExŒC
D 1 C .1 p/ ExŒC :

Then, rearranging terms gives

1 D ExŒC .1 p/ ExŒC D p ExŒC ;

and thus
ExŒC D 1=p:
The general principle here is well-worth remembering.

Mean Time to Failure

If a system independently fails at each time step with probability p, then the
expected number of steps up to the first failure is 1=p.

So, for example, if there is a 1% chance that the program crashes at the end of
each hour, then the expected time until the program crashes is 1=0:01 D 100 hours.
As a further example, suppose a couple insists on having children until they get
a boy, then how many baby girls should they expect before their first boy? Assume
for simplicity that there is a 50% chance that a child will be a boy and that the
genders of siblings are mutually independent.
This is really a variant of the previous problem. The question, “How many hours
until the program crashes?” is mathematically the same as the question, “How
many children must the couple have until they get a boy?” In this case, a crash
corresponds to having a boy, so we should set p D 1=2. By the preceding analysis,
the couple should expect a baby boy after having 1=p D 2 children. Since the last
of these will be a boy, they should expect just one girl. So even in societies where
couples pursue this commitment to boys, the expected population will divide evenly
between boys and girls.
There is a simple intuitive argument that explains the mean time to failure for-
mula (19.8). Suppose the system is restarted after each failure. This makes the
mean time to failure the same as the mean time between successive repeated fail-
ures. Now if the probability of failure at a given step is p, then after n steps we
expect to have pn failures. Now, by definition, the average number of steps be-
tween failures is equal to np=p, namely, 1=p.
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19.4. Great Expectations 825

For the record, we’ll state a formal version of this result. A random variable
like C that counts steps to first failure is said to have a geometric distribution with
parameter p.
Definition 19.4.6. A random variable C has a geometric distribution with param-
eter p iff codomain.C / D ZC and

PrŒC D i D .1 p/i 1
p:

Lemma 19.4.7. If a random variable C has a geometric distribution with param-
eter p, then
1
ExŒC D : (19.8)
p

19.4.7 Expected Returns in Gambling Games
Some of the most interesting examples of expectation can be explained in terms of
gambling games. For straightforward games where you win w dollars with proba-
bility p and you lose x dollars with probability 1 p, it is easy to compute your
expected return or winnings. It is simply

pw .1 p/x dollars:

For example, if you are flipping a fair coin and you win $1 for heads and you lose $1
for tails, then your expected winnings are

1 1
1 1 1 D 0:
2 2

In such cases, the game is said to be fair since your expected return is zero.

Splitting the Pot
We’ll now look at a different game which is fair—but only on first analysis.
It’s late on a Friday night in your neighborhood hangout when two new biker
dudes, Eric and Nick, stroll over and propose a simple wager. Each player will
put $2 on the bar and secretly write “heads” or “tails” on their napkin. Then you
will flip a fair coin. The $6 on the bar will then be “split”—that is, be divided
equally—among the players who correctly predicted the outcome of the coin toss.
Pot splitting like this is a familiar feature in poker games, betting pools, and lotter-
ies.
This sounds like a fair game, but after your regrettable encounter with strange
dice (Section 17.3), you are definitely skeptical about gambling with bikers. So
before agreeing to play, you go through the four-step method and write out the
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826 Chapter 19 Random Variables

you guess Eric guesses Nick guesses your probability
right? right? right? payoff

yes 1=2 $0 1=8

yes 1=2
no 1=2 $1 1=8

yes 1=2 $1 1=8
yes 1=2
no 1=2

no 1=2 $4 1=8

yes 1=2 �$2 1=8

yes 1=2
no 1=2
no 1=2 �$2 1=8

yes 1=2 �$2 1=8
no 1=2

no 1=2 $0 1=8

Figure 19.6 The tree diagram for the game where three players each wager $2
and then guess the outcome of a fair coin toss. The winners split the pot.
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19.4. Great Expectations 827

tree diagram to compute your expected return. The tree diagram is shown in Fig-
ure 19.6.
The “payoff” values in Figure 19.6 are computed by dividing the $6 pot1 among
those players who guessed correctly and then subtracting the $2 that you put into
the pot at the beginning. For example, if all three players guessed correctly, then
your payoff is $0, since you just get back your $2 wager. If you and Nick guess
correctly and Eric guessed wrong, then your payoff is
6
2 D 1:
2
In the case that everyone is wrong, you all agree to split the pot and so, again, your
payoff is zero.
To compute your expected return, you use equation (19.3):
1 1 1 1
ExŒpayoff D 0 C1 C1 C4
8 8 8 8
1 1 1 1
C . 2/ C . 2/ C . 2/ C 0
8 8 8 8
D 0:

This confirms that the game is fair. So, for old time’s sake, you break your solemn
vow to never ever engage in strange gambling games.

The Impact of Collusion
Needless to say, things are not turning out well for you. The more times you play
the game, the more money you seem to be losing. After 1000 wagers, you have
lost over $500. As Nick and Eric are consoling you on your “bad luck,” you do a
back-of-the-envelope calculation and decide that the probability of losing $500 in
1000 fair $2 wagers is very, very small.
Now it is possible of course that you are very, very unlucky. But it is more likely
that something fishy is going on. Somehow the tree diagram in Figure 19.6 is not a
good model of the game.
The “something” that’s fishy is the opportunity that Nick and Eric have to collude
against you. The fact that the coin flip is fair certainly means that each of Nick and
Eric can only guess the outcome of the coin toss with probability 1=2. But when
you look back at the previous 1000 bets, you notice that Eric and Nick never made
the same guess. In other words, Nick always guessed “tails” when Eric guessed
“heads,” and vice-versa. Modelling this fact now results in a slightly different tree
diagram, as shown in Figure 19.7.
1 The money invested in a wager is commonly referred to as the pot.
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828 Chapter 19 Random Variables

you guess Eric guesses Nick guesses your probability
right? right? right? payoff

yes 0 $0 0

yes 1=2
no 1 $1 1=4

yes 1 $1 1=4
yes 1=2
no 1=2

no 0 $4 0

yes 0 �$2 0

yes 1=2
no 1=2
no 1 �$2 1=4

yes 1 �$2 1=4
no 1=2

no 0 $0 0

Figure 19.7 The revised tree diagram reflecting the scenario where Nick always
guesses the opposite of Eric.
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19.4. Great Expectations 829

The payoffs for each outcome are the same in Figures 19.6 and 19.7, but the
probabilities of the outcomes are different. For example, it is no longer possible
for all three players to guess correctly, since Nick and Eric are always guessing
differently. More importantly, the outcome where your payoff is $4 is also no
longer possible. Since Nick and Eric are always guessing differently, one of them
will always get a share of the pot. As you might imagine, this is not good for you!
When we use equation (19.3) to compute your expected return in the collusion
scenario, we find that
1 1
ExŒpayoff D 0 0 C 1 C1 C40
4 4
1 1
C . 2/ 0 C . 2/ C . 2/ C 0 0
4 4
1
D :
2
So watch out for these biker dudes! By colluding, Nick and Eric have made it so
that you expect to lose $.50 every time you play. No wonder you lost $500 over the
course of 1000 wagers.

How to Win the Lottery
Similar opportunities to collude arise in many betting games. For example, consider
the typical weekly football betting pool, where each participant wagers $10 and the
participants that pick the most games correctly split a large pot. The pool seems
fair if you think of it as in Figure 19.6. But, in fact, if two or more players collude
by guessing differently, they can get an “unfair” advantage at your expense!
In some cases, the collusion is inadvertent and you can profit from it. For ex-
ample, many years ago, a former MIT Professor of Mathematics named Herman
Chernoff figured out a way to make money by playing the state lottery. This was
surprising since the state usually takes a large share of the wagers before paying the
winners, and so the expected return from a lottery ticket is typically pretty poor. So
how did Chernoff find a way to make money? It turned out to be easy!
In a typical state lottery,
all players pay $1 to play and select 4 numbers from 1 to 36,
the state draws 4 numbers from 1 to 36 uniformly at random,
the states divides 1/2 of the money collected among the people who guessed
correctly and spends the other half redecorating the governor’s residence.
This is a lot like the game you played with Nick and Eric, except that there are
more players and more choices. Chernoff discovered that a small set of numbers
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830 Chapter 19 Random Variables

was selected by a large fraction of the population. Apparently many people think
the same way; they pick the same numbers not on purpose as in the previous game
with Nick and Eric, but based on the Red Sox winning average or today’s date. The
result is as though the players were intentionally colluding to lose. If any one of
them guessed correctly, then they’d have to split the pot with many other players.
By selecting numbers uniformly at random, Chernoff was unlikely to get one of
these favored sequences. So if he won, he’d likely get the whole pot! By analyzing
actual state lottery data, he determined that he could win an average of 7 cents on
the dollar. In other words, his expected return was not $:50 as you might think,
but C$:07.2 Inadvertent collusion often arises in betting pools and is a phenomenon
that you can take advantage of.

19.5 Linearity of Expectation
Expected values obey a simple, very helpful rule called Linearity of Expectation.
Its simplest form says that the expected value of a sum of random variables is the
sum of the expected values of the variables.

Theorem 19.5.1. For any random variables R1 and R2 ,

ExŒR1 C R2 D ExŒR1 C ExŒR2 :

Proof. Let T WWD R1 C R2 . The proof follows straightforwardly by rearranging
terms in equation (19.2) in the definition of expectation:
X
ExŒT WWD T .!/ PrŒ!
!2S
X
D .R1 .!/ C R2 .!// PrŒ! (def of T )
!2S
X X
D R1 .!/ PrŒ! C R2 .!/ PrŒ! (rearranging terms)
!2S !2S
D ExŒR1 C ExŒR2 : (by (19.2))

A small extension of this proof, which we leave to the reader, implies
2 Most lotteries now offer randomized tickets to help smooth out the distribution of selected se-
quences.
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19.5. Linearity of Expectation 831

Theorem 19.5.2. For random variables R1 , R2 and constants a1 ; a2 2 R,

ExŒa1 R1 C a2 R2 D a1 ExŒR1 C a2 ExŒR2 :

In other words, expectation is a linear function. A routine induction extends the
result to more than two variables:
Corollary 19.5.3 (Linearity of Expectation). For any random variables R1 ; : : : ; Rk
and constants a1 ; : : : ; ak 2 R,
2 3
X k k
X
Ex 4 ai Ri 5 D ai ExŒRi :
i D1 i D1

The great thing about linearity of expectation is that no independence is required.
This is really useful, because dealing with independence is a pain, and we often
need to work with random variables that are not known to be independent.
As an example, let’s compute the expected value of the sum of two fair dice.

19.5.1 Expected Value of Two Dice
What is the expected value of the sum of two fair dice?
Let the random variable R1 be the number on the first die, and let R2 be the
number on the second die. We observed earlier that the expected value of one die
is 3.5. We can find the expected value of the sum using linearity of expectation:

ExŒR1 C R2 D ExŒR1 C ExŒR2 D 3:5 C 3:5 D 7:

Assuming that the dice were independent, we could use a tree diagram to prove
that this expected sum is 7, but this would be a bother since there are 36 cases. And
without assuming independence, it’s not apparent how to apply the tree diagram
approach at all. But notice that we did not have to assume that the two dice were
independent. The expected sum of two dice is 7—even if they are controlled to act
together in some way—as long as each individual controlled die remains fair.

19.5.2 Sums of Indicator Random Variables
Linearity of expectation is especially useful when you have a sum of indicator ran-
dom variables. As an example, suppose there is a dinner party where n men check
their hats. The hats are mixed up during dinner, so that afterward each man receives
a random hat. In particular, each man gets his own hat with probability 1=n. What
is the expected number of men who get their own hat?
Letting G be the number of men that get their own hat, we want to find the
expectation of G. But all we know about G is that the probability that a man gets
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832 Chapter 19 Random Variables

his own hat back is 1=n. There are many different probability distributions of hat
permutations with this property, so we don’t know enough about the distribution of
G to calculate its expectation directly using equation (19.2) or (19.3). But linearity
of expectation lets us sidestep this issue.
We’ll use a standard, useful trick to apply linearity, namely, we’ll express G as
a sum of indicator variables. In particular, let Gi be an indicator for the event that
the i th man gets his own hat. That is, Gi D 1 if the i th man gets his own hat, and
Gi D 0 otherwise. The number of men that get their own hat is then the sum of
these indicator random variables:
G D G1 C G2 C C Gn : (19.9)
These indicator variables are not mutually independent. For example, if n 1 men
all get their own hats, then the last man is certain to receive his own hat. But again,
we don’t need to worry about this dependence, since linearity holds regardless.
Since Gi is an indicator random variable, we know from Lemma 19.4.2 that
ExŒGi D PrŒGi D 1 D 1=n: (19.10)
By Linearity of Expectation and equation (19.9), this means that
ExŒG D ExŒG1 C G2 C C Gn
D ExŒG1 C ExŒG2 C C ExŒGn
n
‚ …„ ƒ
1 1 1
D C C C
n n n
D 1:
So even though we don’t know much about how hats are scrambled, we’ve figured
out that on average, just one man gets his own hat back, regardless of the number
of men with hats!
More generally, Linearity of Expectation provides a very good method for com-
puting the expected number of events that will happen.
Theorem 19.5.4. Given any collection of events A1 ; A2 ; : : : ; An , the expected
number of events that will occur is
n
X
PrŒAi :
i D1

For example, Ai could be the event that the i th man gets the right hat back. But
in general, it could be any subset of the sample space, and we are asking for the
expected number of events that will contain a random sample point.
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19.5. Linearity of Expectation 833

Proof. Define Ri to be the indicator random variable for Ai , where Ri .!/ D 1 if
w 2 Ai and Ri .!/ D 0 if w … Ai . Let R D R1 C R2 C C Rn . Then
n
X
ExŒR D ExŒRi (by Linearity of Expectation)
i D1
X n
D PrŒRi D 1 (by Lemma 19.4.2)
i D1
X n
D PrŒAi : (def of indicator variable)
i D1

So whenever you are asked for the expected number of events that occur, all you
have to do is sum the probabilities that each event occurs. Independence is not
needed.

19.5.3 Expectation of a Binomial Distribution
Suppose that we independently flip n biased coins, each with probability p of com-
ing up heads. What is the expected number of heads?
Let J be the random variable denoting the number of heads. Then J has a
binomial distribution with parameters n, p, and
!
n k
PrŒJ D k D p .1 p/n k :
k

Applying equation (19.3), this means that
n n
!
X X n k
ExŒJ D k PrŒJ D k D k p .1 p/n k
: (19.11)
k
kD0 kD0

This sum looks a tad nasty, but linearity of expectation leads to an easy derivation
of a simple closed form. We just express J as a sum of indicator random variables,
which is easy. Namely, let Ji be the indicator random variable for the i th coin
coming up heads, that is,
(
1 if the i th coin is heads
Ji WWD
0 if the i th coin is tails:
Then the number of heads is simply
J D J1 C J2 C C Jn :
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834 Chapter 19 Random Variables

By Theorem 19.5.4,
n
X
ExŒJ D PrŒJi D pn: (19.12)
i D1

That really was easy. If we flip n mutually independent coins, we expect to get
pn heads. Hence the expected value of a binomial distribution with parameters n
and p is simply pn.
But what if the coins are not mutually independent? It doesn’t matter—the an-
swer is still pn because Linearity of Expectation and Theorem 19.5.4 do not as-
sume any independence.
If you are not yet convinced that Linearity of Expectation and Theorem 19.5.4
are powerful tools, consider this: without even trying, we have used them to prove
a complicated looking identity, namely,
n
!
X n k
k p .1 p/n k D pn; (19.13)
k
kD0

which follows by combining equations (19.11) and (19.12) (see also Exercise 19.28).
The next section has an even more convincing illustration of the power of linear-
ity to solve a challenging problem.

19.5.4 The Coupon Collector Problem
Every time we purchase a kid’s meal at Taco Bell, we are graciously presented with
a miniature “Racin’ Rocket” car together with a launching device which enables us
to project our new vehicle across any tabletop or smooth floor at high velocity.
Truly, our delight knows no bounds.
There are different colored Racin’ Rocket cars. The color of car awarded to
us by the kind server at the Taco Bell register appears to be selected uniformly and
independently at random. What is the expected number of kid’s meals that we must
purchase in order to acquire at least one of each color of Racin’ Rocket car?
The same mathematical question shows up in many guises: for example, what
is the expected number of people you must poll in order to find at least one person
with each possible birthday? The general question is commonly called the coupon
collector problem after yet another interpretation.
A clever application of linearity of expectation leads to a simple solution to the
coupon collector problem. Suppose there are five different colors of Racin’ Rocket
cars, and we receive this sequence:

blue green green red blue orange blue orange gray.
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19.5. Linearity of Expectation 835

Let’s partition the sequence into 5 segments:

blue green green red blue orange blue orange gray :
„ƒ‚… „ƒ‚… „ ƒ‚ … „ ƒ‚ … „ ƒ‚ …
X0 X1 X2 X3 X4

The rule is that a segment ends whenever we get a new kind of car. For example, the
middle segment ends when we get a red car for the first time. In this way, we can
break the problem of collecting every type of car into stages. Then we can analyze
each stage individually and assemble the results using linearity of expectation.
In the general case there are n colors of Racin’ Rockets that we’re collecting.
Let Xk be the length of the kth segment. The total number of kid’s meals we must
purchase to get all n Racin’ Rockets is the sum of the lengths of all these segments:

T D X0 C X1 C C Xn 1:

Now let’s focus our attention on Xk , the length of the kth segment. At the
beginning of segment k, we have k different types of car, and the segment ends
when we acquire a new type. When we own k types, each kid’s meal contains a
type that we already have with probability k=n. Therefore, each meal contains a
new type of car with probability 1 k=n D .n k/=n. Thus, the expected number
of meals until we get a new kind of car is n=.n k/ by the Mean Time to Failure
rule. This means that
n
ExŒXk D :
n k
Linearity of expectation, together with this observation, solves the coupon col-
lector problem:

ExŒT D ExŒX0 C X1 C C Xn 1

D ExŒX0 C ExŒX1 C C ExŒXn 1
n n n n n
D C C C C C
n 0 n 1 3 2 1

1 1 1 1 1
Dn C C C C C
n n 1 3 2 1

1 1 1 1 1
Dn C C C C C
1 2 3 n 1 n
D nHn (19.14)
n ln n:

Cool! It’s those Harmonic Numbers again.
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836 Chapter 19 Random Variables

We can use equation (19.14) to answer some concrete questions. For example,
the expected number of die rolls required to see every number from 1 to 6 is:
6H6 D 14:7 : : : :
And the expected number of people you must poll to find at least one person with
each possible birthday is:
365H365 D 2364:6 : : : :

19.5.5 Infinite Sums
Linearity of expectation also works for an infinite number of random variables
provided that the variables satisfy an absolute convergence criterion.
Theorem 19.5.5 (Linearity of Expectation). Let R0 , R1 , . . . , be random variables
such that
X1
ExŒ jRi j
i D0
converges. Then
1 1
" #
X X
Ex Ri D ExŒRi :
i D0 i D0

Proof. Let T WWD 1
P
i D0 Ri .
We leave it to the reader to verify that, under the given convergence hypothesis,
all the sums in the following derivation are absolutely convergent, which justifies
rearranging them as follows:
1
X 1 X
X
ExŒRi D Ri .s/ PrŒs (Def. 19.4.1)
i D0 i D0 s2S
1
XX
D Ri .s/ PrŒs (exchanging order of summation)
s2S i D0
"1 #
X X
D Ri .s/ PrŒs (factoring out PrŒs)
s2S i D0
X
D T .s/ PrŒs (Def. of T )
s2S
D ExŒT (Def. 19.4.1)
" 1 #
X
D Ex Ri : (Def. of T ):
i D0
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19.5. Linearity of Expectation 837

19.5.6 A Gambling Paradox
One of the simplest casino bets is on “red” or “black” at the roulette table. In each
play at roulette, a small ball is set spinning around a roulette wheel until it lands in
a red, black, or green colored slot. The payoff for a bet on red or black matches the
bet; for example, if you bet $10 on red and the ball lands in a red slot, you get back
your original $10 bet plus another matching $10.
The casino gets its advantage from the green slots, which make the probability
of both red and black each less than 1/2. In the US, a roulette wheel has 2 green
slots among 18 black and 18 red slots, so the probability of red is 18=38 0:473.
In Europe, where roulette wheels have only 1 green slot, the odds for red are a little
better—that is, 18=37 0:486—but still less than even.
Of course you can’t expect to win playing roulette, even if you had the good
fortune to gamble against a fair roulette wheel. To prove this, note that with a fair
wheel, you are equally likely win or lose each bet, so your expected win on any
spin is zero. Therefore if you keep betting, your expected win is the sum of your
expected wins on each bet: still zero.
Even so, gamblers regularly try to develop betting strategies to win at roulette
despite the bad odds. A well known strategy of this kind is bet doubling, where
you bet, say, $10 on red and keep doubling the bet until a red comes up. This
means you stop playing if red comes up on the first spin, and you leave the casino
with a $10 profit. If red does not come up, you bet $20 on the second spin. Now if
the second spin comes up red, you get your $20 bet plus $20 back and again walk
away with a net profit of $20 10 D $10. If red does not come up on the second
spin, you next bet $40 and walk away with a net win of $40 20 10 D $10 if red
comes up on on the third spin, and so on.
Since we’ve reasoned that you can’t even win against a fair wheel, this strat-
egy against an unfair wheel shouldn’t work. But wait a minute! There is a 0.486
probability of red appearing on each spin of the wheel, so the mean time until a red
occurs is less than three. What’s more, red will come up eventually with probability
one, and as soon as it does, you leave the casino $10 ahead. In other words, by bet
doubling you are certain to win $10, and so your expectation is $10, not zero!
Something’s wrong here.

19.5.7 Solution to the Paradox
The argument claiming the expectation is zero against a fair wheel is flawed by an
implicit, invalid use of linearity of expectation for an infinite sum.
To explain this carefully, let Bn be the number of dollars you win on your nth
bet, where Bn is defined to be zero if red comes up before the nth spin of the wheel.
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838 Chapter 19 Random Variables

Now the dollar amount you win in any gambling session is
1
X
Bn ;
nD1

and your expected win is
1
" #
X
Ex Bn : (19.15)
nD1
Moreover, since we’re assuming the wheel is fair, it’s true that ExŒBn D 0, so
1
X 1
X
ExŒBn D 0 D 0: (19.16)
nD1 nD1

The flaw in the argument that you can’t win is the implicit appeal to linearity of
expectation to conclude that the expectation (19.15) equals the sum of expectations
in (19.16). This is a case where linearity of expectation fails to hold—even though
the expectation (19.15) is 10 and the sum (19.16) of expectations converges. The
problem is that the expectation of the sum of the absolute values of the bets di-
verges, so the condition required for infinite linearity fails. In particular, under bet
doubling your nth bet is 10 2n 1 dollars while the probability that you will make
an nth bet is 2 n . So

ExŒjBn j D 10 2n 1
2 n
D 20:

Therefore the sum
1
X
ExŒjBn j D 20 C 20 C 20 C
nD1

diverges rapidly.
So the presumption that you can’t beat a fair game, and the argument we offered
to support this presumption, are mistaken: by bet doubling, you can be sure to walk
away a winner. Probability theory has led to an apparently absurd conclusion.
But probability theory shouldn’t be rejected because it leads to this absurd con-
clusion. If you only had a finite amount of money to bet with—say enough money
to make k bets before going bankrupt—then it would be correct to calculate your
expection by summing B1 C B2 C C Bk , and your expectation would be zero
for the fair wheel and negative against an unfair wheel. In other words, in order
to follow the bet doubling strategy, you need to have an infinite bankroll. So it’s
absurd to assume you could actually follow a bet doubling strategy, and it’s entirely
reasonable that an absurd assumption leads to an absurd conclusion.
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19.5. Linearity of Expectation 839

19.5.8 Expectations of Products
While the expectation of a sum is the sum of the expectations, the same is usually
not true for products. For example, suppose that we roll a fair 6-sided die and
denote the outcome with the random variable R. Does ExŒR R D ExŒR ExŒR?
We know that ExŒR D 3 12 and thus ExŒR2 D 12 14 . Let’s compute ExŒR2 to
see if we get the same result.
6
X 2 X
Ex R2 D R .!/ PrŒw D i 2 PrŒRi D i
!2S i D1
12 22 32 4 2 52 62
D C C C C C D 15 1=6 ¤ 12 1=4:
6 6 6 6 6 6
That is,
ExŒR R ¤ ExŒR ExŒR:
So the expectation of a product is not always equal to the product of the expecta-
tions.
There is a special case when such a relationship does hold however; namely,
when the random variables in the product are independent.
Theorem 19.5.6. For any two independent random variables R1 , R2 ,
ExŒR1 R2 D ExŒR1 ExŒR2 :
The proof follows by rearrangement of terms in the sum that defines ExŒR1 R2 .
Details appear in Problem 19.26.
Theorem 19.5.6 extends routinely to a collection of mutually independent vari-
ables.
Corollary 19.5.7. [Expectation of Independent Product]
If random variables R1 ; R2 ; : : : ; Rk are mutually independent, then
2 3
Y k k
Y
Ex 4 Ri 5 D ExŒRi :
i D1 i D1

Problems for Section 19.2
Practice Problems
Problem 19.1.
Let IA and IB be the indicator variables for events A and B. Prove that IA and IB
are independent iff A and B are independent.
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840 Chapter 19 Random Variables

Hint: Let A1 WWD A and A0 WWD A, so the event ŒIA D c is the same as Ac for
c 2 f0; 1g; likewise for B 1 ; B 0 .

Homework Problems
Problem 19.2.
Let R, S and T be random variables with the same codomain V .
(a) Suppose R is uniform—that is,
1
PrŒR D b D ;
jV j

for all b 2 V —and R is independent of S. Originally this text had the following
argument:

The probability that R D S is the same as the probability that R takes
whatever value S happens to have, therefore
1
PrŒR D S D : (19.17)
jV j

Are you convinced by this argument? Write out a careful proof of (19.17).
Hint: The event ŒR D S is a disjoint union of events
[
ŒR D S D ŒR D b AND S D b:
b2V

(b) Let S T be the random variable giving the values of S and T .3 Now suppose
R has a uniform distribution, and R is independent of S T . How about this
argument?

The probability that R D S is the same as the probability that R equals
the first coordinate of whatever value S T happens to have, and this
probability remains equal to 1=jV j by independence. Therefore the
event ŒR D S is independent of ŒS D T .

Write out a careful proof that ŒR D S is independent of ŒS D T .
3 That is, S T W S ! V V where

.S T /.!/ WWD .S.!/; T .!//

for every outcome ! 2 S.
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19.5. Linearity of Expectation 841

.1; 1; 1/; .2; 1; 1/; .1; 2; 3/; .2; 2; 3/; .1; 3; 2/; .2; 3; 2/:

Verify that

1. R is independent of S T ,
2. The event ŒR D S is not independent of ŒS D T .
3. S and T have a uniform distribution.

Problem 19.3.
Let R, S and T be mutually independent indicator variables.
In general, the event that S D T is not independent of R D S. We can explain
this intuitively as follows: suppose for simplicity that S is uniform, that is, equally
likely to be 0 or 1. This implies that S is equally likely as not to equal R, that is
PrŒR D S D 1=2; likewise, PrŒS D T D 1=2.
Now suppose further that both R and T are more likely to equal 1 than to equal
0. This implies that R D S makes it more likely than not that S D 1, and knowing
that S D 1, makes it more likely than not that S D T . So knowing that RDS
makes it more likely than not that S D T , that is, Pr S D T j R D S > 1=2.
Now prove rigorously (without any appeal to intuition) that the events ŒR D S
and ŒS D T are independent iff either R is uniform4 , or T is uniform, or S is
constant5 .

Problems for Section 19.3
Practice Problems
Problem 19.4.
Suppose R, S and T be mutually independent random variables on the same prob-
ability space with uniform distribution on the range Œ1; 3.
Let M D maxfR; S; T g. Compute the values of the probability density function
PDFM of M .
4 That is, PrŒR D 1 D 1=2.
5 That is, PrŒS D 1 is one or zero.
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842 Chapter 19 Random Variables

Class Problems

Guess the Bigger Number Game

Team 1:

Write two different integers between 0 and 7 on separate pieces of paper.

Put the papers face down on a table.

Team 2:

Turn over one paper and look at the number on it.

Either stick with this number or switch to the other (unseen) number.

Team 2 wins if it chooses the larger number; else, Team 1 wins.

Problem 19.5.
The analysis in Section 19.3.3 implies that Team 2 has a strategy that wins 4/7 of
the time no matter how Team 1 plays. Can Team 2 do better? The answer is “no,”
because Team 1 has a strategy that guarantees that it wins at least 3/7 of the time,
no matter how Team 2 plays. Describe such a strategy for Team 1 and explain why
it works.

Problem 19.6.
Suppose you have a biased coin that has probability p of flipping heads. Let J be
the number of heads in n independent coin flips. So J has the general binomial
distribution: !
n k n k
PDFJ .k/ D p q
k
where q WWD 1 p.
(a) Show that

PDFJ .k 1/ < PDFJ .k/ for k < np C p;
PDFJ .k 1/ > PDFJ .k/ for k > np C p:
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19.5. Linearity of Expectation 843

(b) Conclude that the maximum value of PDFJ is asymptotically equal to
1
p :
2 npq

Hint: For the asymptotic estimate, it’s ok to assume that np is an integer, so by
part (a), the maximum value is PDFJ .np/. Use Stirling’s Formula.

Problem 19.7.
Let R1 ; R2 ; : : : ; Rm , be mutually independent random variables with uniform dis-
tribution on Œ1; n. Let M WWD maxfRi j i 2 Œ1; m g.
(a) Write a formula for PDFM .1/.

(b) More generally, write a formula for PrŒM k.

Homework Problems
Problem 19.8.
A drunken sailor wanders along main street, which conveniently consists of the
points along the x axis with integer coordinates. In each step, the sailor moves
one unit left or right along the x axis. A particular path taken by the sailor can be
described by a sequence of “left” and “right” steps. For example, hleft,left,righti
describes the walk that goes left twice then goes right.
We model this scenario with a random walk graph whose vertices are the integers
and with edges going in each direction between consecutive integers. All edges are
labelled 1=2.
The sailor begins his random walk at the origin. This is described by an initial
distribution which labels the origin with probability 1 and all other vertices with
probability 0. After one step, the sailor is equally likely to be at location 1 or 1,
so the distribution after one step gives label 1/2 to the vertices 1 and 1 and labels
all other vertices with probability 0.
(a) Give the distributions after the 2nd, 3rd, and 4th step by filling in the table of
probabilities below, where omitted entries are 0. For each row, write all the nonzero
entries so they have the same denominator.
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844 Chapter 19 Random Variables

location
-4 -3 -2 -1 0 1 2 3 4
initially 1
after 1 step 1=2 0 1=2
after 2 steps ? ? ? ? ?
after 3 steps ? ? ? ? ? ? ?
after 4 steps ? ? ? ? ? ? ? ? ?
(b)

1. What is the final location of a t-step path that moves right exactly i times?
2. How many different paths are there that end at that location?
3. What is the probability that the sailor ends at this location?

(c) Let L be the random variable giving the sailor’s location after t steps, and let
B WWD.LCt /=2. Use the answer to part (b) to show that B has an unbiased binomial
density function.

(d) Again let L be the random variable giving the sailor’s location after t steps,
where t is even. Show that
p
t 1
PrŒjLj < < :
2 2
p
So there is a better than even chance that the sailor ends up at least t =2 steps from
where he started.
Hint: Work in terms of B. Then you can use an estimate that bounds the binomial
distribution. Alternatively, observe that the origin is the most likely final location
and then use the asymptotic estimate
r
2
PrŒL D 0 D PrŒB D t =2 :
t

Problems for Section 19.4
Practice Problems
Problem 19.9.
Bruce Lee, on a movie that didn’t go public, is practicing by breaking 5 boards with
his fists. He is able to break a board with probability 0.8—he is practicing with his
left fist, that’s why it’s not 1—and he breaks each board independently.
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19.5. Linearity of Expectation 845

(a) What is the probability that Bruce breaks exactly 2 out of the 5 boards that are
placed before him?

(b) What is the probability that Bruce breaks at most 3 out of the 5 boards that are
placed before him?

Problem 19.10.
A news article reporting on the departure of a school official from California to
Alabama dryly commented that this move would raise the average IQ in both states.
Explain.

Class Problems
Problem 19.11.
Here’s a dice game with maximum payoff k: make three independent rolls of a fair
die, and if you roll a six

no times, then you lose 1 dollar;

exactly once, then you win 1 dollar;

exactly twice, then you win 2 dollars;

all three times, then you win k dollars.

For what value of k is this game fair?6

Problem 19.12. (a) Suppose we flip a fair coin and let NTT be the number of flips
until the first time two consecutive Tails appear. What is ExŒNTT ?
Hint: Let D be the tree diagram for this process. Explain why D can be described
by the tree in Figure 19.8. Use the Law of Total Expectation 19.4.5.

(b) Let NTH be the number of flips until a Tail immediately followed by a Head
comes up. What is ExŒNTH ?

(c) Suppose we now play a game: flip a fair coin until either TT or TH occurs.
You win if TT comes up first, and lose if TH comes up first. Since TT takes 50%
6 This game is actually offered in casinos with k D 3, where it is called Carnival Dice.
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846 Chapter 19 Random Variables

T
H T

D
D

Figure 19.8 Sample space tree for coin toss until two consecutive tails.

longer on average to turn up, your opponent agrees that he has the advantage. So
you tell him you’re willing to play if you pay him $5 when he wins, and he pays
you with a mere 20% premium—that is $6—when you win.
If you do this, you’re sneakily taking advantage of your opponent’s untrained intu-
ition, since you’ve gotten him to agree to unfair odds. What is your expected profit
per game?

Problem 19.13.
Let T be a positive integer valued random variable such that
1
PDFT .n/ D ;
an2
where
X 1
a WWD :
C
n2
n2Z

(a) Prove that ExŒT is infinite.
p
(b) Prove that ExŒ T is finite.

Exam Problems
Problem 19.14.
A record of who beat whom in a round-robin tournament can be described with a
tournament digraph, where the vertices correspond to players and there is an edge
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19.5. Linearity of Expectation 847

D
B H T
C H T
H T
D
D
D

Figure 19.9 Outcome Tree for Flipping Until HHH

hx ! yi iff x beat y in their game. A ranking of the players is a path that includes
all the players. A tournament digraph may in general have one or more rankings.7
Suppose we construct a random tournament digraph by letting each of the players
in a match be equally likely to win and having results of all the matches be mutually
independent. Find a formula for the expected number of rankings in a random 10-
player tournament. Conclude that there is a 10-vertex tournament digraph with
more than 7000 rankings.
This problem is an instance of the probabilistic method. It uses probability to
prove the existence of an object without constructing it.

Problem 19.15.
A coin with probability p of flipping Heads and probability q WWD 1 p of flipping
tails is repeatedly flipped until three consecutive Heads occur. The outcome tree D
for this setup is illustrated in Figure 19.9.
Let e.S / be the expected number of flips starting at the root of subtree S of D.
So we’re interested in finding e.D/.
Write a small system of equations involving e.D/; e.B/, and e.C / that could be
solved to find e.D/. You do not need to solve the equations.

7 It has a unique ranking iff it is a DAG, see Problem 10.10.
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848 Chapter 19 Random Variables

A
B H T C
H T H T

C B

Figure 19.10 Outcome Tree for Flipping Until HH or TT

Problem 19.16.
A coin with probability p of flipping Heads and probability q WWD 1 p of flipping
tails is repeatedly flipped until two consecutive flips match—that is, until HH or
TT occurs. The outcome tree A for this setup is illustrated in Figure 19.10.
Let e.T / be the expected number of flips starting at the root of subtree T of A.
So we’re interested in finding e.A/.
Write a small system of equations involving e.A/; e.B/, and e.C / that could be
solved to find e.A/. You do not need to solve the equations.

Homework Problems
Problem 19.17.
We are given a random vector of n distinct numbers. We then determine the maxi-
mum of these numbers using the following procedure:
Pick the first number. Call it the current maximum. Go through the rest of the
vector (in order) and each time we come across a number (call it x) that exceeds
our current maximum, we update the current maximum with x.
What is the expected number of times we update the current maximum?
Hint: Let Xi be the indicator variable for the event that the i th element in the
vector is larger than all the previous elements.
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19.5. Linearity of Expectation 849

Problem 19.18 (Deviations from the mean).
Let B be a random variable with unbiased binomial distribution, nemely,
!
n
PrŒB D k D 2 n:
k

Assume n is even. Prove the following Lemma about the expected absolute devia-
tion of B from its mean:

Lemma. !
n n
ExŒjB ExŒBj D n :
2 2nC1

Problems for Section 19.5
Practice Problems
Problem 19.19.
MIT students sometimes delay doing laundry until they finish their problem sets.
Assume all random values described below are mutually independent.
(a) A busy student must complete 3 problem sets before doing laundry. Each
problem set requires 1 day with probability 2=3 and 2 days with probability 1=3.
Let B be the number of days a busy student delays laundry. What is ExŒB?
Example: If the first problem set requires 1 day and the second and third problem
sets each require 2 days, then the student delays for B D 5 days.

(b) A relaxed student rolls a fair, 6-sided die in the morning. If he rolls a 1, then he
does his laundry immediately (with zero days of delay). Otherwise, he delays for
one day and repeats the experiment the following morning. Let R be the number
of days a relaxed student delays laundry. What is ExŒR?
Example: If the student rolls a 2 the first morning, a 5 the second morning, and a 1
the third morning, then he delays for R D 2 days.

(c) Before doing laundry, an unlucky student must recover from illness for a num-
ber of days equal to the product of the numbers rolled on two fair, 6-sided dice.
Let U be the expected number of days an unlucky student delays laundry. What is
ExŒU ?
Example: If the rolls are 5 and 3, then the student delays for U D 15 days.
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850 Chapter 19 Random Variables

(d) A student is busy with probability 1=2, relaxed with probability 1=3, and un-
lucky with probability 1=6. Let D be the number of days the student delays laundry.
What is ExŒD?

Problem 19.20.
Each Math for Computer Science final exam will be graded according to a rigorous
procedure:

With probability 4=7 the exam is graded by a TA,with probability 2=7 it is
graded by a lecturer, and with probability 1=7, it is accidentally dropped
behind the radiator and arbitrarily given a score of 84.

TAs score an exam by scoring each problem individually and then taking the
sum.

– There are ten true/false questions worth 2 points each. For each, full
credit is given with probability 3=4, and no credit is given with proba-
bility 1=4.
– There are four questions worth 15 points each. For each, the score is
determined by rolling two fair dice, summing the results, and adding 3.
– The single 20 point question is awarded either 12 or 18 points with
equal probability.

Lecturers score an exam by rolling a fair die twice, multiplying the results,
and then adding a “general impression”score.

– With probability 4=10, the general impression score is 40.
– With probability 3=10, the general impression score is 50.
– With probability 3=10, the general impression score is 60.

Assume all random choices during the grading process are independent.
(a) What is the expected score on an exam graded by a TA?

(b) What is the expected score on an exam graded by a lecturer?

19.5. Linearity of Expectation 851

Class Problems
Problem 19.21.
A classroom has sixteen desks in a 4 4 arrangement as shown below.

If there is a girl in front, behind, to the left, or to the right of a boy, then the two
flirt. One student may be in multiple flirting couples; for example, a student in a
corner of the classroom can flirt with up to two others, while a student in the center
can flirt with as many as four others. Suppose that desks are occupied mutually in-
dependently by boys and girls with equal probability. What is the expected number
of flirting couples? Hint: Linearity.

Problem 19.22.
Here are seven propositions:

x1 OR x3 OR x7
x5 OR x6 OR x7
x2 OR x4 OR x6
x4 OR x5 OR x7
x3 OR x5 OR x8
x9 OR x8 OR x2
x3 OR x9 OR x4

Note that:

1. Each proposition is the disjunction (OR) of three terms of the form xi or the
form xi .
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852 Chapter 19 Random Variables

2. The variables in the three terms in each proposition are all different.

Suppose that we assign true/false values to the variables x1 ; : : : ; x9 indepen-
dently and with equal probability.
(a) What is the expected number of true propositions?
Hint: Let Ti be an indicator for the event that the i -th proposition is true.

(b) Use your answer to prove that for any set of 7 propositions satisfying the
conditions 1. and 2., there is an assignment to the variables that makes all 7 of the
propositions true.

Problem 19.23.
A literal is a propositional variable or its negation. A k-clause is an OR of k literals,
with no variable occurring more than once in the clause. For example,

P OR Q OR R OR V;

is a 4-clause, but
V OR Q OR X OR V;
is not, since V appears twice.
Let S be a sequence of n distinct k-clauses involving v variables. The variables
in different k-clauses may overlap or be completely different, so k v nk.
A random assignment of true/false values will be made independently to each of
the v variables, with true and false assignments equally likely. Write formulas in n,
k and v in answer to the first two parts below.
(a) What is the probability that the last k-clause in S is true under the random
assignment?

(b) What is the expected number of true k-clauses in S?

(c) A set of propositions is satisfiable iff there is an assignment to the variables
that makes all of the propositions true. Use your answer to part (b) to prove that if
n < 2k , then S is satisfiable.

Problem 19.24.
There are n students who are both taking Math for Computer Science (MCS) and
Introduction to Signal Processing (SP) this term. To make it easier on themselves,
the Professors in charge of these classes have decided to randomly permute their
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19.5. Linearity of Expectation 853

class lists and then assign students grades based on their rank in the permutation
(just as many students have suspected). Assume the permutations are equally likely
and independent of each other. What is the expected number of students that have
in rank in SP that is higher by k than their rank in MCS?
Hint: Let Xr be the indicator variable for the rth ranked student in CS having a
rank in SP of at least r C k.

Problem 19.25.
A man has a set of n keys, one of which fits the door to his apartment. He tries the
keys randomly until he finds the key that fits. Let T be the number of times he tries
keys until he finds the right key.
(a) Suppose each time he tries a key that does not fit the door, he simply puts it
back. This means he might try the same ill-fitting key several times before he finds
the right key. What is ExŒT ?
Hint: Mean time to failure.

Now suppose he throws away each ill-fitting key that he tries. That is, he chooses
keys randomly from among those he has not yet tried. This way he is sure to find
the right key within n tries.
(b) If he hasn’t found the right key yet and there are m keys left, what is the
probability that he will find the right key on the next try?

(c) Given that he did not find the right key on his first k 1 tries, verify that the
probability that he does not find it on the kth trial is given by
n k
Pr T > k j T > k 1 D :
n .k 1/

(d) Prove that
n k
PrŒT > k D : (19.18)
n
Hint: This can be argued directly, but if you don’t see how, induction using part (c)
will work.

(e) Conclude that in this case
nC1
ExŒT D :
2
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854 Chapter 19 Random Variables

Problem 19.26.
Justify each line of the following proof that if R1 and R2 are independent, then

ExŒR1 R2 D ExŒR1 ExŒR2 :

Proof.

ExŒR1 R2
X
D r PrŒR1 R2 D r
r2range.R1 R2 /
X
D r1 r2 PrŒR1 D r1 and R2 D r2
ri 2range.Ri /
X X
D r1 r2 PrŒR1 D r1 and R2 D r2
r1 2range.R1 / r2 2range.R2 /
X X
D r1 r2 PrŒR1 D r1 PrŒR2 D r2
r1 2range.R1 / r2 2range.R2 /
0 1
X X
D @r1 PrŒR1 D r1 r2 PrŒR2 D r2 A
r1 2range.R1 / r2 2range.R2 /
X
D r1 PrŒR1 D r1 ExŒR2
r1 2range.R1 /
X
D ExŒR2 r1 PrŒR1 D r1
r1 2range.R1 /

D ExŒR2 ExŒR1 :

Problem 19.27.
A gambler bets on the toss of a fair coin: if the toss is Heads, the gambler gets back
the amount he bet along with an additional the amount equal to his bet. Otherwise
he loses the amount bet. For example, the gambler bets $10 and wins, he gets back
$20 for a net profit of $10. If he loses, he gets back nothing for a net profit of
$10—that is, a net loss of $10.
Gamblers often try to develop betting strategies to beat the odds is such a game.
A well known strategy of this kind is bet doubling, namely, bet $10 on red, and
keep doubling the bet until a red comes up. So if the gambler wins his first $10
bet, he stops playing and leaves with his $10 profit. If he loses the first bet, he bets
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19.5. Linearity of Expectation 855

$20 on the second toss. Now if the second toss is Heads, he gets his $20 bet plus
$20 back and again walks away with a net profit of 20 10 D $10. If he loses the
second toss, he bets $40 on the third toss, and so on.
You would think that any such strategy will be doomed: in a fair game your
expected win by definition is zero, so no strategy should have nonzero expectation.
We can make this reasoning more precise as follows:

Let Wn be a random variable equal to the amount won in the nth
coin toss. So with the bet doubling strategy starting with a $10 bet,
W1 D ˙10 with equal probability. If the betting ends before the nth
bet, define Wn D 0. So W2 is zero with probability 1/2, is 10 with
probability 1/4, and is 10 with probability 1/4. Now letting W be the
amount won when the gambler stops betting, we have

W D W1 C W2 C C Wn C :

Furthermore, since each toss is fair,

ExŒWn D 0

for all n > 0. Now by linearity of expectation, we have

ExŒW D ExŒW1 CExŒW2 C CExŒWn C D 0C0C C0C D 0;
(19.19)
confirming that with fair tosses, the expected win is zero.

But wait a minute!
(a) Explain why the gambler is certain to win eventually if he keeps betting.

(b) Prove that when the gambler finally wins a bet, his net profit is $10.

(c) Since the gambler’s profit is always $10 when he wins, and he is certain to
win, his expected profit is also $10. That is

ExŒW D 10;

contradicting (19.19). So what’s wrong with the reasoning that led to the false
conclusion (19.19)?
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856 Chapter 19 Random Variables

Homework Problems
Problem 19.28.
Applying linearity of expectation to the binomial distribution fn;p immediately
yielded the identity 19.13:
n
!
X n k
ExŒfn;p WWD k p .1 p/n k D pn: (*)
k
kD0

Though it might seem daunting to prove this equation without appeal to linearity, it
is, after all, pretty similar to the binomial identity, and this connection leads to an
immediate alternative algebraic derivation.
(a) Starting with the binomial identity for .x C y/n , prove that
n
!
n 1
X n k n k
x n.x C y/ D k x y : (**)
k
kD0

(b) Now conclude equation (*).

Problem 19.29.
A coin will be flipped repeatedly until the sequence TTH (tail/tail/head) comes
up. Successive flips are independent, and the coin has probability p of coming up
heads. Let NTTH be the number of coin flips until TTH first appears. What value of
p minimizes ExŒNTTH ?

Problem 19.30.
(A true story from World War Two.)
The army needs to test n soldiers for a disease. There is a blood test that accu-
rately determines when a blood sample contains blood from a diseased soldier. The
army presumes, based on experience, that the fraction of soldiers with the disease
is approximately equal to some small number p.
Approach (1) is to test blood from each soldier individually; this requires n tests.
Approach (2) is to randomly group the soldiers into g groups of k soldiers, where
n D gk. For each group, blend the k blood samples of the people in the group,
and test the blended sample. If the group-blend is free of the disease, we are done
with that group after one test. If the group-blend tests positive for the disease, then
someone in the group has the disease, and we to test all the people in the group for
a total of k C 1 tests on that group.
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19.5. Linearity of Expectation 857

Since the groups are chosen randomly, each soldier in the group has the disease
with probability p, and it is safe to assume that whether one soldier has the disease
is independent of whether the others do.
(a) What is the expected number of tests in Approach (2) as a function of the
number of soldiers n, the disease fraction p, and the group size k?

(b) Show how to choose k so that the expected number of tests using Approach (2)
p
is approximately n p. Hint: Since p is small, you may assume that .1 p/k 1
and ln.1 p/ p.

(c) What fraction of the work does Approach (2) expect to save over Approach
(1) in a million-strong army of whom approximately 1% are diseased?

(d) Can you come up with a better scheme by using multiple levels of grouping,
that is, groups of groups?

Problem 19.31.
A wheel-of-fortune has the numbers from 1 to 2n arranged in a circle. The wheel
has a spinner, and a spin randomly determines the two numbers at the opposite ends
of the spinner. How would you arrange the numbers on the wheel to maximize the
expected value of:
(a) the sum of the numbers chosen? What is this maximum?

(b) the product of the numbers chosen? What is this maximum?
Hint: For part (b), verify that the sum of the products of numbers oppposite each
other is maximized when successive integers are on the opposite ends of the spin-
ner, that is, 1 is opposite 2, 3 is opposite 4, 5 is opposite 6, . . . .

Problem 19.32.
Let R and S be independent random variables, and f and g be any functions such
that domain.f / D codomain.R/ and domain.g/ D codomain.S /. Prove that f .R/
and g.S / are also independent random variables.
Hint: The event Œf .R/ D a is the disjoint union of all the events ŒR D r for r
such that f .r/ D a.

Problem 19.33.
Peeta bakes between 1 and 2n loaves of bread to sell every day. Each day he rolls
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858 Chapter 19 Random Variables

a fair, n-sided die to get a number from 1 to n, then flips a fair coin. If the coin is
heads, he bakes m loaves of bread , where m is the number on the die that day, and
if the coin is tails, he bakes 2m loaves.
(a) For any positive integer k 2n, what is the probability that Peeta will make k
loaves of bread on any given day? (Hint: you can express your solution by cases.)

(b) What is the expected number of loaves that Peeta would bake on any given
day?

(c) Continuing this process, Peeta bakes bread every day for 30 days. What is the
expected total number of loaves that Peeta would bake?

Exam Problems
Problem 19.34.
A box initially contains n balls, all colored black. A ball is drawn from the box at
random.

If the drawn ball is black, then a biased coin with probability, p > 0, of
coming up heads is flipped. If the coin comes up heads, a white ball is put
into the box; otherwise the black ball is returned to the box.

If the drawn ball is white, then it is returned to the box.

This process is repeated until the box contains n white balls.
Let D be the number of balls drawn until the process ends with the box full of
white balls. Prove that ExŒD D nHn =p, where Hn is the nth Harmonic number.
Hint: Let Di be the number of draws after the i th white ball until the draw when
the .i C 1/st white ball is put into the box.

Problem 19.35.
A gambler bets $10 on “red” at a roulette table (the odds of red are 18/38, slightly
less than even) to win $10. If he wins, he gets back twice the amount of his bet,
and he quits. Otherwise, he doubles his previous bet and continues.
For example, if he loses his first two bets but wins his third bet, the total spent
on his three bets is 10 C 20 C 40 dollars, but he gets back 2 40 dollars after his
win on the third bet, for a net profit of $10.
(a) What is the expected number of bets the gambler makes before he wins?

(b) What is his probability of winning?
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19.5. Linearity of Expectation 859

(d) You can beat a biased game by bet doubling, but bet doubling is not feasible
because it requires an infinite bankroll. Verify this by proving that the expected size
of the gambler’s last bet is infinite.

Problem 19.36.
Six pairs of cards with ranks 1–6 are shuffled and laid out in a row, for example,

1 2 3 3 4 6 1 4 5 5 2 6

In this case, there are two adjacent pairs with the same value, the two 3’s and the
two 5’s. What is the expected number of adjacent pairs with the same value?

Problem 19.37.
There are six kinds of cards, three of each kind, for a total of eighteen cards. The
cards are randonly shuffled and laid out in a row, for example,

1 2 5 5 5 1 4 6 2 6 6 2 1 4 3 3 3 4

In this case, there are two adjacent triples of the same kind, the three 3’s and the
three 5’s.
(a) Derive a formula for the probability that the 4th, 5th, and 6th consecutive cards
will be the same kind—that is, all 1’s or all 2’s or. . . all 6’s?

(b) Let p WWD PrŒ4th, 5th and 6th cards match—that is, p is the correct answer to
part (a). Write a simple formula for the expected number of matching triples in
terms of p.
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20 Deviation from the Mean
In the previous chapter, we took it for granted that expectation is useful and de-
veloped a bunch of techniques for calculating expected values. But why should we
care about this value? After all, a random variable may never take a value anywhere
near its expectation.
The most important reason to care about the mean value comes from its con-
nection to estimation by sampling. For example, suppose we want to estimate the
average age, income, family size, or other measure of a population. To do this,
we determine a random process for selecting people—say, throwing darts at cen-
sus lists. This process makes the selected person’s age, income, and so on into a
random variable whose mean equals the actual average age or income of the pop-
ulation. So, we can select a random sample of people and calculate the average
of people in the sample to estimate the true average in the whole population. But
when we make an estimate by repeated sampling, we need to know how much con-
fidence we should have that our estimate is OK, and how large a sample is needed
to reach a given confidence level. The issue is fundamental to all experimental
science. Because of random errors—noise—repeated measurements of the same
quantity rarely come out exactly the same. Determining how much confidence
to put in experimental measurements is a fundamental and universal scientific is-
sue. Technically, judging sampling or measurement accuracy reduces to finding the
probability that an estimate deviates by a given amount from its expected value.
Another aspect of this issue comes up in engineering. When designing a sea wall,
you need to know how strong to make it to withstand tsunamis for, say, at least a
century. If you’re assembling a computer network, you might need to know how
many component failures it should tolerate to likely operate without maintenance
for at least a month. If your business is insurance, you need to know how large a
financial reserve to maintain to be nearly certain of paying benefits for, say, the next
three decades. Technically, such questions come down to finding the probability of
extreme deviations from the mean.
This issue of deviation from the mean is the focus of this chapter.

20.1 Markov’s Theorem
Markov’s theorem gives a generally coarse estimate of the probability that a random
variable takes a value much larger than its mean. It is an almost trivial result by
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862 Chapter 20 Deviation from the Mean

itself, but it actually leads fairly directly to much stronger results.
The idea behind Markov’s Theorem can be explained by considering the quantity
known as intelligence quotient, IQ, which remains in wide use despite doubts about
its legitimacy. IQ was devised so that its average measurement would be 100. This
immediately implies that at most 1/3 of the population can have an IQ of 300 or
more, because if more than a third had an IQ of 300, then the average would have to
be more than .1=3/ 300 D 100. So, the probability that a randomly chosen person
has an IQ of 300 or more is at most 1/3. By the same logic, we can also conclude
that at most 2/3 of the population can have an IQ of 150 or more.
Of course, these are not very strong conclusions. No IQ of over 300 has ever
been recorded; and while many IQ’s of over 150 have been recorded, the fraction
of the population that actually has an IQ that high is very much smaller than 2/3.
But though these conclusions are weak, we reached them using just the fact that the
average IQ is 100—along with another fact we took for granted, that IQ is never
negative. Using only these facts, we can’t derive smaller fractions, because there
are nonnegative random variables with mean 100 that achieve these fractions. For
example, if we choose a random variable equal to 300 with probability 1/3 and 0
with probability 2/3, then its mean is 100, and the probability of a value of 300 or
more really is 1/3.
Theorem 20.1.1 (Markov’s Theorem). If R is a nonnegative random variable, then
for all x > 0
ExŒR
PrŒR x : (20.1)
x
Proof. Let y vary over the range of R. Then for any x > 0

X
ExŒR WWD y PrŒR D y
y
X X X
y PrŒR D y x PrŒR D y D x PrŒR D y
yx yx yx

D x PrŒR x; (20.2)
where the first inequality follows from the fact that R 0.
Dividing the first and last expressions in (20.2) by x gives the desired result.
Our focus is deviation from the mean, so it’s useful to rephrase Markov’s Theo-
rem this way:
Corollary 20.1.2. If R is a nonnegative random variable, then for all c 1
1
PrŒR c ExŒR : (20.3)
c
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20.1. Markov’s Theorem 863

This Corollary follows immediately from Markov’s Theorem(20.1.1) by letting
x be c ExŒR.

20.1.1 Applying Markov’s Theorem
Let’s go back to the Hat-Check problem of Section 19.5.2. Now we ask what
the probability is that x or more men get the right hat, this is, what the value of
PrŒG x is.
We can compute an upper bound with Markov’s Theorem. Since we know
ExŒG D 1, Markov’s Theorem implies
ExŒG 1
PrŒG x D :
x x
For example, there is no better than a 20% chance that 5 men get the right hat,
regardless of the number of people at the dinner party.
The Chinese Appetizer problem is similar to the Hat-Check problem. In this
case, n people are eating different appetizers arranged on a circular, rotating Chi-
nese banquet tray. Someone then spins the tray so that each person receives a
random appetizer. What is the probability that everyone gets the same appetizer as
before?
There are n equally likely orientations for the tray after it stops spinning. Ev-
eryone gets the right appetizer in just one of these n orientations. Therefore, the
correct answer is 1=n.
But what probability do we get from Markov’s Theorem? Let the random vari-
able R be the number of people that get the right appetizer. Then of course ExŒR D
1, so applying Markov’s Theorem, we find:
ExŒR 1
PrŒR n D :
n n
So for the Chinese appetizer problem, Markov’s Theorem is precisely right!
Unfortunately, Markov’s Theorem is not always so accurate. For example, it
gives the same 1=n upper limit for the probability that everyone gets their own hat
back in the Hat-Check problem, where the probability is actually 1=.nŠ/. So for
Hat-Check, Markov’s Theorem gives a probability bound that is way too large.

20.1.2 Markov’s Theorem for Bounded Variables
Suppose we learn that the average IQ among MIT students is 150 (which is not
true, by the way). What can we say about the probability that an MIT student has
an IQ of more than 200? Markov’s theorem immediately tells us that no more than
150=200 or 3=4 of the students can have such a high IQ. Here, we simply applied
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864 Chapter 20 Deviation from the Mean

Markov’s Theorem to the random variable R equal to the IQ of a random MIT
student to conclude:
ExŒR 150 3
PrŒR > 200 D D :
200 200 4
But let’s observe an additional fact (which may be true): no MIT student has an
IQ less than 100. This means that if we let T WWD R 100, then T is nonnegative
and ExŒT D 50, so we can apply Markov’s Theorem to T and conclude:

ExŒT 50 1
PrŒR > 200 D PrŒT > 100 D D :
100 100 2
So only half, not 3/4, of the students can be as amazing as they think they are. A
bit of a relief!
In fact, we can get better bounds applying Markov’s Theorem to R b instead of
R for any lower bound b on R (see Problem 20.3). Similarly, if we have any upper
bound u on a random variable S , then u S will be a nonnegative random variable,
and applying Markov’s Theorem to u S will allow us to bound the probability
that S is much less than its expectation.

20.2 Chebyshev’s Theorem
We’ve seen that Markov’s Theorem can give a better bound when applied to R b
rather than R. More generally, a good trick for getting stronger bounds on a ran-
dom variable R out of Markov’s Theorem is to apply the theorem to some cleverly
chosen function of R. Choosing functions that are powers of the absolute value of
R turns out to be especially useful. In particular, since jRjz is nonnegative for any
real number z, Markov’s inequality also applies to the event Œ jRjz x z . But for
positive x; z > 0 this event is equivalent to the event Œ jRj x for , so we have:

Lemma 20.2.1. For any random variable R and positive real numbers x; z,

ExŒ jRjz
PrŒjRj x :
xz
Rephrasing (20.2.1) in terms of jR ExŒR j, the random variable that measures
R’s deviation from its mean, we get

ExŒ.jR ExŒRj/z
PrŒ jR ExŒR j x : (20.4)
xz
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20.2. Chebyshev’s Theorem 865

When z is positive and even, .R ExŒR/z is nonnegative, so the absolute value
on the right-hand side of the inequality (20.4) is redundant. The case when z D 2
turns out to be so important that the numerator of the right-hand side has been given
a name:
Definition 20.2.2. The variance of a random variable R is:
VarŒR WWD Ex .R ExŒR/2 :

Variance is also known as mean square deviation.
The restatement of (20.4) for z D 2 is known as Chebyshev’s Theorem.1
Theorem 20.2.3 (Chebyshev). Let R be a random variable and x 2 RC . Then
VarŒR
PrŒjR ExŒR j x :
x2
The expression ExŒ.R ExŒR/2 for variance is a bit cryptic; the best approach
is to work through it from the inside out. The innermost expression R ExŒR is
precisely the deviation of R above its mean. Squaring this, we obtain .R ExŒR/2 .
This is a random variable that is near 0 when R is close to the mean and is a large
positive number when R deviates far above or below the mean. So if R is always
close to the mean, then the variance will be small. If R is often far from the mean,
then the variance will be large.

20.2.1 Variance in Two Gambling Games
The relevance of variance is apparent when we compare the following two gam-
bling games.
Game A: We win $2 with probability 2=3 and lose $1 with probability 1=3.
Game B: We win $1002 with probability 2=3 and lose $2001 with probability
1=3.
Which game is better financially? We have the same probability, 2/3, of winning
each game, but that does not tell the whole story. What about the expected return for
each game? Let random variables A and B be the payoffs for the two games. For
example, A is 2 with probability 2/3 and -1 with probability 1/3. We can compute
the expected payoff for each game as follows:
2 1
ExŒA D 2
C . 1/ D 1;
3 3
2 1
ExŒB D 1002 C . 2001/ D 1:
3 3
1 There are Chebyshev Theorems in several other disciplines, but Theorem 20.2.3 is the only one
we’ll refer to.
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866 Chapter 20 Deviation from the Mean

The expected payoff is the same for both games, but the games are very different.
This difference is not apparent in their expected value, but is captured by variance.
We can compute the VarŒA by working “from the inside out” as follows:
with probability 23

1
A ExŒA D
2 with probability 13
1 with probability 32

.A ExŒA/2 D
4 with probability 13
2 1
ExŒ.A ExŒA/2 D 1 C 4
3 3
VarŒA D 2:
Similarly, we have for VarŒB:
2

1001 with probability 3
B ExŒB D 1
2002 with probability 3
2

2 1; 002; 001 with probability 3
.B ExŒB/ D 1
4; 008; 004 with probability 3
2 1
ExŒ.B ExŒB/2 D 1; 002; 001 C 4; 008; 004
3 3
VarŒB D 2; 004; 002:
The variance of Game A is 2 and the variance of Game B is more than two
million! Intuitively, this means that the payoff in Game A is usually close to the
expected value of $1, but the payoff in Game B can deviate very far from this
expected value.
High variance is often associated with high risk. For example, in ten rounds of
Game A, we expect to make $10, but could conceivably lose $10 instead. On the
other hand, in ten rounds of game B, we also expect to make $10, but could actually
lose more than $20,000!

20.2.2 Standard Deviation
In Game B above, the deviation from the mean is 1001 in one outcome and -2002
in the other. But the variance is a whopping 2,004,002. The happens because the
“units” of variance are wrong: if the random variable is in dollars, then the expec-
tation is also in dollars, but the variance is in square dollars. For this reason, people
often describe random variables using standard deviation instead of variance.
Definition 20.2.4. The standard deviation R of a random variable R is the square
root of the variance:
p q
R WWD VarŒR D ExŒ.R ExŒR/2 :
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20.2. Chebyshev’s Theorem 867

mean

O.¢/

Figure 20.1 The standard deviation of a distribution indicates how wide the
“main part” of it is.

So the standard deviation is the square root of the mean square deviation, or
the root mean square for short. It has the same units—dollars in our example—as
the original random variable and as the mean. Intuitively, it measures the average
deviation from the mean, since we can think of the square root on the outside as
canceling the square on the inside.
Example 20.2.5. The standard deviation of the payoff in Game B is:
p p
B D VarŒB D 2; 004; 002 1416:

The random variable B actually deviates from the mean by either positive 1001
or negative 2002, so the standard deviation of 1416 describes this situation more
closely than the value in the millions of the variance.
For bell-shaped distributions like the one illustrated in Figure 20.1, the standard
deviation measures the “width” of the interval in which values are most likely to
fall. This can be more clearly explained by rephrasing Chebyshev’s Theorem in
terms of standard deviation, which we can do by substituting x D cR in (20.1):

Corollary 20.2.6. Let R be a random variable, and let c be a positive real number.
1
PrŒjR ExŒRj cR : (20.5)
c2
Now we see explicitly how the “likely” values of R are clustered in an O.R /-
sized region around ExŒR, confirming that the standard deviation measures how
spread out the distribution of R is around its mean.
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868 Chapter 20 Deviation from the Mean

The IQ Example
The standard standard deviation of IQ’s regularly turns out to be about 15 even
across different populations. This additional fact along with the national average
IQ being 100 allows a better determination of the occurrence of IQ’s of 300 or
more.
Let the random variable R be the IQ of a random person. So ExŒR D 100,
R D 15 and R is nonnegative. We want to compute PrŒR 300.
We have already seen that Markov’s Theorem 20.1.1 gives a coarse bound, namely,
1
PrŒR 300 :
3
Now we apply Chebyshev’s Theorem to the same problem:
VarŒR 152 1
PrŒR 300 D PrŒjR 100j 200 2
D 2
:
200 200 178
So Chebyshev’s Theorem implies that at most one person in 178 has an IQ of 300
or more. We have gotten a much tighter bound using additional information—the
variance of R—than we could get knowing only the expectation.

20.3 Properties of Variance
Variance is the average of the square of the distance from the mean. For this rea-
son, variance is sometimes called the “mean square deviation.” Then we take its
square root to get the standard deviation—which in turn is called “root mean square
deviation.”
But why bother squaring? Why not study the actual distance from the mean,
namely, the absolute value of R ExŒR, instead of its root mean square? The
answer is that variance and standard deviation have useful properties that make
them much more important in probability theory than average absolute deviation.
In this section, we’ll describe some of those properties. In the next section, we’ll
see why these properties are important.

20.3.1 A Formula for Variance
Applying linearity of expectation to the formula for variance yields a convenient
alternative formula.
Lemma 20.3.1.
VarŒR D ExŒR2 Ex2 ŒR;
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20.3. Properties of Variance 869

for any random variable R.

Here we use the notation Ex2 ŒR as shorthand for .ExŒR/2 .

Proof. Let D ExŒR. Then

VarŒR D ExŒ.R ExŒR/2 (Def 20.2.2 of variance)
2
D ExŒ.R / (def of )
2 2
D ExŒR 2R C
2
D ExŒR 2 ExŒR C 2 (linearity of expectation)
2 2 2
D ExŒR 2 C (def of )
D ExŒR2 2
D ExŒR2 Ex2 ŒR: (def of )

A simple and very useful formula for the variance of an indicator variable is an
immediate consequence.

Corollary 20.3.2. If B is a Bernoulli variable where p WWD PrŒB D 1, then

VarŒB D p p 2 D p.1 p/: (20.6)

Proof. By Lemma 19.4.2, ExŒB D p. But B only takes values 0 and 1, so B 2 D B
and equation (20.6) follows immediately from Lemma 20.3.1.

20.3.2 Variance of Time to Failure
According to Section 19.4.6, the mean time to failure is 1=p for a process that fails
during any given hour with probability p. What about the variance?
By Lemma 20.3.1,
VarŒC D ExŒC 2 .1=p/2 (20.7)
so all we need is a formula for ExŒC 2 .
Reasoning about C using conditional expectation worked nicely in Section 19.4.6
to find mean time to failure, and a similar approach works for C 2 . Namely, the ex-
pected value of C 2 is the probability p of failure in the first hour times 12 , plus
the probability .1 p/ of non-failure in the first hour times the expected value of
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870 Chapter 20 Deviation from the Mean

.C C 1/2 . So
ExŒC 2 D p 12 C .1 p/ ExŒ.C C 1/2

2 2
D p C .1 p/ ExŒC C C 1
p

2
D p C .1 p/ ExŒC 2 C .1 p/ C1 ; so
p

2
p ExŒC 2 D p C .1 p/ C1
p
p 2 C .1 p/.2 C p/
D and
p
2 p
ExŒC 2 D
p2
Combining this with (20.7) proves
Lemma 20.3.3. If failures occur with probability p independently at each step, and
C is the number of steps until the first failure,2 then
1 p
VarŒC D : (20.8)
p2

20.3.3 Dealing with Constants
It helps to know how to calculate the variance of aR C b:
Theorem 20.3.4. [Square Multiple Rule for Variance] Let R be a random variable
and a a constant. Then
VarŒaR D a2 VarŒR: (20.9)
Proof. Beginning with the definition of variance and repeatedly applying linearity
of expectation, we have:
VarŒaR WWD ExŒ.aR ExŒaR/2
D ExŒ.aR/2 2aR ExŒaR C Ex2 ŒaR
D ExŒ.aR/2 ExŒ2aR ExŒaR C Ex2 ŒaR
D a2 ExŒR2 2 ExŒaR ExŒaR C Ex2 ŒaR
D a2 ExŒR2 a2 Ex2 ŒR
D a2 ExŒR2 Ex2 ŒR

D a2 VarŒR (Lemma 20.3.1)
2 That is, C has the geometric distribution with parameter p according to Definition 19.4.6.
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20.3. Properties of Variance 871

It’s even simpler to prove that adding a constant does not change the variance, as
the reader can verify:

Theorem 20.3.5. Let R be a random variable, and b a constant. Then

VarŒR C b D VarŒR: (20.10)

Recalling that the standard deviation is the square root of variance, this implies
that the standard deviation of aR C b is simply jaj times the standard deviation of
R:

Corollary 20.3.6.
.aRCb/ D jaj R :

20.3.4 Variance of a Sum
In general, the variance of a sum is not equal to the sum of the variances, but
variances do add for independent variables. In fact, mutual independence is not
necessary: pairwise independence will do. This is useful to know because there are
some important situations, such as Birthday Matching in Section 17.4, that involve
variables that are pairwise independent but not mutually independent.

Theorem 20.3.7. If R and S are independent random variables, then

VarŒR C S D VarŒR C VarŒS : (20.11)

Proof. We may assume that ExŒR D 0, since we could always replace R by
R ExŒR in equation (20.11); likewise for S. This substitution preserves the
independence of the variables, and by Theorem 20.3.5, does not change the vari-
ances.
But for any variable T with expectation zero, we have VarŒT D ExŒT 2 , so we
need only prove
ExŒ.R C S /2 D ExŒR2 C ExŒS 2 : (20.12)
But (20.12) follows from linearity of expectation and the fact that

ExŒRS D ExŒR ExŒS (20.13)
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872 Chapter 20 Deviation from the Mean

since R and S are independent:

ExŒ.R C S /2 D ExŒR2 C 2RS C S 2
D ExŒR2 C 2 ExŒRS C ExŒS 2
D ExŒR2 C 2 ExŒR ExŒS C ExŒS 2 (by (20.13))
D ExŒR2 C 2 0 0 C ExŒS 2
D ExŒR2 C ExŒS 2 :

It’s easy to see that additivity of variance does not generally hold for variables
that are not independent. For example, if R D S , then equation (20.11) becomes
VarŒR CR D VarŒRCVarŒR. By the Square Multiple Rule, Theorem 20.3.4, this
holds iff 4 VarŒR D 2 VarŒR, which implies that VarŒR D 0. So equation (20.11)
fails when R D S and R has nonzero variance.
The proof of Theorem 20.3.7 carries over to the sum of any finite number of
variables (Problem 20.18), so we have:

Theorem 20.3.8. [Pairwise Independent Additivity of Variance] If R1 ; R2 ; : : : ; Rn
are pairwise independent random variables, then

VarŒR1 C R2 C C Rn D VarŒR1 C VarŒR2 C C VarŒRn : (20.14)

Now we have a simple way of computing the variance Pn of a variable J that has
an .n; p/-binomial distribution. We know that J D kD1 Ik where the Ik are
mutually independent indicator variables with PrŒIk D 1 D p. The variance of
each Ik is p.1 p/ by Corollary 20.3.2, so by linearity of variance, we have

Lemma 20.3.9 (Variance of the Binomial Distribution). If J has the .n; p/-binomial
distribution, then
VarŒJ D n VarŒIk D np.1 p/: (20.15)

20.3.5 Matching Birthdays
We saw in Section 17.4 that in a class of 95 students, it is virtually certain that
at least one pair of students will have the same birthday. In fact, several pairs of
students are likely to have the same birthday. How many matched birthdays should
we expect, and how likely are we to see that many matches in a random group of
students?
Having matching birthdays for different pairs of students are not mutually inde-
pendent events. If Alice matches Bob and Alice matches Carol, it’s certain that Bob
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20.3. Properties of Variance 873

and Carol match as well! So the events that various pairs of students have matching
birthdays are not even three-way independent.
But knowing that Alice’s birthday matches Bob’s tells us nothing about who
Carol matches. This means that the events that a pair of people have matching
birthdays are pairwise independent (see Problem 19.2). So pairwise independent
additivity of variance, Theorem 20.3.8, will allow us to calculate the variance of
the number of birthday pairs and then apply Chebyshev’s bound to estimate the
liklihood of seeing some given number of matching pairs.
In particular, suppose there are n students and d days in the year, and let M be the
number of pairs of students with matching birthdays. Namely, let B1 ; B2 ; : : : ; Bn
be the birthdays of n independently chosen people, and let Ei;j be the indicator
variable for the event that the i th and j th people chosen have the same birthdays,
that is, the event ŒBi D Bj . So in our probability model, the Bi ’s are mutually
independent variables, and the Ei;j ’s are pairwise independent. Also, the expecta-
tions of Ei;j for i ¤ j equals the probability that Bi D Bj , namely, 1=d .
Now the number M of matching pairs of birthdays among the n choices is simply
the sum of the Ei;j ’s: X
M D Ei;j : (20.16)
1i <j n

Linearity of expectation make it easy to calculate the expected number of pairs of
students with matching birthdays.
2 3 !
X X n 1
ExŒM D Ex 4 Ei;j 5 D ExŒEi;j D :
2 d
1i <j n 1i <j n

Similarly, pairwise independence makes it easy to calculate the variance.
2 3
X
VarŒM D Var 4 Ei;j 5
1i <j n
X
D VarŒEi;j (Theorem 20.3.8)
1i <j n
!
n 1 1
D 1 : (Corollary 20.3.2)
2 d d

In particular, for a class of n D 95 students with d D 365 possible birthdays, we
have ExŒM 12:23 and VarŒM 12:23.1 1=365/ < 12:2. So by Chebyshev’s
Theorem
12:2
PrŒjM ExŒM j x < 2 :
x
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874 Chapter 20 Deviation from the Mean

Letting x D 7, we conclude that there is a better than 75% chance that in a class of
95 students, the number of pairs of students with the same birthday will be within
7 of 12.23, that is, between 6 and 19.

20.4 Estimation by Random Sampling
Democratic politicians were astonished in 2010 when their early polls of sample
voters showed Republican Scott Brown was favored by a majority of voters and so
would win the special election to fill the Senate seat that the late Democrat Teddy
Kennedy had occupied for over 40 years. Based on their poll results, they mounted
an intense, but ultimately unsuccessful, effort to save the seat for their party.

20.4.1 A Voter Poll
Suppose at some time before the election that p was the fraction of voters favoring
Scott Brown. We want to estimate this unknown fraction p. Suppose we have
some random process for selecting voters from registration lists that selects each
voter with equal probability. We can define an indicator variable K by the rule that
K D 1 if the random voter most prefers Brown, and K D 0 otherwise.
Now to estimate p, we take a large number n of random choices of voters3 and
count the fraction who favor Brown. That is, we define variables K1 ; K2 ; : : : ,
where Ki is interpreted to be the indicator variable for the event that the i th cho-
sen voter prefers Brown. Since our choices are made independently, the Ki ’s are
independent. So formally, we model our estimation process by assuming we have
mutually independent indicator variables K1 ; K2 ; : : : ; each with the same proba-
bility p of being equal to 1. Now let Sn be their sum, that is,
n
X
Sn WWD Ki : (20.17)
i D1

The variable Sn =n describes the fraction of sampled voters who favor Scott Brown.
Most people intuitively, and correctly, expect this sample fraction to give a useful
approximation to the unknown fraction p.
So we will use the sample value Sn =n as our statistical estimate of p. We know
that Sn has a binomial distribution with parameters n and p; we can choose n, but
3 We’re choosing a random voter n times with replacement. We don’t remove a chosen voter from
the set of voters eligible to be chosen later; so we might choose the same voter more than once!
We would get a slightly better estimate if we required n different people to be chosen, but doing so
complicates both the selection process and its analysis for little gain.
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20.4. Estimation by Random Sampling 875

p is unknown.

How Large a Sample?
Suppose we want our estimate to be within 0:04 of the fraction p at least 95% of
the time. This means we want
ˇ ˇ
ˇ Sn ˇ
Pr ˇˇ p ˇˇ 0:04 0:95 : (20.18)
n
So we’d better determine the number n of times we must poll voters so that in-
equality (20.18) will hold. Chebyshev’s Theorem offers a simple way to determine
such a n.
Sn is binomially distributed. Equation (20.15), combined with the fact that p.1
p/ is maximized when p D 1 p, that is, when p D 1=2 (check for yourself!),
gives
1 n
VarŒSn D n.p.1 p// n D : (20.19)
4 4
Next, we bound the variance of Sn =n:
2
Sn 1
Var D VarŒSn (Square Multiple Rule for Variance (20.9))
n n
2
1 n
(by (20.19))
n 4
1
D (20.20)
4n
Using Chebyshev’s bound and (20.20) we have:
ˇ ˇ
ˇ Sn ˇ VarŒSn =n 1 156:25
Pr ˇ ˇ p ˇ 0:04
ˇ
2
2
D (20.21)
n .0:04/ 4n.0:04/ n
To make our our estimate with 95% confidence, we want the right-hand side
of (20.21) to be at most 1/20. So we choose n so that
156:25 1
;
n 20
that is,
n 3; 125:
Section 20.6.2 describes how to get tighter estimates of the tails of binomial
distributions that lead to a bound on n that is about four times smaller than the
one above. But working through this example using only the variance illustrates
an approach to estimation that is applicable to arbitrary random variables, not just
binomial variables.
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876 Chapter 20 Deviation from the Mean

20.4.2 Pairwise Independent Sampling
The reasoning we used above to analyze voter polling and matching birthdays is
very similar. We summarize it in slightly more general form with a basic result
called the Pairwise Independent Sampling Theorem. In particular, we do not need
to restrict ourselves to sums of zero-one valued variables, or to variables with the
same distribution. For simplicity, we state the Theorem for pairwise independent
variables with possibly different distributions but with the same mean and variance.
Theorem 20.4.1 (Pairwise Independent Sampling). Let G1 ; : : : ; Gn be pairwise
independent variables with the same mean and deviation . Define
n
X
Sn WWD Gi : (20.22)
i D1

Then ˇ ˇ
ˇ Sn ˇ 1 2
Pr ˇˇ ˇ x
ˇ :
n n x
Proof. We observe first that the expectation of Sn =n is :
Pn
Sn i D1 Gi
Ex D Ex (def of Sn )
n n
Pn
ExŒGi
D i D1 (linearity of expectation)
Pn n

D i D1
n
n
D D :
n
The second important property of Sn =n is that its variance is the variance of Gi
divided by n:
2
Sn 1
Var D VarŒSn (Square Multiple Rule for Variance (20.9))
n n
" n #
1 X
D 2 Var Gi (def of Sn )
n
i D1
n
1 X
D VarŒGi (pairwise independent additivity)
n2
i D1
1 2
D 2 n 2 D : (20.23)
n n
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20.5. Confidence in an Estimation 877

This is enough to apply Chebyshev’s Theorem and conclude:

Var Sn =n
ˇ ˇ
ˇ Sn ˇ
Pr ˇˇ ˇ x
ˇ : (Chebyshev’s bound)
n x2
2 =n
D (by (20.23))
x2
1 2
D :
n x

The Pairwise Independent Sampling Theorem provides a quantitative general
statement about how the average of independent samples of a random variable ap-
proaches the mean. In particular, it proves what is known as the Law of Large
Numbers:4 by choosing a large enough sample size, we can get arbitrarily accurate
estimates of the mean with confidence arbitrarily close to 100%.

Corollary 20.4.2. [Weak Law of Large Numbers] Let G1 ; : : : ; Gn be pairwise in-
dependent variables with the same mean , and the same finite deviation, and let
Pn
Gi
Sn WWD i D1 :
n
Then for every > 0,
lim PrŒjSn j D 1:
n!1

20.5 Confidence in an Estimation
So Chebyshev’s Bound implies that sampling 3,125 voters will yield a fraction that,
95% of the time, is within 0.04 of the actual fraction of the voting population who
prefer Brown.
Notice that the actual size of the voting population was never considered because
it did not matter. People who have not studied probability theory often insist that
the population size should influence the sample size. But our analysis shows that
polling a little over 3000 people people is always sufficient, regardless of whether
there are ten thousand, or a million, or a billion voters. You should think about
an intuitive explanation that might persuade someone who thinks population size
matters.
4 This is the Weak Law of Large Numbers. As you might suppose, there is also a Strong Law, but
it’s outside the scope of 6.042.
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878 Chapter 20 Deviation from the Mean

Now suppose a pollster actually takes a sample of 3,125 random voters to esti-
mate the fraction of voters who prefer Brown, and the pollster finds that 1250 of
them prefer Brown. It’s tempting, but sloppy, to say that this means:

False Claim. With probability 0.95, the fraction p of voters who prefer Brown is
1250=3125 ˙ 0:04. Since 1250=3125 0:04 > 1=3, there is a 95% chance that
more than a third of the voters prefer Brown to all other candidates.

As already discussed in Section 18.9, what’s objectionable about this statement
is that it talks about the probability or “chance” that a real world fact is true, namely
that the actual fraction p of voters favoring Brown is more than 1/3. But p is what
it is, and it simply makes no sense to talk about the probability that it is something
else. For example, suppose p is actually 0.3; then it’s nonsense to ask about the
probability that it is within 0.04 of 1250/3125. It simply isn’t.
This example of voter preference is typical: we want to estimate a fixed, un-
known real-world quantity. But being unknown does not make this quantity a ran-
dom variable, so it makes no sense to talk about the probability that it has some
property.
A more careful summary of what we have accomplished goes this way:

We have described a probabilistic procedure for estimating the value
of the actual fraction p. The probability that our estimation procedure
will yield a value within 0.04 of p is 0.95.

This is a bit of a mouthful, so special phrasing closer to the sloppy language is
commonly used. The pollster would describe his conclusion by saying that

At the 95% confidence level, the fraction of voters who prefer Brown
is 1250=3125 ˙ 0:04.

So confidence levels refer to the results of estimation procedures for real-world
quantities. The phrase “confidence level” should be heard as a reminder that some
statistical procedure was used to obtain an estimate. To judge the credibility of the
estimate, it may be important to examine how well this procedure was performed.
More important, the confidence assertion above can be rephrased as

Either the fraction of voters who prefer Brown is 1250=3125 ˙ 0:04
or something unlikely (probability 1/20) happened.

If our experience led us to judge that having the preference fraction actually be in
this particular interval was unlikely, then this level of confidence would justifiably
remain unconvincing.
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20.6. Sums of Random Variables 879

20.6 Sums of Random Variables
If all you know about a random variable is its mean and variance, then Cheby-
shev’s Theorem is the best you can do when it comes to bounding the probabil-
ity that the random variable deviates from its mean. In some cases, however, we
know more—for example, that the random variable has a binomial distribution—
and then it is possible to prove much stronger bounds. Instead of polynomially
small bounds such as 1=c 2 , we can sometimes even obtain exponentially small
bounds such as 1=e c . As we will soon discover, this is the case whenever the ran-
dom variable T is the sum of n mutually independent random variables T1 , T2 , . . . ,
Tn where 0 Ti 1. A random variable with a binomial distribution is just one
of many examples of such a T .

20.6.1 A Motivating Example
Fussbook is a new social networking site oriented toward unpleasant people. Like
all major web services, Fussbook has a load balancing problem: it receives lots of
forum posts that computer servers have to process. If any server is assigned more
work than it can complete in a given interval, then it is overloaded and system
performance suffers. That would be bad, because Fussbook users are not a tolerant
bunch. So balancing the work load across mutliple servers is vital.
An early idea was to assign each server an alphabetic range of forum topics.
(“That oughta work!”, one programmer said.) But after the computer handling the
“privacy” and “preferred text editor” threads melted from overload, the drawback
of an ad hoc approach was clear: it’s easy to miss something that will mess up your
plan.
If the length of every task were known in advance, then finding a balanced distri-
bution would be a kind of “bin packing” problem. Such problems are hard to solve
exactly, but approximation algorithms can come close. Unfortunately, in this case
task lengths are not known in advance, which is typical of workload problems in
the real world.
So the load balancing problem seems sort of hopeless, because there is no data
available to guide decisions. So the programmers of Fussbook gave up and just
randomly assigned posts to computers. Imagine their surprise when the system
stayed up and hasn’t crashed yet!
As it turns out, random assignment not only balances load reasonably well, but
also permits provable performance guarantees. In general, a randomized approach
to a problem is worth considering when a deterministic solution is hard to compute
or requires unavailable information.
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880 Chapter 20 Deviation from the Mean

Specifically, Fussbook receives 24,000 forum posts in every 10-minute interval.
Each post is assigned to one of several servers for processing, and each server
works sequentially through its assigned tasks. It takes a server an average of 1=4
second to process a post. Some posts, such as pointless grammar critiques and snide
witticisms, are easier, but no post—not even the most protracted harangues—takes
more than one full second.
Measuring workload in seconds, this means a server is overloaded when it is
assigned more than 600 units of work in a given 600 second interval. Fussbook’s
average processing load of 24;000 1=4 D 6000 seconds per interval would keep
10 computers running at 100% capacity with perfect load balancing. Surely, more
than 10 servers are needed to cope with random fluctuations in task length and
imperfect load balance. But would 11 be enough? . . . or 15, 20, 100? We’ll answer
that question with a new mathematical tool.

20.6.2 The Chernoff Bound
The Chernoff5 bound is a hammer that you can use to nail a great many problems.
Roughly, the Chernoff bound says that certain random variables are very unlikely
to significantly exceed their expectation. For example, if the expected load on
a processor is just a bit below its capacity, then that processor is unlikely to be
overloaded, provided the conditions of the Chernoff bound are satisfied.
More precisely, the Chernoff Bound says that the sum of lots of little, indepen-
dent, random variables is unlikely to significantly exceed the mean of the sum. The
Markov and Chebyshev bounds lead to the same kind of conclusion but typically
provide much weaker bounds. In particular, the Markov and Chebyshev bounds are
polynomial, while the Chernoff bound is exponential.
Here is the theorem. The proof will come later in Section 20.6.6.

Theorem 20.6.1 (Chernoff Bound). Let T1 ; : : : Tn be mutually independent ran-
dom variables such that 0 Ti 1 for all i . Let T D T1 C C Tn . Then for all
c 1,
PrŒT c ExŒT e ˇ .c/ ExŒT (20.24)
where ˇ.c/ WWD c ln c c C 1.

The Chernoff bound applies only to distributions of sums of independent random
variables that take on values in the real interval Œ0; 1. The binomial distribution is
the most well-known distribution that fits these criteria, but many others are possi-
ble, because the Chernoff bound allows the variables in the sum to have differing,
5 Yes, this is the same Chernoff who figured out how to beat the state lottery—this guy knows a

thing or two.
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20.6. Sums of Random Variables 881

arbitrary, or even unknown distributions over the range Œ0; 1. Furthermore, there is
no direct dependence on either the number of random variables in the sum or their
expectations. In short, the Chernoff bound gives strong results for lots of problems
based on little information—no wonder it is widely used!

20.6.3 Chernoff Bound for Binomial Tails
The Chernoff bound can be applied in easy steps, though the details can be daunting
at first. Let’s walk through a simple example to get the hang of it: bounding the
probability that the number of heads that come up in 1000 independent tosses of a
coin exceeds the expectation by 20% or more. Let Ti be an indicator variable for
the event that the i th coin is heads. Then the total number of heads is

T D T1 C C T1000 :

The Chernoff bound requires that the random variables Ti be mutually independent
and take on values in the range Œ0; 1. Both conditions hold here. In this example
the Ti ’s only take the two values 0 and 1, since they’re indicators.
The goal is to bound the probability that the number of heads exceeds its expec-
tation by 20% or more; that is, to bound PrŒT c ExŒT where c = 1:2. To that
end, we compute ˇ.c/ as defined in the theorem:

ˇ.c/ D c ln.c/ c C 1 D 0:0187 : : : :

If we assume the coin is fair, then ExŒT D 500. Plugging these values into the
Chernoff bound gives:

Pr T 1:2 ExŒT e ˇ .c/: ExŒT

.0:0187::: /500
De < 0:0000834:

So the probability of getting 20% or more extra heads on 1000 coins is less than 1
in 10,000.
The bound rapidly becomes much smaller as the number of coins increases, be-
cause the expected number of heads appears in the exponent of the upper bound.
For example, the probability of getting at least 20% extra heads on a million coins
is at most
e .0:0187::: /500000 < e 9392 ;
which is an inconceivably small number.
Alternatively, the bound also becomes stronger for larger deviations. For exam-
ple, suppose we’re interested in the odds of getting 30% or more extra heads in
1000 tosses, rather than 20%. In that case, c D 1:3 instead of 1:2. Consequently,
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882 Chapter 20 Deviation from the Mean

the parameter ˇ.c/ rises from 0:0187 to about 0:0410, which may not seem sig-
nificant, but because ˇ.c/ appears in the exponent of the upper bound, the final
probability decreases from around 1 in 10,000 to about 1 in a billion!

20.6.4 Chernoff Bound for a Lottery Game
Pick-4 is a lottery game in which you pay $1 to pick a 4-digit number between 0000
and 9999. If your number comes up in a random drawing, then you win $5,000.
Your chance of winning is 1 in 10,000. If 10 million people play, then the expected
number of winners is 1000. When there are exactly 1000 winners, the lottery keeps
$5 million of the $10 million paid for tickets. The lottery operator’s nightmare is
that the number of winners is much greater—especially at the point where more
than 2000 win and the lottery must pay out more than it received. What is the
probability that will happen?
Let Ti be an indicator for the event that the i th player wins. Then T D T1 C C
Tn is the total number of winners. If we assume6 that the players’ picks and the
winning number are random, independent and uniform, then the indicators Ti are
independent, as required by the Chernoff bound.
Since 2000 winners would be twice the expected number, we choose c D 2,
compute ˇ.c/ D 0:386 : : : , and plug these values into the Chernoff bound:

PrŒT 2000 D Pr T 2 ExŒT
k ExŒT .0:386::: /1000
e De
386
<e :

So there is almost no chance that the lottery operator pays out more than it took in.
In fact, the number of winners won’t even be 10% higher than expected very often.
To prove that, let c D 1:1, compute ˇ.c/ D 0:00484 : : : , and plug in again:

Pr T 1:1 ExŒT e k ExŒT

.0:00484/1000
De < 0:01:

So the Pick-4 lottery may be exciting for the players, but the lottery operator has
little doubt as to the outcome!
6 As we noted in Chapter 19, human choices are often not uniform and they can be highly de-

pendent. For example, lots of people will pick an important date. The lottery folks should not get
too much comfort from the analysis that follows, unless they assign random 4-digit numbers to each
player.
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20.6. Sums of Random Variables 883

20.6.5 Randomized Load Balancing
Now let’s return to Fussbook and its load balancing problem. Specifically, we need
to determine a number m of servers that makes it very unlikely that any server is
overloaded by being assigned more than 600 seconds of work in a given interval.
To begin, let’s find the probability that the first server is overloaded. Letting T be
the number of seconds of work assigned to the first server, this means we want an
upper bound on PrŒT 600. Let Ti be the number of seconds that the first server
spends on the i th task: then Ti is zero if the task isPassigned to another machine,
and otherwise Ti is the length of the task. So T D niD1 Ti is the total number of
seconds of work assigned to the first server, where n D 24;000.
The Chernoff bound is applicable only if the Ti are mutually independent and
take on values in the range Œ0; 1. The first condition is satisfied if we assume that
assignment of a post to a server is independent of the time required to process the
post. The second condition is satisfied because we know that no post takes more
than 1 second to process; this is why we chose to measure work in seconds.
In all, there are 24,000 tasks, each with an expected length of 1/4 second. Since
tasks are assigned to the m servers at random, the expected load on the first server
is:
24;000 tasks 1=4 second per task
ExŒT D
m servers
D 6000=m seconds: (20.25)

So if there are fewer than 10 servers, then the expected load on the first server is
greater than its capacity, and we can expect it to be overloaded. If there are exactly
10 servers, then the server is expected to run for 6000=10 D 600 seconds, which is
100% of its capacity.
Now we can use the Chernoff bound based on the number of servers to bound
the probability that the first server is overloaded. We have from (20.25)

600 D c ExŒT where c WWD m=10;

so by the Chernoff bound
.c ln.c/ cC1/6000=m
PrŒT 600 D PrŒT c ExŒT e ;

The probability that some server is overloaded is at most m times the probability
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884 Chapter 20 Deviation from the Mean

that the first server is overloaded, by the Union Bound in Section 17.5.2. So
m
X
PrŒsome server is overloaded PrŒserver i is overloaded
i D1
D m PrŒthe first server is overloaded
.c ln.c/ cC1/6000=m
me ;
where c D m=10. Some values of this upper bound are tabulated below:
m D 11 W 0:784 : : :
m D 12 W 0:000999 : : :
m D 13 W 0:0000000760 : : : :
These values suggest that a system with m D 11 machines might suffer immediate
overload, m D 12 machines could fail in a few days, but m D 13 should be fine for
a century or two!

20.6.6 Proof of the Chernoff Bound
The proof of the Chernoff bound is somewhat involved. In fact, Chernoff himself
couldn’t come up with it: his friend, Herman Rubin, showed him the argument.
Thinking the bound not very significant, Chernoff did not credit Rubin in print. He
felt pretty bad when it became famous!7

Proof. (of Theorem 20.6.1)
For clarity, we’ll go through the proof “top down.” That is, we’ll use facts that
are proved immediately afterward.
The key step is to exponentiate both sides of the inequality T c ExŒT and
then apply the Markov bound:
PrŒT c ExŒT D PrŒc T c c ExŒT
ExŒc T
(Markov Bound)
c c ExŒT
e .c 1/ ExŒT
(Lemma 20.6.2 below)
c c ExŒT
e .c 1/ ExŒT .c ln.c/ cC1/ ExŒT
D De :
e c ln.c/ ExŒT

7 See “A Conversation with Herman Chernoff,” Statistical Science 1996, Vol. 11, No. 4, pp 335–
350.
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20.6. Sums of Random Variables 885

Algebra aside, there is a brilliant idea in this proof: in this context, exponenti-
ating somehow supercharges the Markov bound. This is not true in general! One
unfortunate side-effect of this supercharging is that we have to bound some nasty
expectations involving exponentials in order to complete the proof. This is done in
the two lemmas below, where variables take on values as in Theorem 20.6.1.

Lemma 20.6.2. h i
Ex c T e .c 1/ ExŒT
:

Proof.
h i h i
Ex c T D Ex c T1 CCTn (def of T )
h i
D Ex c T1 c Tn
h i
D Ex c T1 ExŒc Tn (independent product Cor 19.5.7)
e .c 1/ ExŒT1
e .c 1/ ExŒTn
(Lemma 20.6.3 below)
.c 1/.ExŒT1 CCExŒTn /
De
D e .c 1/ ExŒT1 CCTn
(linearity of ExŒ)
D e .c 1/ ExŒT
:

The third equality depends on the fact that functions of independent variables are
also independent (see Lemma 19.2.2).

Lemma 20.6.3.
ExŒc Ti e .c 1/ ExŒTi

Proof. All summations below range over values v taken by the random variable Ti ,
which are all required to be in the interval Œ0; 1.
X
ExŒc Ti D c v PrŒTi D v (def of ExŒ)
X
.1 C .c 1/v/ PrŒTi D v (convexity—see below)
X
D PrŒTi D v C .c 1/v PrŒTi D v
X X
D PrŒTi D v C .c 1/ v PrŒTi D v
D 1 C .c 1/ ExŒTi
.c 1/ ExŒTi
e (since 1 C z e z ):
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886 Chapter 20 Deviation from the Mean

The second step relies on the inequality

c v 1 C .c 1/v;

which holds for all v in Œ0; 1 and c 1. This follows from the general principle
that a convex function, namely c v , is less than the linear function 1 C .c 1/v
between their points of intersection, namely v D 0 and 1. This inequality is why
the variables Ti are restricted to the real interval Œ0; 1.

20.6.7 Comparing the Bounds
Suppose that we have a collection of mutually independent events A1 , A2 , . . . , An ,
and we want to know how many of the events are likely to occur.
Let Ti be the indicator random variable for Ai and define

pi D PrŒTi D 1 D Pr Ai

for 1 i n. Define
T D T1 C T2 C C Tn
to be the number of events that occur.
We know from Linearity of Expectation that

ExŒT D ExŒT1 C ExŒT2 C C ExŒTn
n
X
D pi :
i D1

This is true even if the events are not independent.
By Theorem 20.3.8, we also know that

VarŒT D VarŒT1 C VarŒT2 C C VarŒTn
n
X
D pi .1 pi /;
i D1

and thus that v
u n
uX
T D t pi .1 pi /:
i D1

This is true even if the events are only pairwise independent.
Markov’s Theorem tells us that for any c > 1,
1
PrŒT c ExŒT :
c
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20.6. Sums of Random Variables 887

Chebyshev’s Theorem gives us the stronger result that
1
PrŒjT ExŒT j cT :
c2
The Chernoff Bound gives us an even stronger result, namely, that for any c > 0,
.c ln.c/ cC1/ ExŒT
PrŒT ExŒT c ExŒT e :

In this case, the probability of exceeding the mean by c ExŒT decreases as an
exponentially small function of the deviation.
By considering the random variable n T , we can also use the Chernoff Bound
to prove that the probability that T is much lower than ExŒT is also exponentially
small.

20.6.8 Murphy’s Law
If the expectation of a random variable is much less than 1, then Markov’s Theorem
implies that there is only a small probability that the variable has a value of 1 or
more. On the other hand, a result that we call Murphy’s Law8 says that if a random
variable is an independent sum of 0–1-valued variables and has a large expectation,
then there is a huge probability of getting a value of at least 1.

Theorem 20.6.4 (Murphy’s Law). Let A1 , A2 , . . . , An be mutually independent
events. Let Ti be the indicator random variable for Ai and define

T WWD T1 C T2 C C Tn

to be the number of events that occur. Then
ExŒT
PrŒT D 0 e :
8 This is in reference and deference to the famous saying that “If something can go wrong, it

probably will.”
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888 Chapter 20 Deviation from the Mean

Proof.

PrŒT D 0 D PrŒA1 \ A2 \ : : : \ An (T D 0 iff no Ai occurs)
n
Y
D PrŒAi (independence of Ai )
i D1
Y n
D .1 PrŒAi /
i D1
n
Y
PrŒAi x
e (since 1 xe )
i D1
Pn
PrŒAi
De i D1
Pn
ExŒTi
De iD1 (since Ti is an indicator for Ai )
ExŒT
De (linearity of expectation)

For example, given any set of mutually independent events, if you expect 10 of
them to happen, then at least one of them will happen with probability at least 1
e 10 . The probability that none of them happen is at most e 10 < 1=22000.
So if there are a lot of independent things that can go wrong and their probabil-
ities sum to a number much greater than 1, then Theorem 20.6.4 proves that some
of them surely will go wrong.
This result can help to explain “coincidences,” “miracles,” and crazy events that
seem to have been very unlikely to happen. Such events do happen, in part, because
there are so many possible unlikely events that the sum of their probabilities is
greater than one. For example, someone does win the lottery.
In fact, if there are 100,000 random tickets in Pick-4, Theorem 20.6.4 says that
the probability that there is no winner is less than e 10 < 1=22000. More generally,
there are literally millions of one-in-a-million possible events and so some of them
will surely occur.

20.7 Really Great Expectations
Making independent tosses of a fair coin until some desired pattern comes up is a
simple process you should feel solidly in command of by now, right? So how about
a bet about the simplest such process—tossing until a head comes up? Ok, you’re
wary of betting with us, but how about this: we’ll let you set the odds.
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20.7. Really Great Expectations 889

20.7.1 Repeating Yourself
Here’s the bet: you make independent tosses of a fair coin until a head comes up.
Then you will repeat the process. If a second head comes up in the same or fewer
tosses than the first, you have to start over yet again. You keep starting over until
you finally toss a run of tails longer than your first one. The payment rules are that
you will pay me 1 cent each time you start over. When you win by finally getting a
run of tails longer than your first one, I will pay you some generous amount. Notice
by the way that you’re certain to win—whatever your initial run of tails happened
to be, a longer run will eventually occur again with probability 1!
For example, if your first tosses are TTTH, then you will keep tossing until you
get a run of 4 tails. So your winning flips might be

TTTHTHTTHHTTHTHTTTHTHHHTTTT:

In this run there are 10 heads, which means you had to start over 9 times. So you
would have paid me 9 cents by the time you finally won by tossing 4 tails. Now
you’ve won, and I’ll pay you generously —how does 25 cents sound? Maybe you’d
rather have $1? How about $1000?
Of course there’s a trap here. Let’s calculate your expected winnings.
Suppose your initial run of tails had length k. After that, each time a head comes
up, you have to start over and try to get k C1 tails in a row. If we regard your getting
k C 1 tails in a row as a “failed” try, and regard your having to start over because a
head came up too soon as a “successful” try, then the number of times you have to
start over is the number of tries till the first failure. So the expected number of tries
will be the mean time to failure, which is 2kC1 , because the probability of tossing
k C 1 tails in a row is 2 .kC1/ .
Let T be the length of your initial run of tails. So T D k means that your initial
tosses were Tk H. Let R be the number of times you repeat trying to beat your
original run of tails. The number of cents you expect to finish with is the number
of cents in my generous payment minus ExŒR. It’s now easy to calculate ExŒR by
conditioning on the value of T :
X X X
ExŒR D ExŒR j T D k PrŒT D k D 2kC1 2 .kC1/ D 1 D 1:
k2N k2N k2N

So you can expect to pay me an infinite number of cents before winning my
“generous” payment. No amount of generosity can make this bet fair! In fact this
particular example is a special case of an astonishingly general one: the expected
waiting time for any random variable to achieve a larger value is infinite.
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890 Chapter 20 Deviation from the Mean

Problems for Section 20.1
Practice Problems
Problem 20.1.
The vast majority of people have an above average number of fingers. Which of
the following statements explain why this is true? Explain your reasoning.

1. Most people have a super secret extra bonus finger of which they are un-
aware.

2. A pedantic minority don’t count their thumbs as fingers, while the majority
of people do.

3. Polydactyly is rarer than amputation.

4. When you add up the total number of fingers among the world’s population
and then divide by the size of the population, you get a number less than ten.

5. This follows from Markov’s Theorem, since no one has a negative number
of fingers.

6. Missing fingers are more common than extra ones.

Class Problems
Problem 20.2.
A herd of cows is stricken by an outbreak of cold cow disease. The disease lowers
a cow’s body temperature from normal levels, and a cow will die if its temperature
goes below 90 degrees F. The disease epidemic is so intense that it lowered the
average temperature of the herd to 85 degrees. Body temperatures as low as 70
degrees, but no lower, were actually found in the herd.
(a) Use Markov’s Bound 20.1.1 to prove that at most 3/4 of the cows could sur-
vive.

(b) Suppose there are 400 cows in the herd. Show that the bound from part (a)
is the best possible by giving an example set of temperatures for the cows so that
the average herd temperature is 85 and 3/4 of the cows will have a high enough
temperature to survive.

(c) Notice that the results of part (b) are purely arithmetic facts about averages,
not about probabilities. But you verified the claim in part (a) by applying Markov’s
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20.7. Really Great Expectations 891

bound on the deviation of a random variable. Justify this approach by regarding
the temperature T of a cow as a random variable. Carefully specify the probability
space on which T is defined: what are the sample points? what are their proba-
bilities? Explain the precise connection between properties of T and average herd
temperature that justifies the application of Markov’s Bound.

Homework Problems
Problem 20.3.
If R is a nonnegative random variable, then Markov’s Theorem gives an upper
bound on PrŒR x for any real number x > ExŒR. If b is a lower bound on R,
then Markov’s Theorem can also be applied to R b to obtain a possibly different
bound on PrŒR x.
(a) Show that if b > 0, applying Markov’s Theorem to R b gives a smaller
upper bound on PrŒR x than simply applying Markov’s Theorem directly to R.

(b) What value of b 0 in part (a) gives the best bound?

Exam Problems
Problem 20.4.
A herd of cows is stricken by an outbreak of hot cow disease. The disease raises
the normal body temperature of a cow, and a cow will die if its temperature goes
above 90 degrees. The disease epidemic is so intense that it raised the average
temperature of the herd to 120 degrees. Body temperatures as high as 140 degrees,
but no higher, were actually found in the herd.
(a) Use Markov’s Bound 20.1.1 to prove that at most 2/5 of the cows could have
survived.

(b) Notice that the conclusion of part (a) is a purely arithmetic facts about aver-
ages, not about probabilities. But you verified the claim of part (a) by applying
Markov’s bound on the deviation of a random variable. Justify this approach by
explaining how to define a random variable T for the temperature of a cow. Care-
fully specify the probability space on which T is defined: what are the outcomes?
what are their probabilities? Explain the precise connection between properties of
T , average herd temperature, and fractions of the herd with various temperatures,
that justify application of Markov’s Bound.
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892 Chapter 20 Deviation from the Mean

Problems for Section 20.2
Exam Problems
Problem 20.5.
There is a herd of cows whose average body temperature turns out to be 100 de-
grees. Our thermometer produces such sensitive readings that no two cows have
exactly the same body temperature. The herd is stricken by an outbreak of wacky
cow disease, which will eventually kill any cow whose body temperature differs
from the average by 10 degrees or more.
It turns out that the collection-variance of all the body temperatures is 20, where
the collection-variance CVar.A/ of set A of numbers is

.a /2
P
CVar.A/ WWD a2A ;
jAj

where is the average value of the numbers in A. (In other words, CVar.A/ is A’s
average square deviation from its mean.)
(a) Apply the Chebyshev bound to the temperature T of a random cow to show
that at most 20% of the cows will be killed by this disease outbreak.

The conclusion of part (a) about a certain fraction of the herd was derived by
bounding the deviation of a random variable. We can justify this approach by
explaining how to define a suitable probability space in which, the temperature
T of a cow is a random variable.
(b) Carefully specify the probability space on which T is defined: what are the
outcomes? what are their probabilities?

(d) Show that ExŒT equals the average temperature of the herd.

(e) Show that VarŒT equals the collection variance of the herd.
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20.7. Really Great Expectations 893

Problems for Section 20.3
Practice Problems
Problem 20.6.
Suppose 120 students take a final exam and the mean of their scores is 90. You
have no other information about the students and the exam, that is, you should not
assume that the highest possible score is 100. You may, however, assume that exam
scores are nonnegative.
(a) State the best possible upper bound on the number of students who scored at
least 180.

(b) Now suppose somebody tells you that the lowest score on the exam is 30.
Compute the new best possible upper bound on the number of students who scored
at least 180.

Problem 20.7.
Suppose you flip a fair coin 100 times. The coin flips are all mutually independent.
(a) What is the expected number of heads?

(b) What upper bound does Markov’s Theorem give for the probability that the
number of heads is at least 70?

(d) What upper bound does Chebyshev’s Theorem give for the probability that the
number of heads is either less than 30 or greater than 70?

Problem 20.8.
Albert has a gambling problem. He plays 240 hands of draw poker, 120 hands of
black jack, and 40 hands of stud poker per day. He wins a hand of draw poker with
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894 Chapter 20 Deviation from the Mean

probability 1/6, a hand of black jack with probability 1/2, and a hand of stud poker
with probability 1/5. Let W be the expected number of hands that Albert wins in a
day.
(a) What is ExŒW ?

(b) What would the Markov bound be on the probability that Albert will win at
least 216 hands on a given day?

(c) Assume the outcomes of the card games are pairwise independent. What is
VarŒW ? You may answer with a numerical expression that is not completely eval-
uated.

(d) What would the Chebyshev bound be on the probability that Albert will win
at least 216 hands on a given day? You may answer with a numerical expression
that includes the constant v D VarŒW .

Class Problems
Problem 20.9.
The hat-check staff has had a long day serving at a party, and at the end of the party
they simply return the n checked hats in a random way such that the probability
that any particular person gets their own hat back is 1=n.
Let Xi be the indicator variable for the i th person getting their own hat back. Let
Sn be the total number of people who get their own hat back.
(a) What is the expected number of people who get their own hat back?

(b) Write a simple formula for ExŒXi Xj for i ¤ j .

Hint: What is Pr Xj D 1 j Xi D 1 ?

(d) Show that ExŒ.Sn /2 D 2. Hint: .Xi /2 D Xi .

(e) What is the variance of Sn ?
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20.7. Really Great Expectations 895

(f) Show that there is at most a 1% chance that more than 10 people get their own
hat back.

Problem 20.10.
For any random variable R with mean and standard deviation the Chebyshev
bound says that for any real number x > 0,
2
PrŒjR j x :
x
Show that for any real number and real numbers x > 0, there is an R for
which the Chebyshev bound is tight, that is,
2
PrŒjR j x D : (20.26)
x
Hint: First assume D 0 and let R take only the values 0; x and x.

Problem 20.11.
A computer program crashes at the end of each hour of use with probability 1=p,
if it has not crashed already. Let H be the number of hours until the first crash.
(a) What is the Chebyshev bound on

PrŒjH .1=p/j > x=p

where x > 0?

(b) Conclude from part (a) that for a 2,
1 p
PrŒH > a=p
.a 1/2

Hint: Check that jH .1=p/j > .a 1/=p iff H > a=p.

Conclude that for any fixed p > 0, the probability that H > a=p is an asymptoti-
cally smaller function of a than the Chebyshev bound of part (b).
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896 Chapter 20 Deviation from the Mean

Problem 20.12.
Let R be a positive integer valued random variable.
(a) If ExŒR D 2, how large can VarŒR be?

(b) How large can ExŒ1=R be?

Problem 20.13.
A man has a set of n keys, one of which fits the door to his apartment. He tries the
keys randomly throwing away each ill-fitting key that he tries until he finds the key
that fits. That is, he chooses keys randomly from among those he has not yet tried.
This way he is sure to find the right key within n tries.
Let T be the number of times he tries keys until he finds the right key. Prob-
lem 19.25 shows that
nC1
ExŒT D :
2
Write a closed formula for VarŒT .

Homework Problems
Problem 20.14.
A man has a set of n keys, one of which fits the door to his apartment. He tries a
key at random, and if it does not fit the door, he simply puts it back; so he might try
the same ill-fitting key several times. He continues until he finds the one right key
that fits.
Let T be the number of times he tries keys until he finds the right key.
(a) Explain why

ExŒT D n and VarŒT D n.n 1/:

Let
fn .a/ WWD PrŒT an:
(b) Use the Chebyshev Bound to show that for any fixed n > 1,

1
fn .a/ D ‚ : (20.27)
a2
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20.7. Really Great Expectations 897

(c) Derive an upper bound for fn .a/ that for any fixed n > 1 is asymptoticaly
smaller than Chebyshev’s bound (20.27).
You may assume that n is large enough to use the approximation
1 cn

1
1 c
n e

Problem 20.15.
There is a fair coin and a biased coin that flips heads with probability 3=4. You are
given one of the coins, but you don’t know which. To determine which coin was
picked, your strategy will be to choose a number n and flip the picked coin n times.
If the number of heads flipped is closer to .3=4/n than to .1=2/n, you will guess
that the biased coin had been picked and otherwise you will guess that the fair coin
had been picked.
(a) Use the Chebyshev Bound to find a value n so that with probability 0.95 your
strategy makes the correct guess, no matter which coin was picked.

(b) Suppose you had access to a computer program that would generate, in the
form of a plot or table, the full binomial-.n; p/ probability density and cumulative
distribution functions. How would you find the minimum number of coin flips
needed to infer the identity of the chosen coin with probability 0.95? How would
you expect the number n determined this way to compare to the number obtained
in part(a)? (You do not need to determine the numerical value of this minimum n,
but we’d be interested to know if you did.)

(c) Now that we have determined the proper number n, we will assert that the
picked coin was the biased one whenever the number of Heads flipped is greater
than .5=8/n, and we will be right with probability 0.95. What, if anything, does
this imply about

Pr picked coin was biased j # Heads flipped .5=8/n ‹

Problem 20.16.
The expected absolute deviation of a real-valued random variable R with mean ,
is defined to be
ExŒ jR j :
Prove that the expected absolute deviation is always less than or equal to the stan-
dard deviation . (For simplicity, you may assume that R is defined on a finite
sample space.)
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898 Chapter 20 Deviation from the Mean

Hint: Suppose the sample space outcomes are !1 ; !2 ; : : : ; !n , and let
p
p WWD .p1 ; p2 ; : : : ; pn / where pi D PrŒ!i ;
p
r WWD .r1 ; r2 ; : : : ; rn / where ri D jR.!i / j PrŒ!i :

As usual, let v w WWD niD1 vi ui denote the dot product of n-vectors v; w, and let
P
p
jvj be the norm of v, namely, v v.
Then verify that

jpj D 1; jrj D ; and ExŒ jR j D r p:

Problem 20.17.
Prove the following “one-sided” version of the Chebyshev bound for deviation
above the mean:
Lemma (One-sided Chebyshev bound).
VarŒR
PrŒR ExŒR x :
x 2 C VarŒR
Hint: Let Sa WWD .R ExŒR C a/2 , for 0 a 2 R. So R ExŒR x
implies Sa .x C a/2 . Apply Markov’s bound to PrŒSa .x C a/2 . Choose a to
minimize this last bound.

Problem 20.18.
Prove the pairwise independent additivity of variance Theorem 20.3.8: If R1 ; R2 ; : : : ; Rn
are pairwise independent random variables, then

VarŒR1 C R2 C C Rn D VarŒR1 C VarŒR2 C C VarŒRn : (*)

Hint: Why is it OK to assume ExŒRi D 0?

Exam Problems
Problem 20.19.
You are playing a game where you get n turns. Each of your turns involves flipping
a coin a number of times. On the first turn, you have 1 flip, on the second turn you
have two flips, and so on until your nth turn when you flip the coin n times. All the
flips are mutually independent.
The coin you are using is biased to flip Heads with probability p. You win a turn
if you flip all Heads. Let W be the number of winning turns.
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20.7. Really Great Expectations 899

(a) Write a closed-form (no summations) expression for ExŒW .

(b) Write a closed-form expression for VarŒW .

Problem 20.20.
Let Kn be the complete graph with n vertices. Each of the edges of the graph
will be randomly assigned one of the colors red, green, or blue. The assignments
of colors to edges are mutually independent, and the probability of an edge being
assigned red is r, blue is b, and green is g (so r C b C g D 1).
A set of three vertices in the graph is called a triangle. A triangle is monochro-
matic if the three edges connecting the vertices are all the same color.
(a) Let m be the probability that any given triangle T is monochromatic. Write a
simple formula for m in terms of r; b; and g.

(b) Let IT be the indicator variable for whether T is monochromatic. Write simple
formulas in terms of m; r; b; and g for ExŒIT and VarŒIT .
Let T and U be distinct triangles.
(c) What is the probability that T and U are both monochromatic if they do not
share an edge?. . . if they do share an edge?
1
Now assume r D b D g D .
3
(d) Show that IT and IU are independent random variables.

(e) Let M be the number of monochromatic triangles. Write simple formulas in
terms of n and m for ExŒM and VarŒM .

(f) Let WWD ExŒM . Use Chebyshev’s Bound to prove that
p 1
PrŒjM j > log :
log

(g) Conclude that
p
lim PrŒjM j > log D 0
n!1
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900 Chapter 20 Deviation from the Mean

Problem 20.21.
You have a biased coin which flips Heads with probability p. You flip the coin n
times. The coin flips are all mutually independent. Let H be the number of Heads.
(a) Write a simple expression in terms of p and n for ExŒH , the expected number
of Heads.

(b) Write a simple expression in terms of p and n for VarŒH , the variance of the
number of Heads.

(c) Write a simple expression in terms of p for the upper bound that Markov’s
Theorem gives for the probability that the number of Heads is larger than the ex-
pected number by at least 1% of the number of flips, that is, by n=100.

(d) Show that the bound Chebyshev’s Theorem gives for the probability that H
differs from ExŒH by at least n=100 is

p.1 p/
1002 :
n

(e) The bound in part (d) implies that if you flip at least m times for a certain
number m, then there is a 95% chance that the proportion of Heads among these m
flips will be within 0.01 of p. Write a simple expression for m in terms of p.

Problem 20.22.
A classroom has sixteen desks in a 4 4 arrangement as shown below.
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20.7. Really Great Expectations 901

If two desks are next to each other, vertically or horizontally, they are called an
adjacent pair. So there are three horizontally adjacent pairs in each row, for a total
of twelve horizontally adjacent pairs. Likewise, there are twelve vertically adjacent
pairs. An adjacent pair D of desks is said to have a flirtation when there is a boy at
one desk and a girl at the other desk.
(a) Suppose boys and girls are assigned to desks in some unknown probabilistic
way. What is the Markov bound on the probability that the number of flirtations is
at least 33 1/3% more than expected?

Suppose that boys and girls are actually assigned to desks mutually indepen-
dently, with probability p of a desk being occupied by a boy, where 0 < p < 1.
(b) Express the expected number of flirtations in terms of p.
Hint: Let ID be the indicator variable for a flirtation at D.

Different pairs D and E of adjacent desks are said to overlap when they share
a desk. For example, the first and second pairs in each row overlap, and so do the
second and third pairs, but the first and third pairs do not overlap.
(c) Prove that if D and E overlap, and p D 1=2, then ID and IE are independent.

(d) When p D 1=2, what is the variance of the number of flirtations?
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902 Chapter 20 Deviation from the Mean

(e) What upper bound does Chebyshev’s Theorem give on the probability that the
number of heads is either less than 30 or greater than 70?

(f) Let D and E be pairs of adjacent desks that overlap. Prove that if p ¤ 1=2,
then FD and FE are not independent.

(g) Find four pairs of desks D1 ; D2 ; D3 ; D4 and explain why FD1 ; FD2 ; FD3 ; FD4
are not mutually independent (even if p D 1=2).

Problems for Section 20.5
Class Problems
Problem 20.23.
A recent Gallup poll found that 35% of the adult population of the United States
believes that the theory of evolution is “well-supported by the evidence.” Gallup
polled 1928 Americans selected uniformly and independently at random. Of these,
675 asserted belief in evolution, leading to Gallup’s estimate that the fraction of
Americans who believe in evolution is 675=1928 0:350. Gallup claims a margin
of error of 3 percentage points, that is, he claims to be confident that his estimate is
within 0.03 of the actual percentage.
(a) What is the largest variance an indicator variable can have?

(b) Use the Pairwise Independent Sampling Theorem to determine a confidence
level with which Gallup can make his claim.

(c) Gallup actually claims greater than 99% confidence in his estimate. How
might he have arrived at this conclusion? (Just explain what quantity he could
calculate; you do not need to carry out a calculation.)

(d) Accepting the accuracy of all of Gallup’s polling data and calculations, can
you conclude that there is a high probability that the percentage of adult Americans
who believe in evolution is 35 ˙ 3 percent?

Problem 20.24.
Let B1 ; B2 ; : : : ; Bn be mutually independent random variables with a uniform dis-
tribution on the integer interval Œ1::d . Let Ei;j be the indicator variable for the
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20.7. Really Great Expectations 903

event ŒBi D Bj .
Let M equal the number of events ŒBi D Bj that are true, where 1 i < j
n. So X
M D Ei;j :
1i <j n
It was observed in Section 17.4 (and proved in Problem 19.2) that PrŒBi D
Bj D 1=d for i ¤ j and that the random variables Ei;j , where 1 i < j n,
are pairwise independent.
(a) What are ExŒEi;j and VarŒEi;j for i ¤ j ?

(b) What are ExŒM and VarŒM ?

(c) In a 6.01 class of 500 students, the youngest student was born 15 years ago
and the oldest 35 years ago. Show that more than half the time, there will be will be
between 12 and 23 pairs of students who have the same birth date. (For simplicity,
assume that the distribution of birthdays is uniform over the 7305 days in the two
decade interval from 35 years ago to 15 years ago.)
Hint: Let D be the number of pairs of students in the class who have the same birth
date. Note that jD ExŒDj < 6 IFF D 2 Œ12::23.

Problem 20.25.
A defendent in traffic court is trying to beat a speeding ticket on the grounds that—
since virtually everybody speeds on the turnpike—the police have unconstitutional
discretion in giving tickets to anyone they choose. (By the way, we don’t recom-
mend this defense :-).)
To support his argument, the defendent arranged to get a random sample of trips
by 3,125 cars on the turnpike and found that 94% of them broke the speed limit
at some point during their trip. He says that as a consequence of sampling theory
(in particular, the Pairwise Independent Sampling Theorem), the court can be 95%
confident that the actual percentage of all cars that were speeding is 94 ˙ 4%.
The judge observes that the actual number of car trips on the turnpike was never
considered in making this estimate. He is skeptical that, whether there were a
thousand, a million, or 100,000,000 car trips on the turnpike, sampling only 3,125
is sufficient to be so confident.
Suppose you were were the defendent. How would you explain to the judge
why the number of randomly selected cars that have to be checked for speeding
does not depend on the number of recorded trips? Remember that judges are not
trained to understand formulas, so you have to provide an intuitive, nonquantitative
explanation.
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904 Chapter 20 Deviation from the Mean

Problem 20.26.
The proof of the Pairwise Independent Sampling Theorem 20.4.1 was given for
a sequence R1 ; R2 ; : : : of pairwise independent random variables with the same
mean and variance.
The theorem generalizes straighforwardly to sequences of pairwise independent
random variables, possibly with different distributions, as long as all their variances
are bounded by some constant.
Theorem (Generalized Pairwise Independent Sampling). Let X1 ; X2 ; : : : be a se-
quence of pairwise independent random variables such that VarŒXi b for some
b 0 and all i 1. Let
X1 C X2 C C Xn
An WWD ;
n
n WWD ExŒAn :

Then for every > 0,
b 1
PrŒjAn :
n j (20.28)
2 n
(a) Prove the Generalized Pairwise Independent Sampling Theorem.

(b) Conclude that the following holds:
Corollary (Generalized Weak Law of Large Numbers). For every > 0,

lim PrŒjAn n j D 1:
n!1

Exam Problems
Problem 20.27.
You work for the president and you want to estimate the fraction p of voters in the
entire nation that will prefer him in the upcoming elections. You do this by random
sampling. Specifically, you select a random voter and ask them who they are going
to vote for. You do this n times, with each voter selected with uniform probability
and independently of other selections. Finally, you use the fraction P of voters
who said they will vote for the President as an estimate for p.
(a) Our theorems about sampling and distributions allow us to calculate how con-
fident we can be that the random variable P takes a value near the constant p. This
calculation uses some facts about voters and the way they are chosen. Indicate the
true facts among the following:

1. Given a particular voter, the probability of that voter preferring the President
is p.
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20.7. Really Great Expectations 905

2. The probability that some voter is chosen more than once in the random sam-
ple goes to one as n increases.
3. The probability that some voter is chosen more than once in the random sam-
ple goes to zero as the population of voters grows.
4. All voters are equally likely to be selected as the third in the random sample
of n voters (assuming n 3).
5. The probability that the second voter in the random sample will favor the
President, given that the first voter prefers the President, is greater than p.
6. The probability that the second voter in the random sample will favor the
President, given that the second voter is from the same state as the first, may
not equal p.

(b) Suppose that according to your calculations, the following is true about your
polling:
PrŒjP pj 0:04 0:95:

You do the asking, you count how many said they will vote for the President, you
divide by n, and find the fraction is 0.53. Among the following, Indicate the legiti-
mate things you might say in a call to the President:

1. Mr. President, p D 0:53!
2. Mr. President, with probability at least 95 percent, p is within 0.04 of 0.53.
3. Mr. President, either p is within 0.04 of 0.53 or something very strange (5-
in-100) has happened.
4. Mr. President, we can be 95% confident that p is within 0.04 of 0.53.

Problem 20.28.
Yesterday, the programmers at a local company wrote a large program. To estimate
the fraction b of lines of code in this program that are buggy, the QA team will
take a small sample of lines chosen randomly and independently (so it is possible,
though unlikely, that the same line of code might be chosen more than once). For
each line chosen, they can run tests that determine whether that line of code is
buggy, after which they will use the fraction of buggy lines in their sample as their
estimate of the fraction b.
The company statistician can use estimates of a binomial distribution to calculate
a value s for a number of lines of code to sample which ensures that with 97%
confidence, the fraction of buggy lines in the sample will be within 0.006 of the
actual fraction b of buggy lines in the program.
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906 Chapter 20 Deviation from the Mean

Mathematically, the program is an actual outcome that already happened. The
random sample is a random variable defined by the process for randomly choosing
s lines from the program. The justification for the statistician’s confidence depends
on some properties of the program and how the random sample of s lines of code
from the program are chosen. These properties are described in some of the state-
ments below. Indicate which of these statements are true, and explain your answers.

1. The probability that the ninth line of code in the program is buggy is b.

2. The probability that the ninth line of code chosen for the random sample is
defective is b.

3. All lines of code in the program are equally likely to be the third line chosen
in the random sample.

4. Given that the first line chosen for the random sample is buggy, the probabil-
ity that the second line chosen will also be buggy is greater than b.

5. Given that the last line in the program is buggy, the probability that the next-
to-last line in the program will also be buggy is greater than b.

6. The expectation of the indicator variable for the last line in the random sam-
ple being buggy is b.

7. Given that the first two lines of code selected in the random sample are the
same kind of statement—they might both be assignment statements, or both
be conditional statements, or both loop statements,. . . —the probability that
the first line is buggy may be greater than b.

8. There is zero probability that all the lines in the random sample will be dif-
ferent.

Problem 20.29.
Let G1 ; G2 ; G3 ; : : : ; be an infinite sequence of pairwise independent random vari-
ables with the same expectation and the same finite variance. Let
ˇ Pn ˇ
ˇ i D1 Gi ˇ
f .n; / WWD Pr ˇ ˇ ˇ :
ˇ
n
The Weak Law of Large Numbers can be expressed as a logical formula of the
form:
8 > 0 Q1 Q2 : : : Œf .n; / 1 ı
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20.7. Really Great Expectations 907

where Q1 Q2 : : : is a sequence of quantifiers from among:
8n 9n 8n0 9n0 8n n0 9n n0
8ı > 0 9ı > 0 8ı 0 9ı 0
Here the n and n0 range over nonnegative integers, and ı and range over real
numbers.
Write out the proper sequence Q1 Q2 : : :

Problems for Section 20.6
Practice Problems
Problem 20.30.
A gambler plays 120 hands of draw poker, 60 hands of black jack, and 20 hands of
stud poker per day. He wins a hand of draw poker with probability 1/6, a hand of
black jack with probability 1/2, and a hand of stud poker with probability 1/5.
(a) What is the expected number of hands the gambler wins in a day?

(b) What would the Markov bound be on the probability that the gambler will win
at least 108 hands on a given day?

(c) Assume the outcomes of the card games are pairwise, but possibly not mutu-
ally, independent. What is the variance in the number of hands won per day? You
may answer with a numerical expression that is not completely evaluated.

(d) What would the Chebyshev bound be on the probability that the gambler will
win at least 108 hands on a given day? You may answer with a numerical expres-
sion that is not completely evaluated.

(e) Assuming outcomes of the card games are mutually independent, show that
the probability that the gambler will win at least 108 hands on a given day is much
smaller than the bound in part (d). Hint: e 1 2 ln 2 0:7
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908 Chapter 20 Deviation from the Mean

Class Problems
Problem 20.31.
We want to store 2 billion records into a hash table that has 1 billion slots. Assum-
ing the records are randomly and independently chosen with uniform probability
of being assigned to each slot, two records are expected to be stored in each slot.
Of course under a random assignment, some slots may be assigned more than two
records.
(a) Show that the probability that a given slot gets assigned more than 23 records
is less than e 36 .
Hint: Use Chernoff’s Bound, Theorem 20.6.1,. Note that ˇ.12/ > 18, where
ˇ.c/ WWD c ln c c C 1.

(b) Show that the probability that there is a slot that gets assigned more than 23
records is less than e 15 , which is less than 1=3; 000; 000. Hint: 109 < e 21 ; use
part (a).

Problem 20.32.
Sometimes I forget a few items when I leave the house in the morning. For example,
here are probabilities that I forget various pieces of footwear:

left sock 0:2
right sock 0:1
left shoe 0:1
right shoe 0:3

(a) Let X be the number of these that I forget. What is ExŒX ?

(b) Give a tight upper bound on the probability that I forget one or more items
when no independence assumption is made about forgetting different items.

(d) Now suppose that I forget each item of footwear independently. Use the
Chebyshev Bound to derive an upper bound on the probability that I forget two
or more items.

(e) Use Murphy’s Law, Theorem 20.6.4, to derive a lower bound on the probabil-
ity that I forget one or more items.
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20.7. Really Great Expectations 909

(f) I’m supposed to remember many other items, of course: clothing, watch, back-
pack, notebook, pencil, kleenex, ID, keys, etc. Let X be the total number of items
I remember. Suppose I remember items mutually independently and ExŒX D 36.
Use Chernoff’s Bound to give an upper bound on the probability that I remember
48 or more items.

(g) Give an upper bound on the probability that I remember 108 or more items.

Problem 20.33.
Reasoning based on the Chernoff bound goes a long way in explaining the recent
subprime mortgage collapse. A bit of standard vocabulary about the mortgage
market is needed:

A loan is money lent to a borrower. If the borrower does not pay on the
loan, the loan is said to be in default, and collateral is seized. In the case of
mortgage loans, the borrower’s home is used as collateral.

A bond is a collection of loans, packaged into one entity. A bond can be
divided into tranches, in some ordering, which tell us how to assign losses
from defaults. Suppose a bond contains 1000 loans, and is divided into 10
tranches of 100 bonds each. Then, all the defaults must fill up the lowest
tranche before the affect others. For example, suppose 150 defaults hap-
pened. Then, the first 100 defaults would occur in tranche 1, and the next 50
defaults would happen in tranche 2.

The lowest tranche of a bond is called the mezzanine tranche.

We can make a “super bond” of tranches called a collateralized debt obli-
gation (CDO) by collecting mezzanine tranches from different bonds. This
super bond can then be itself separated into tranches, which are again ordered
to indicate how to assign losses.

(a) Suppose that 1000 loans make up a bond, and the fail rate is 5% in a year.
Assuming mutual independence, give an upper bound for the probability that there
are one or more failures in the second-worst tranche. What is the probability that
there are failures in the best tranche?

(b) Now, do not assume that the loans are independent. Give an upper bound for
the probability that there are one or more failures in the second tranche. What is an
upper bound for the probability that the entire bond defaults? Show that it is a tight
bound. Hint: Use Markov’s theorem.
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910 Chapter 20 Deviation from the Mean

(c) Given this setup (and assuming mutual independence between the loans), what
is the expected failure rate in the mezzanine tranche?

(d) We take the mezzanine tranches from 100 bonds and create a CDO. What is
the expected number of underlying failures to hit the CDO?

(e) We divide this CDO into 10 tranches of 1000 bonds each. Assuming mutual
independence, give an upper bound on the probability of one or more failures in the
best tranche. The third tranche?

(f) Repeat the previous question without the assumption of mutual independence.

Homework Problems
Problem 20.34.
We have two coins: one is a fair coin, but the other produces heads with probability
3=4. One of the two coins is picked, and this coin is tossed n times. Use the
Chernoff Bound to determine the smallest n which allows determination of which
coin was picked with 95% confidence.

Problem 20.35.
An infinite version of Murphy’s Law is that if an infinite number of mutually inde-
pendent events are expected to happen, then the probability that only finitely many
happen is 0. This is known as the first Borel-Cantelli Lemma.
(a) Let A0 ; A1 ; : : : be any infinite sequence of mutually independent events such
that X
PrŒAn D 1: (20.29)
n2N
Prove that PrŒno An occurs D 0.
Hint: Bk the event that no An with n k occurs. So the event that no An occurs is
\
B WWD Bk :
k2N

Apply Murphy’s Law, Theorem 20.6.4, to Bk .

(b) Conclude that PrŒonly finitely many An ’s occur D 0.
Hint: Let Ck be the event that no An with n k occurs. So the event that only
finitely many An ’s occur is [
C WWD Ck :
k2N
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20.7. Really Great Expectations 911

Apply part (a) to Ck .

Problems for Section 20.7
Practice Problems
Problem 20.36.
Let R be a positive integer valued random variable such that
1
PDFR .n/ D ;
cn3
where
1
X 1
c WWD :
n3
nD1

(a) Prove that ExŒR is finite.

(b) Prove that VarŒR is infinite.
A joking way to phrase the point of this example is “the square root of infinity may
be finite.”
p Namely, let T WWD R2 ; then part (b) implies that ExŒT D 1 while
ExŒ T < 1 by (a).

Class Problems
Problem 20.37.
You have a biased coin with nonzero probability p < 1 of tossing a Head. You
toss until a Head comes up. Then, similar to the example in Section 20.7, you
keep tossing until you get another Head preceded by a run of consecutive Tails
whose length is within 10 of your original run. That is, if you began by tossing k
tails followed by a Head, then you continue tossing until you get a run of at least
maxfk 10; 0g consecutive Tails.
(a) Let H be the number of Heads that you toss until you get the required run of
Tails. Prove that the expected value of H is infinite.

(b) Let r < 1 be a positive real number. Instead of waiting for a run of Tails of
length k 10 when your original run was length k, just wait for a run of length at
least rk. Show that in this case, the expected number of Heads is finite.
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912 Chapter 20 Deviation from the Mean

Exam Problems
Problem 20.38.
You have a random process for generating a positive integer K. The behavior of
your process each time you use it is (mutually) independent of all its other uses. You
use your process to generate an integer, and then use your procedure repeatedly
until you generate an integer as big as your first one. Let R be the number of
additional integers you have to generate.
(a) State and briefly explain a simple closed formula for ExŒR j K D k in terms
of PrŒK k.

Suppose PrŒK D k D ‚.k 4 /.

(b) Show that PrŒK k D ‚.k 3 /.

21 Random Walks
Random Walks are used to model situations in which an object moves in a se-
quence of steps in randomly chosen directions. For example, physicists use three-
dimensional random walks to model Brownian motion and gas diffusion. In this
chapter we’ll examine two examples of random walks. First, we’ll model gambling
as a simple 1-dimensional random walk—a walk along a straight line. Then we’ll
explain how the Google search engine used random walks through the graph of
world-wide web links to determine the relative importance of websites.

21.1 Gambler’s Ruin
Suppose a gambler starts with an initial stake of n dollars and makes a sequence of
$1 bets. If he wins an individual bet, he gets his money back plus another $1. If he
loses the bet, he loses the $1.
We can model this scenario as a random walk between integer points on the real
line. The position on the line at any time corresponds to the gambler’s cash-on-
hand, or capital. Walking one step to the right corresponds to winning a $1 bet
and thereby increasing his capital by $1. Similarly, walking one step to the left
corresponds to losing a $1 bet.
The gambler plays until either he runs out of money or increases his capital to a
target amount of T dollars. The amount T n is defined to be his intended profit.
If he reaches his target, he will have won his intended profit and is called an
overall winner. If his capital reaches zero before reaching his target, he will have
lost n dollars; this is called going broke or being ruined. We’ll assume that the
gambler has the same probability p of winning each individual $1 bet, and that the
bets are mutually independent. We’d like to find the probability that the gambler
wins.
The gambler’s situation as he proceeds with his $1 bets is illustrated in Fig-
ure 21.1. The random walk has boundaries at 0 and T . If the random walk ever
reaches either of these boundary values, then it terminates.
In an unbiased game, the individual bets are fair: the gambler is equally likely
to win or lose each bet—that is, p D 1=2. The gambler is more likely to win if
p > 1=2 and less likely to win if p < 1=2; these random walks are called biased.
We want to determine the probability that the walk terminates at boundary T —the
probability that the gambler wins. We’ll do this in Section 21.1.1. But before we
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914 Chapter 21 Random Walks

T=n+m

gambler’s bet outcomes:
capital WLLWLWWLLL
n

time
Figure 21.1 A graph of the gambler’s capital versus time for one possible se-
quence of bet outcomes. At each time step, the graph goes up with probabil-
ity p and down with probability 1 p. The gambler continues betting until the
graph reaches either 0 or T . If he starts with $n, his intended profit is $m where
T D n C m.

derive the probability, let’s examine what it turns out to be.
Let’s begin by supposing the gambler plays an unbiased game starting with $100
and will play until he goes broke or reaches a target of 200 dollars. Since he starts
equidistant from his target and bankruptcy in this case, it’s clear by symmetry that
his probability of winning is 1/2.
We’ll show below that starting with n dollars and aiming for a target of T n
dollars, the probability the gambler reaches his target before going broke is n=T .
For example, suppose he wants to win the same $100, but instead starts out with
$500. Now his chances are pretty good: the probability of his making the 100
dollars is 5=6. And if he started with one million dollars still aiming to win $100
dollars he almost certain to win: the probability is 1M=.1M C 100/ > :9999.
So in the unbiased game, the larger the initial stake relative to the target, the
higher the probability the gambler will win, which makes some intuitive sense. But
note that although the gambler now wins nearly all the time, when he loses, he
loses big. Bankruptcy costs him $1M, while when he wins, he wins only $100.
The gambler’s average win remains zero dollars, which is what you’d expect when
making fair bets.
Another useful way to describe this scenario is as a game between two players.
Say Albert starts with $500, and Eric starts with $100. They flip a fair coin, and
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21.1. Gambler’s Ruin 915

every time a Head appears, Albert wins $1 from Eric, and vice versa for Tails.
They play this game until one person goes bankrupt. This problem is identical to
the Gambler’s Ruin problem with n D 500 and T D 100 C 500 D 600. The
probability of Albert winning is 500=600 D 5=6.
Now suppose instead that the gambler chooses to play roulette in an American
casino, always betting $1 on red. Because the casino puts two green numbers on its
roulette wheels, the probability of winning a single bet is a little less than 1/2. The
casino has an advantage, but the bets are close to fair, and you might expect that
starting with $500, the gambler has a reasonable chance of winning $100—the 5/6
probability of winning in the unbiased game surely gets reduced, but perhaps not
too drastically.
This mistaken intuition is how casinos stay in business. In fact, the gambler’s
odds of winning $100 by making $1 bets against the “slightly” unfair roulette wheel
are less than 1 in 37,000. If that’s surprising to you, it only gets weirder from here:
1 in 37,000 is in fact an upper bound on the gambler’s chance of winning regardless
of his starting stake. Whether he starts with $5000 or $5 billion, he still has almost
no chance of winning!

21.1.1 The Probability of Avoiding Ruin
We will determine the probability that the gambler wins using an idea of Pascal’s
dating back to the beginnings probability theory in the mid-seventeenth century.
Pascal viewed the walk as a two-player game between Albert and Eric as de-
scribed above. Albert starts with a stack of n chips and Eric starts with a stack of
m D T n chips. At each bet, Albert wins Eric’s top chip with probability p and
loses his top chip to Eric with probability q WWD 1 p. They play this game until
one person goes bankrupt.
Pascal’s ingenious idea was to alter the worth of the chips to make the game
fair regardless of p. Specifically, Pascal assigned Albert’s bottom chip a worth of
r WWDq=p and then assigned successive chips up his stack worths equal to r 2 ; r 3 ; : : :
up to his top chip with worth r n . Eric’s top chip gets assigned worth r nC1 , and the
successive chips down his stack are worth r nC2 ; r nC3 ; : : : down to his bottom chip
worth r nCm .
The expected payoff of Albert’s first bet is worth

nC1 n n q
r p r q D r p r n q D 0:
p

so this assignment makes the first bet a fair one in terms of worth. Moreover,
whether Albert wins or loses the bet, the successive chip worths counting up Al-
bert’s stack and then down Eric’s remain r; r 2 ; : : : ; r n ; : : : ; r nCm , ensuring by the
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916 Chapter 21 Random Walks

same reasoning that every bet has fair worth. So, Albert’s expected worth at the
end of the game is the sum of the expectations of the worth of each bet, which is
0.1
When Albert wins all of Eric’s chips his total gain is worth
nCm
X
ri ;
i DnC1
Pn i.
and when he loses all his chips to Eric, his total loss is worth i D1 r Letting wn
be Albert’s probability of winning, we now have
nCm
X n
X
0 D ExŒworth of Albert’s payoff D wn ri .1 wn / ri :
i DnC1 i D1

In the truly fair game when r D 1, we have 0 D mwn n.1 wn /, so wn D
n=.n C m/, as claimed above.
In the biased game with r ¤ 1, we have

r nCm r n rn 1
0Dr wn r .1 wn /:
r 1 r 1
Solving for wn gives
rn 1 rn 1
wn D D (21.1)
r nCm 1 rT 1
We have now proved

Theorem 21.1.1. In the Gambler’s Ruin game with initial capital n, target T , and
probability p of winning each individual bet,
n 1
8
ˆ
ˆ for p D ;
<T
ˆ 2
PrŒthe gambler wins D (21.2)
ˆ
ˆ r n 1 1
for p ¤ ;
ˆ
:
rT 1 2
where r WWD q=p.
1 Herewe’re legitimately appealing to infinite linearity, since the payoff amounts remain bounded
independent of the number of bets.
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21.1. Gambler’s Ruin 917

21.1.2 A Recurrence for the Probability of Winning
Fortunately, you don’t need to be as ingenuious Pascal in order to handle Gambler’s
Ruin, because linear recurrences offer a methodical approach to the basic problems.
The probability that the gambler wins is a function of his initial capital n his
target T n and the probability p that the wins an individual one dollar bet. For
fixed p and T , let wn be the gambler’s probability of winning when his initial
capital is n dollars. For example, w0 is the probability that the gambler will win
given that he starts off broke and wT is the probability he will win if he starts off
with his target amount, so clearly

w0 D 0; (21.3)
wT D 1: (21.4)

Otherwise, the gambler starts with n dollars, where 0 < n < T . Now suppose
the gambler wins his first bet. In this case, he is left with n C 1 dollars and becomes
a winner with probability wnC1 . On the other hand, if he loses the first bet, he is
left with n 1 dollars and becomes a winner with probability wn 1 . By the Total
Probability Rule, he wins with probability wn D pwnC1 C qwn 1 . Solving for
wnC1 we have
wn
wnC1 D rwn 1 (21.5)
p
where r is q=p as in Section 21.1.1.
This recurrence holds only for n C 1 T , but there’s no harm in using (21.5) to
define wnC1 for all n C 1 > 1. Now, letting

W .x/ WWD w0 C w1 x C w2 x 2 C

be the generating function for the wn , we derive from (21.5) and (21.3) using our
generating function methods that
w1 x
W .x/ D : (21.6)
rx 2 x=p C 1
But it’s easy to check that the denominator factors:
x
rx 2 C 1 D .1 x/.1 rx/:
p

Now if p ¤ q, then using partial fractions we conclude that
A B
W .x/ D C ; (21.7)
1 x 1 rx
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918 Chapter 21 Random Walks

for some constants A; B. To solve for A; B, note that by (21.6) and (21.7),

w1 x D A.1 rx/ C B.1 x/;

so letting x D 1, we get A D w1 =.1 r/, and letting x D 1=r, we get B D
w1 =.r 1/. Therefore,

w1 1 1
W .x/ D ;
r 1 1 rx 1 x

which implies
rn 1
wn D w1 : (21.8)
r 1
Finally, we can use (21.8) to solve for w1 by letting n D T to get
r 1
w1 D :
rT 1
Plugging this value of w1 into (21.8), we arrive at the solution:

rn 1
wn D ;
rT 1
matching Pascal’s result (21.1).
In the unbiased case where p D q, we get from (21.6) that
w1 x
W .x/ D ;
.1 x/2
and again can use partial fractions to match Pascal’s result (21.2).

21.1.3 A simpler expression for the biased case
The expression (21.1) for the probability that the Gambler wins in the biased game
is a little hard to interpret. There is a simpler upper bound which is nearly tight
when the gambler’s starting capital is large and the game is biased against the
gambler. Then r > 1, both the numerator and denominator in (21.1) are positive,
and the numerator is smaller. This implies that
T n
rn 1
wn < T D
r r

and gives:
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21.1. Gambler’s Ruin 919

Corollary 21.1.2. In the Gambler’s Ruin game with initial capital n, target T , and
probability p < 1=2 of winning each individual bet,
T n
1
PrŒthe gambler wins < (21.9)
r

where r WWD q=p > 1.

So the gambler gains his intended profit before going broke with probability at
most 1=r raised to the intended profit power. Notice that this upper bound does
not depend on the gambler’s starting capital, but only on his intended profit. This
has the amazing consequence we announced above: no matter how much money he
starts with, if he makes $1 bets on red in roulette aiming to win $100, the probability
that he wins is less than

18=38 100
100
9 1
D < :
20=38 10 37; 648

The bound (21.9) decreases exponentially as the intended profit increases. So,
for example, doubling his intended profit will square his probability of winning.
In this case, the probability that the gambler’s stake goes up 200 dollars before he
goes broke playing roulette is at most
2
200 100 2 1
.9=10/ D ..9=10/ / < ;
37; 648

which is about 1 in 1.4 billion.

Intuition
Why is the gambler so unlikely to make money when the game is only slightly
biased against him? To answer this intuitively, we can identify two forces at work
on the gambler’s wallet. First, the gambler’s capital has random upward and down-
ward swings from runs of good and bad luck. Second, the gambler’s capital will
have a steady, downward drift, because the negative bias means an average loss of
a few cents on each $1 bet. The situation is shown in Figure 21.2.
Our intuition is that if the gambler starts with, say, a billion dollars, then he is
sure to play for a very long time, so at some point there should be a lucky, upward
swing that puts him $100 ahead. But his capital is steadily drifting downward. If
the gambler does not have a lucky, upward swing early on, then he is doomed.
After his capital drifts downward by tens and then hundreds of dollars, the size of
the upward swing the gambler needs to win grows larger and larger. And as the
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920 Chapter 21 Random Walks

w
upward
n swing
(too late)

gambler’s downward
wealth drift

0
time

Figure 21.2 In a biased random walk, the downward drift usually dominates
swings of good luck.

size of the required swing grows, the odds that it occurs decrease exponentially. As
a rule of thumb, drift dominates swings in the long term.
We can quantify these drifts and swings. After k rounds for k min.m; n/, the
number of wins by our player has a binomial distribution with parameters p < 1=2
and k. His expected win on any single bet is p q D 2p 1 dollars, so his expected
capital is n k.1 2p/. Now to be a winner, his actual number of wins must exceed
the expected number by m C k.1 2p/. But from p the formula (20.15), the binomial
distribution has a standard deviation of only kp.1 p/ . So for the gambler to
win, he needs his number of wins to deviate by
m C k.1 2p/ p
p D ‚. k/
kp.1 2p/
times its standard deviation. In our study of binomial tails, we saw that this was
extremely unlikely.
In a fair game, there is no drift; swings are the only effect. In the absence of
downward drift, our earlier intuition is correct. If the gambler starts with a trillion
dollars then almost certainly there will eventually be a lucky swing that puts him
$100 ahead.

21.1.4 How Long a Walk?
Now that we know the probability wn that the gambler is a winner in both fair and
unfair games, we consider how many bets he needs on average to either win or go
broke. A linear recurrence approach works here as well.
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21.1. Gambler’s Ruin 921

For fixed p and T , let en be the expected number of bets until the game ends
when the gambler’s initial capital is n dollars. Since the game is over in zero steps
if n D 0 or T , the boundary conditions this time are e0 D eT D 0.
Otherwise, the gambler starts with n dollars, where 0 < n < T . Now by the
conditional expectation rule, the expected number of steps can be broken down
into the expected number of steps given the outcome of the first bet weighted by
the probability of that outcome. But after the gambler wins the first bet, his capital
is n C 1, so he can expect to make another enC1 bets. That is,
ExŒ#bets starting with $n j gambler wins first bet D 1 C enC1 :
Similarly, after the gambler loses his first bet, he can expect to make another en 1
bets:
ExŒ#bets starting with $n j gambler loses first bet D 1 C en 1:

So we have
en D p ExŒ#bets starting with $n j gambler wins first bet C q ExŒ#bets starting with $n j gambler loses
D p.1 C enC1 / C q.1 C en 1/ D penC1 C qen 1 C 1:
This yields the linear recurrence
1 q 1
enC1 D en en 1 : (21.10)
p p p
The routine solution of this linear recurrence yields:
Theorem 21.1.3. In the Gambler’s Ruin game with initial capital n, target T , and
probability p of winning each bet,
8
1
n.T n/ for p D ;
ˆ
ˆ
2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
<
ExŒnumber of bets D wn T n for p ¤ 1 (21.11)
ˆ
ˆ
ˆ p q 2
where wn D .r n 1/=.r T 1/
ˆ
ˆ
ˆ
ˆ
ˆ
D PrŒthe gambler wins:
ˆ
:

In the unbiased case, (21.11) can be rephrased simply as
ExŒnumber of fair bets D initial capital intended profit: (21.12)
For example, if the gambler starts with $10 dollars and plays until he is broke or
ahead $10, then 10 10 D 100 bets are required on average. If he starts with $500
and plays until he is broke or ahead $100, then the expected number of bets until
the game is over is 500 100 D 50; 000. This simple formula (21.12) cries out for
an intuitive proof, but we have not found one (where are you, Pascal?).
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922 Chapter 21 Random Walks

21.1.5 Quit While You Are Ahead
Suppose that the gambler never quits while he is ahead. That is, he starts with
n > 0 dollars, ignores any target T , but plays until he is flat broke. Call this the
unbounded Gambler’s ruin game. It turns out that if the game is not favorable, that
is, p 1=2, the gambler is sure to go broke. In particular, this holds in an unbiased
game with p D 1=2.

Lemma 21.1.4. If the gambler starts with one or more dollars and plays a fair
unbounded game, then he will go broke with probability 1.

Proof. If the gambler has initial capital n and goes broke in a game without reach-
ing a target T , then he would also go broke if he were playing and ignored the
target. So the probability that he will lose if he keeps playing without stopping at
any target T must be at least as large as the probability that he loses when he has a
target T > n.
But we know that in a fair game, the probability that he loses is 1 n=T . This
number can be made arbitrarily close to 1 by choosing a sufficiently large value of
T . Hence, the probability of his losing while playing without any target has a lower
bound arbitrarily close to 1, which means it must in fact be 1.

So even if the gambler starts with a million dollars and plays a perfectly fair
game, he will eventually lose it all with probability 1. But there is good news: if
the game is fair, he can “expect” to play forever:

Lemma 21.1.5. If the gambler starts with one or more dollars and plays a fair
unbounded game, then his expected number of plays is infinite.

A proof appears in Problem 21.2.
So even starting with just one dollar, the expected number of plays before going
broke is infinite! This sounds reassuring—you can go about your business without
worrying about being doomed, because doom will be infinitely delayed. To illus-
trate a situation where you really needn’t worry, think about mean time to failure
with a really tiny probability of failure in any given second—say 10 100 . In this
case you are unlikely to fail any time much sooner than many lifetimes of the es-
timated age of the universe, even though you will eventually fail with probability
one.
But in general, you shouldn’t feel reassured by an infinite expected time to go
broke. For example, think about a variant Gambler’s Ruin game which works as
follows: run one second of the process that has a 10 100 of failing in any second.
If it does not fail, then you go broke immediately. Otherwise, you play a fair, un-
bounded Gambler’s Ruin game. Now there is an overwhelming probability, namely,
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21.2. Random Walks on Graphs 923

1 10 100 , that you will go broke immediately. But there is a 10 100 probability
that you will wind up playing fair Gambler’s Ruin, so your overall expected time
will be at least 10 100 times the expectation of fair Gambler’s Ruin, namely, it will
still be infinite.
For the actual fair, unbounded Gambler’s Ruin gain starting with one dollar, there
is a a 50% chance the Gambler will go broke after the first bet, and a more than
15=16 chance of going broke within five bets, for example. So infinite expected
time is not much consolation to a Gambler who goes broke quickly with high prob-
ability.

21.2 Random Walks on Graphs
The hyperlink structure of the World Wide Web can be described as a digraph.
The vertices are the web pages with a directed edge from vertex x to vertex y if
x has a link to y. A digraph showing part of the website for MIT subject 6.042,
Mathematics for Computer Science, is shown in Figure 21.3.
The web graph is an enormous graph with trillions of vertices. In 1995, two
students at Stanford, Larry Page and Sergey Brin, realized that the structure of
this graph could be very useful in building a search engine. Traditional document
searching programs had been around for a long time and they worked in a fairly
straightforward way. Basically, you would enter some search terms and the search-
ing program would return all documents containing those terms. A relevance score
might also be returned for each document based on the frequency or position that
the search terms appeared in the document. For example, if the search term ap-
peared in the title or appeared 100 times in a document, that document would get a
higher score.
This approach works fine if you only have a few documents that match a search
term. But on the web, there are many billions of documents and millions of matches
to a typical search. For example, on May 2, 2012, a search on Google for “ ‘Mathe-
matics for Computer Science’ text” gave 482,000 hits! Which ones should we look
at first? Just because a page gets a high keyword score—say because it has “Math-
ematics Mathematics : : : Mathematics” copied 200 times across the front of the
document—does not make it a great candidate for attention. The web is filled with
bogus websites that repeat certain words over and over in order to attract visitors.
Google’s enormous market capital in part derives from the revenue it receives
from advertisers paying to appear at the top of search results. That top placement
would not be worth much if Google’s results were as easy to manipulate as keyword
frquencies. Advertisers pay because Google’s ranking method is consistently good
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924 Chapter 21 Random Walks

csail.mit.edu/6.042/classinfo!
stellar.mit.edu/S/course/6/fa15/6.042/gradebook!

csail.mit.edu/6.042/midterm1!

csail.mit.edu/6.042! stellar.mit.edu/S/course/6/fa15/6.042/!

lms.mitx.edu!
csail.mit.edu/6.042/midterm1_histogram!

stellar.mit.edu/S/course/6/fa15/6.042/materials!

csail.mit.edu/6.042/grading_questions! piazza.com! youtube.com/FiAKUX9efxg!

Figure 21.3 Website digraph for MIT subject 6.042
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21.2. Random Walks on Graphs 925

at determining the most relevant web pages. For example, Google demonstrated its
accuracy in our case by giving first rank2 to our 6.042 text.
So how did Google know to pick our text to be first out of 482,000?—because
back in 1995 Larry and Sergey got the idea to allow the digraph structure of the
web to determine which pages are likely to be the most important.

21.2.1 A First Crack at Page Rank
Looking at the web graph, do you have an idea which vertex/page might be the
best to rank first? Assume that all the pages match the search terms for now. Well,
intuitively, we should choose x2 , since lots of other pages point to it. This leads
us to their first idea: try defining the page rank of x to be indegree.x/, the number
of links pointing to x. The idea is to think of web pages as voting for the most
important page—the more votes, the better the rank.
Unfortunately, there are some problems with this idea. Suppose you wanted to
have your page get a high ranking. One thing you could do is to create lots of
dummy pages with links to your page.

There is another problem—a page could become unfairly influential by having
lots of links to other pages it wanted to hype.
2 First rank for some reason was an early version archived at Princeton; the Spring 2010 version

on the MIT Open Courseware site ranked 4th and 5th.
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926 Chapter 21 Random Walks

+1
So this strategy for high ranking would amount to, “vote early, vote often,” which
is no good if you want to build a search engine that’s worth paying fees for. So,
admittedly, their original idea was not so great. It was better than nothing, but
certainly not worth billions of dollars.

21.2.2 Random Walk on the Web Graph
But then Sergey and Larry thought some more and came up with a couple of im-
provements. Instead of just counting the indegree of a vertex, they considered the
probability of being at each page after a long random walk on the web graph. In
particular, they decided to model a user’s web experience as following each link on
a page with uniform probability. For example, if the user is at page x, and there
are three links from page x, then each link is followed with probability 1=3. More
generally, they assigned each edge x ! y of the web graph with a probability
conditioned on being on page x:
1
Pr follow link hx ! yi j at page x WWD :
outdeg.x/
The simulated user experience is then just a random walk on the web graph.
We can also compute the probability of arriving at a particular page y by sum-
ming over all edges pointing to y. We thus have
X
PrŒgo to y D Pr follow link hx ! yi j at page x PrŒat page x
edges hx!yi
X PrŒat x
D (21.13)
outdeg.x/
edges hx!yi

For example, in our web graph, we have
PrŒat x7 PrŒat x2
PrŒgo to x4 D C :
2 1
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21.2. Random Walks on Graphs 927

One can think of this equation as x7 sending half its probability to x2 and the other
half to x4 . The page x2 sends all of its probability to x4 .
There’s one aspect of the web graph described thus far that doesn’t mesh with
the user experience—some pages have no hyperlinks out. Under the current model,
the user cannot escape these pages. In reality, however, the user doesn’t fall off
the end of the web into a void of nothingness. Instead, he restarts his web journey.
Moreover, even if a user does not get stuck at a dead end, they will commonly get
discouraged after following some unproductive path for a while and will decide to
restart.
To model this aspect of the web, Sergey and Larry added a supervertex to the
web graph and added an edge from every page to the supervertex. Moreover, the
supervertex points to every other vertex in the graph with equal probability, allow-
ing the walk to restart from a random place. This ensures that the graph is strongly
connected.
If a page had no hyperlinks, then its edge to the supervertex has to be assigned
probability one. For pages that had some hyperlinks, the additional edge pointing
to the supervertex was assigned some specially given probability. In the original
versions of Page Rank, this probability was arbitrarily set to 0.15. That is, each
vertex with outdegree n 1 got an additional edge pointing to the supervertex
with assigned probability 0.15; its other n outgoing edges were still kept equally
likely, that is, each of the n edges was assigned probability 0:85=n.

21.2.3 Stationary Distribution & Page Rank
The basic idea behind page rank is finding a stationary distribution over the web
graph, so let’s define a stationary distribution.
Suppose each vertex is assigned a probability that corresponds, intuitively, to the
likelihood that a random walker is at that vertex at a randomly chosen time. We
assume that the walk never leaves the vertices in the graph, so we require that
X
PrŒat x D 1: (21.14)
vertices x

Definition 21.2.1. An assignment of probabilities to vertices in a digraph is a sta-
tionary distribution if for all vertices x

PrŒat x D PrŒgo to x at next step

Sergey and Larry defined their page ranks to be a stationary distribution. They
did this by solving the following system of linear equations: find a nonnegative
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928 Chapter 21 Random Walks

number Rank.x/ for each vertex x such that
X Rank.y/
Rank.x/ D ; (21.15)
outdeg.y/
edges hy!xi

corresponding to the intuitive equations given in (21.13). These numbers must also
satisfy the additional constraint corresponding to (21.14):
X
Rank.x/ D 1: (21.16)
vertices x

So if there are n vertices, then equations (21.15) and (21.16) provide a system
of n C 1 linear equations in the n variables Rank.x/. Note that constraint (21.16)
is needed because the remaining constraints (21.15) could be satisfied by letting
Rank.x/ WWD 0 for all x, which is useless.
Sergey and Larry were smart fellows, and they set up their page rank algorithm
so it would always have a meaningful solution. Strongly connected graphs have
unique stationary distributions (Problem 21.12), and their addition of a superver-
tex ensures this. Moreover, starting from any vertex and taking a sufficiently long
random walk on the graph, the probability of being at each page will get closer and
closer to the stationary distribution. Note that general digraphs without superver-
tices may have neither of these properties: there may not be a unique stationary
distribution, and even when there is, there may be starting points from which the
probabilities of positions during a random walk do not converge to the stationary
distribution (Problem 21.8).
Now just keeping track of the digraph whose vertices are trillions of web pages
is a daunting task. That’s why in 2011 Google invested $168,000,000 in a solar
power plant—the electrical power drawn by Google’s servers in 2011 would have
supplied the needs of 200,000 households.3 Indeed, Larry and Sergey named their
system Google after the number 10100 —which is called a “googol”—to reflect the
fact that the web graph is so enormous.
Anyway, now you can see how this text ranked first out of 378,000 matches.
Lots of other universities used our notes and presumably have links to the MIT
Mathematics for Computer Science Open Course Ware site, and the university sites
themselves are legitimate, which ultimately leads to the text getting a high page
rank in the web graph.
3 Google Details, and Defends, Its Use of Electricity, New York Times, September 8, 2011.
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21.2. Random Walks on Graphs 929

Problems for Section 21.1
Practice Problems
Problem 21.1.
Suppose that a gambler is playing a game in which he makes a series of $1 bets.
He wins each one with probability 0.49, and he keeps betting until he either runs
out of money or reaches some fixed goal of T dollars.
Let t .n/ be the expected number of bets the gambler makes until the game ends,
where n is the number of dollars the gambler has when he starts betting. Then the
function t satisfies a linear recurrence of the form

t .n/ D a t .n C 1/ C b t .n 1/ C c

for real constants a, b, c, and 0 < n < T .
(a) What are the values of a, b and c?

(b) What is t .0/?

Class Problems
Problem 21.2.
In a gambler’s ruin scenario, the gambler makes independent $1 bets, where the
probability of winning a bet is p and of losing is q WWD 1 p. The gambler keeps
betting until he goes broke or reaches a target of T dollars.
Suppose T D 1, that is, the gambler keeps playing until he goes broke. Let
r be the probability that starting with n > 0 dollars, the gambler’s stake ever gets
reduced to n 1 dollars.
(a) Explain why
r D q C pr 2 :

(b) Conclude that if p 1=2, then r D 1.

(d) Let t be the expected time for the gambler’s stake to go down by 1 dollar.
Verify that
t D q C p.1 C 2t /:
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930 Chapter 21 Random Walks

Conclude that starting with a 1 dollar stake in a fair game, the gambler can expect
to play forever!

Problem 21.3.
A gambler is placing $1 bets on the “1st dozen” in roulette. This bet wins when a
number from one to twelve comes in, and then the gambler gets his $1 back plus
$2 more. Recall that there are 38 numbers on the roulette wheel.
The gambler’s initial stake in $n and his target is $T . He will keep betting until
he runs out of money (“goes broke”) or reaches his target. Let wn be the probability
of the gambler winning, that is, reaching target $T before going broke.
(a) Write a linear recurrence with boundary conditions for wn . You need not solve
the recurrence.

(b) Let en be the expected number of bets until the game ends. Write a linear
recurrence with boundary conditions for en . You need not solve the recurrence.

Problem 21.4.
In the fair Gambler’s Ruin game with initial stake of n dollars and target of T
dollars, let en be the number of $1 bets the gambler makes until the game ends
(because he reaches his target or goes broke).
(a) Describe constants a; b; c such that

en D aen 1 C ben 2 C c: (21.17)

for 1 < n < T .

(b) Let en be defined by (21.17) for all n > 1, where e0 D 0 and e1 D d for
P1 d . Derive
some constant
n
a closed form (involving d ) for the generating function
E.x/ WWD 0 en x .

(d) Use part (c) to solve for d .

(e) Prove that en D n.T n/.
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21.2. Random Walks on Graphs 931

Problems for Section 21.2
Practice Problems
Problem 21.5.
Consider the following random-walk graphs:

x y

Figure 21.4

w z 0.1

0.9

Figure 21.5

1/2
1/2 1/2
1 a b c d 1

1/2

Figure 21.6

(a) Find d.x/ for a stationary distribution for graph 21.4.

(b) Find d.y/ for a stationary distribution for graph 21.4.

(c) If you start at node x in graph 21.4 and take a (long) random walk, does the
distribution over nodes ever get close to the stationary distribution?

(d) Find d.w/ for a stationary distribution for graph 21.5.
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932 Chapter 21 Random Walks

(e) Find d.z/ for a stationary distribution for graph 21.5.

(f) If you start at node w in graph 21.5 and take a (long) random walk, does the
distribution over nodes ever get close to the stationary distribution? (Hint: try a
few steps and watch what is happening.)

(g) How many stationary distributions are there for graph 21.6?

(h) If you start at node b in graph 21.6 and take a (long) random walk, what will
be the approximate probability that you are at node d ?

Problem 21.6.
A sink in a digraph is a vertex with no edges leaving it. Circle whichever of the
following assertions are true of stable distributions on finite digraphs with exactly
two sinks:

there may not be any

there may be a unique one

there are exactly two

there may be a countably infinite number

there may be a uncountable number

there always is an uncountable number

Problem 21.7.
Explain why there are an uncountable number of stationary distributions for the
following random walk graph.

1/2
1/2 1/2
1 a b c d 1

1/2
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21.2. Random Walks on Graphs 933

x y

Figure 21.7

Class Problems
Problem 21.8. (a) Find a stationary distribution for the random walk graph in Fig-
ure 21.7.

(b) Explain why a long random walk starting at node x in Figure 21.7 will not
converge to a stationary distribution. Characterize which starting distributions will
converge to the stationary one.

w z 0.1

0.9

Figure 21.8

(d) If you start at node w Figure 21.8 and take a (long) random walk, does the
distribution over nodes ever get close to the stationary distribution? You needn’t
prove anything here, just write out a few steps and see what’s happening.

(e) Explain why the random walk graph in Figure 21.9 has an uncountable number
of stationary distributions.

(f) If you start at node b in Figure 21.9 and take a long random walk, the proba-
bility you are at node d will be close to what fraction? Explain.

(g) Give an example of a random walk graph that is not strongly connected but
has a unique stationary distribution. Hint: There is a trivial example.
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934 Chapter 21 Random Walks

1/2
1/2 1/2
1 a b c d 1

1/2

Figure 21.9

Problem 21.9.
We use random walks on a digraph G to model the typical movement pattern of a
Math for CS student right after the final exam.
The student comes out of the final exam located on a particular node of the
graph, corresponding to the exam room. What happens next is unpredictable, as
the student is in a total haze. At each step of the walk, if the student is at node
u at the end of the previous step, they pick one of the edges hu ! vi uniformly at
random from the set of all edges directed out of u, and then walk to the node v.
Let n WWD jV .G/j and define the vector P .j / to be
.j /
P .j / WWD .p1 ; : : : ; pn.j / /
.j /
where pi is the probability of being at node i after j steps.
(a) We will start by looking at a simple graph. If the student starts at node 1 (the
top node) in the following graph, what is P .0/ ; P .1/ ; P .2/ ? Give a nice expression
for P .n/ .

1/2

.j /
(b) Given an arbitrary graph, show how to write an expression for pi in terms
.j 1/
of the pk ’s.

(c) Does your answer to the last part look like any other system of equations
you’ve seen in this course?
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21.2. Random Walks on Graphs 935

(d) Let the limiting distribution vector be
Pk .i /
i D1 P
lim :
k!1 k
What is the limiting distribution of the graph from part a? Would it change if the
start distribution were P .0/ D .1=2; 1=2/ or P .0/ D .1=3; 2=3/?

(e) Let’s consider another directed graph. If the student starts at node 1 with
probability 1/2 and node 2 with probability 1/2, what is P .0/ ; P .1/ ; P .2/ in the
following graph? What is the limiting distribution?

1/3
3

1/3 1/3
1/3 1/3
1/3
1/3 1/3
1/3
1 2
(f) Now we are ready for the real problem. In order to make it home, the poor
Math for student is faced with n doors along a long hall way. Unbeknownst to him,
the door that goes outside to paradise (that is, freedom from the class and more
importantly, vacation!) is at the very end. At each step along the way, he passes
by a door which he opens up and goes through with probability 1/2. Every time he
does this, he gets teleported back to the exam room. Let’s figure out how long it
will take the poor guy to escape from the class. What is P .0/ ; P .1/ ; P .2/ ? What is
the limiting distribution?

1/2
1/2 1/2
... 1
1 1/2 1/2 1/2 1/2
0 1 2 3 n
(g) Show that the expected number T .n/ of teleportations you make back to the
exam room before you escape to the outside world is 2n 1 1.
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936 Chapter 21 Random Walks

Problem 21.10.
Prove that for finite random walk graphs, the uniform distribution is stationary iff
the probabilities of the edges coming into each vertex always sum to 1, namely
X
p.u; v/ D 1; (21.18)
u2into.v/

where into.w/ WWD fv j hv ! wi is an edgeg.

Problem 21.11.
A Google-graph is a random-walk graph such that every edge leaving any given
vertex has the same probability. That is, the probability of each edge hv ! wi is
1= outdeg.v/.
A digraph is symmetric if, whenever hv ! wi is an edge, so is hw ! vi. Given
any finite, symmetric Google-graph, let
outdeg.v/
d.v/ WWD ;
e
where e is the total number of edges in the graph.
(a) If d was used for webpage ranking, how could you hack this to give your page
a high rank? . . . and explain informally why this wouldn’t work for “real” page
rank using digraphs?

(b) Show that d is a stationary distribution.

Homework Problems
Problem 21.12.
A digraph is strongly connected iff there is a directed path between every pair of
distinct vertices. In this problem we consider a finite random walk graph that is
strongly connected.
(a) Let d1 and d2 be distinct distributions for the graph, and define the maximum
dilation of d1 over d2 to be
d1 .x/
WWD max :
x2V d2 .x/
Call a vertex x dilated if d1 .x/=d2 .x/ D . Show that there is an edge hy ! zi
from an undilated vertex y to a dilated vertex z. Hint: Choose any dilated vertex x
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21.2. Random Walks on Graphs 937

0.5

0.5 0.5

0.5

0.5 0.5

1 1 0.5

1 0.5 1

Figure 21.10 Which ones have uniform stationary distribution?

and consider the set D of dilated vertices connected to x by a directed path (going
to x) that only uses dilated vertices. Explain why D ¤ V , and then use the fact
that the graph is strongly connected.

(b) Prove that the graph has at most one stationary distribution. (There always is
a stationary distribution, but we’re not asking you prove this.) Hint: Let d1 be a
stationary distribution and d2 be a different distribution. Let z be the vertex from
part (a). Show that starting from d2 , the probability of z changes at the next step.
That is, db2 .z/ ¤ d2 .z/.

Exam Problems
Problem 21.13.
For which of the graphs in Figure 21.10 is the uniform distribution over nodes a
stationary distribution? The edges are labeled with transition probabilities. Explain
your reasoning.

l
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“mcs” — 2017/3/10 — 22:22 — page 939 — #947

V Recurrences
“mcs” — 2017/3/10 — 22:22 — page 940 — #948
“mcs” — 2017/3/10 — 22:22 — page 941 — #949

Introduction
A recurrence describes a sequence of numbers. Early terms are specified explic-
itly, and later terms are expressed as a function of their predecessors. As a trivial
example, here is a recurrence describing the sequence 1; 2; 3; : : : :

T1 D 1
Tn D Tn 1 C1 (for n 2):

Here, the first term is defined to be 1 and each subsequent term is one more than its
predecessor.
Recurrences turn out to be a powerful tool. In this chapter, we’ll emphasize using
recurrences to analyze the performance of recursive algorithms. However, recur-
rences have other applications in computer science as well, such as enumeration of
structures and analysis of random processes. And, as we saw in Section 14.4, they
also arise in the analysis of problems in the physical sciences.
A recurrence in isolation is not a very useful description of a sequence. Sim-
ple questions such as, “What is the hundredth term?” or “What is the asymptotic
growth rate?” are not in general easy to answer by inspection of the recurrence. So
a typical goal is to solve a recurrence—that is, to find a closed-form expression for
the nth term.
We’ll first introduce two general solving techniques: guess-and-verify and plug-
and-chug. These methods are applicable to every recurrence, but their success re-
quires a flash of insight—sometimes an unrealistically brilliant flash. So we’ll also
introduce two big classes of recurrences, linear and divide-and-conquer, that often
come up in computer science. Essentially all recurrences in these two classes are
solvable using cookbook techniques; you follow the recipe and get the answer. A
drawback is that calculation replaces insight. The “Aha!” moment that is essential
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942 Part V Recurrences

in the guess-and-verify and plug-and-chug methods is replaced by a “Huh” at the
end of a cookbook procedure.
At the end of the chapter, we’ll develop rules of thumb to help you assess many
recurrences without any calculation. These rules can help you distinguish promis-
ing approaches from bad ideas early in the process of designing an algorithm.
Recurrences are one aspect of a broad theme in computer science: reducing a big
problem to progressively smaller problems until easy base cases are reached. This
same idea underlies both induction proofs and recursive algorithms. As we’ll see,
all three ideas snap together nicely. For example, the running time of a recursive
algorithm could be described with a recurrence with induction used to verify the
solution.
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22 Recurrences

22.1 The Towers of Hanoi
There are several methods for solving recurrence equations. The simplest is to
guess the solution and then verify that the guess is correct with an induction proof.
For example, as a alternative to the generating function derivation in Section 16.4.2
of the value of the number Tn of moves in the Tower of Hanoi problem with n disks,
we could have tried guessing. As a basis for a good guess, let’s look for a pattern in
the values of Tn computed above: 1, 3, 7, 15, 31, 63. A natural guess is Tn D 2n 1.
But whenever you guess a solution to a recurrence, you should always verify it with
a proof, typically by induction. After all, your guess might be wrong. (But why
bother to verify in this case? After all, if we’re wrong, its not the end of the. . . no,
let’s check.)

Claim 22.1.1. Tn D 2n 1 satisfies the recurrence:

T1 D 1
Tn D 2Tn 1 C1 (for n 2):

Proof. The proof is by induction on n. The induction hypothesis is that Tn D
2n 1. This is true for n D 1 because T1 D 1 D 21 1. Now assume that
Tn 1 D 2n 1 1 in order to prove that Tn D 2n 1, where n 2:

Tn D 2Tn 1C 1
n 1
D 2.2 1/ C 1
n
D2 1:

The first equality is the recurrence equation, the second follows from the induction
assumption, and the last step is simplification.

Such verification proofs are especially tidy because recurrence equations and
induction proofs have analogous structures. In particular, the base case relies on
the first line of the recurrence, which defines T1 . And the inductive step uses the
second line of the recurrence, which defines Tn as a function of preceding terms.
Our guess is verified. So we can now resolve our remaining questions about the
64-disk puzzle. Since T64 D 264 1, the monks must complete more than 18
billion billion steps before the world ends. Better study for the final.
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944 Chapter 22 Recurrences

22.1.1 The Upper Bound Trap
When the solution to a recurrence is complicated, one might try to prove that some
simpler expression is an upper bound on the solution. For example, the exact so-
lution to the Towers of Hanoi recurrence is Tn D 2n 1. Let’s try to prove the
“nicer” upper bound Tn 2n , proceeding exactly as before.

Proof. (Failed attempt.) The proof is by induction on n. The induction hypothesis
is that Tn 2n . This is true for n D 1 because T1 D 1 21 . Now assume that
Tn 1 2n 1 in order to prove that Tn 2n , where n 2:

Tn D 2Tn 1C 1
n 1
2.2 /C1
n
6 2 IMPLIES Uh-oh!

The first equality is the recurrence relation, the second follows from the induction
hypothesis, and the third step is a flaming train wreck.

The proof doesn’t work! As is so often the case with induction proofs, the ar-
gument only goes through with a stronger hypothesis. This isn’t to say that upper
bounding the solution to a recurrence is hopeless, but this is a situation where in-
duction and recurrences do not mix well.

22.1.2 Plug and Chug
Guess-and-verify is a simple and general way to solve recurrence equations. But
there is one big drawback: you have to guess right. That was not hard for the
Towers of Hanoi example. But sometimes the solution to a recurrence has a strange
form that is quite difficult to guess. Practice helps, of course, but so can some other
methods.
Plug-and-chug is another way to solve recurrences. This is also sometimes called
“expansion” or “iteration.” As in guess-and-verify, the key step is identifying a
pattern. But instead of looking at a sequence of numbers, you have to spot a pattern
in a sequence of expressions, which is sometimes easier. The method consists of
three steps, which are described below and illustrated with the Towers of Hanoi
example.

Step 1: Plug and Chug Until a Pattern Appears
The first step is to expand the recurrence equation by alternately “plugging” (apply-
ing the recurrence) and “chugging” (simplifying the result) until a pattern appears.
Be careful: too much simplification can make a pattern harder to spot. The rule
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22.1. The Towers of Hanoi 945

to remember—indeed, a rule applicable to the whole of college life—is chug in
moderation.

Tn D 2Tn 1 C1
D 2.2Tn 2 C 1/ C 1 plug
D 4Tn 2 C2C1 chug
D 4.2Tn 3 C 1/ C 2 C 1 plug
D 8Tn 3 C4C2C1 chug
D 8.2Tn 4 C 1/ C 4 C 2 C 1 plug
D 16Tn 4 C8C4C2C1 chug

Above, we started with the recurrence equation. Then we replaced Tn 1 with
2Tn 2 C 1, since the recurrence says the two are equivalent. In the third step,
we simplified a little—but not too much! After several similar rounds of plugging
and chugging, a pattern is apparent. The following formula seems to hold:

Tn D 2 k Tn k C 2k 1
C 2k 2
C C 22 C 21 C 20
D 2 k Tn k C 2k 1

Once the pattern is clear, simplifying is safe and convenient. In particular, we’ve
collapsed the geometric sum to a closed form on the second line.

Step 2: Verify the Pattern
The next step is to verify the general formula with one more round of plug-and-
chug.

Tn D 2k Tn k C 2k 1
D 2k .2Tn .kC1/ C 1/ C 2k 1 plug
D 2kC1 Tn .kC1/ C 2kC1 1 chug

The final expression on the right is the same as the expression on the first line,
except that k is replaced by k C 1. Surprisingly, this effectively proves that the
formula is correct for all k. Here is why: we know the formula holds for k D 1,
because that’s the original recurrence equation. And we’ve just shown that if the
formula holds for some k 1, then it also holds for k C 1. So the formula holds
for all k 1 by induction.
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946 Chapter 22 Recurrences

Step 3: Write Tn Using Early Terms with Known Values
The last step is to express Tn as a function of early terms whose values are known.
Here, choosing k D n 1 expresses Tn in terms of T1 , which is equal to 1. Sim-
plifying gives a closed-form expression for Tn :

Tn D 2n 1
T1 C 2n 1
1
n 1 n 1
D2 1C2 1
n
D2 1:

We’re done! This is the same answer we got from guess-and-verify.
Let’s compare guess-and-verify with plug-and-chug. In the guess-and-verify
method, we computed several terms at the beginning of the sequence T1 , T2 , T3 ,
etc., until a pattern appeared. We generalized to a formula for the nth term Tn . In
contrast, plug-and-chug works backward from the nth term. Specifically, we started
with an expression for Tn involving the preceding term Tn 1 , and rewrote this us-
ing progressively earlier terms Tn 2 , Tn 3 , etc. Eventually, we noticed a pattern,
which allowed us to express Tn using the very first term T1 whose value we knew.
Substituting this value gave a closed-form expression for Tn . So guess-and-verify
and plug-and-chug tackle the problem from opposite directions.

22.2 Merge Sort
Algorithms textbooks traditionally claim that sorting is an important, fundamental
problem in computer science. Then they smack you with sorting algorithms until
life as a disk-stacking monk in Hanoi sounds delightful. Here, we’ll cover just one
well-known sorting algorithm, Merge Sort. The analysis introduces another kind of
recurrence.
Here is how Merge Sort works. The input is a list of n numbers, and the output
is those same numbers in nondecreasing order. There are two cases:

If the input is a single number, then the algorithm does nothing, because the
list is already sorted.

Otherwise, the list contains two or more numbers. The first half and the
second half of the list are each sorted recursively. Then the two halves are
merged to form a sorted list with all n numbers.

Let’s work through an example. Suppose we want to sort this list:
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22.2. Merge Sort 947

10, 7, 23, 5, 2, 8, 6, 9.

Since there is more than one number, the first half (10, 7, 23, 5) and the second half
(2, 8, 6, 9) are sorted recursively. The results are 5, 7, 10, 23 and 2, 6, 8, 9. All that
remains is to merge these two lists. This is done by repeatedly emitting the smaller
of the two leading terms. When one list is empty, the whole other list is emitted.
The example is worked out below. In this table, underlined numbers are about to
be emitted.
First Half Second Half Output
5, 7, 10, 23 2, 6, 8, 9
5, 7, 10, 23 6, 8, 9 2
7, 10, 23 6, 8, 9 2, 5
7, 10, 23 8, 9 2, 5, 6
10, 23 8, 9 2, 5, 6, 7
10, 23 9 2, 5, 6, 7, 8
10, 23 2, 5, 6, 7, 8, 9
2, 5, 6, 7, 8, 9, 10, 23
The leading terms are initially 5 and 2. So we output 2. Then the leading terms are
5 and 6, so we output 5. Eventually, the second list becomes empty. At that point,
we output the whole first list, which consists of 10 and 23. The complete output
consists of all the numbers in sorted order.

22.2.1 Finding a Recurrence
A traditional question about sorting algorithms is, “What is the maximum number
of comparisons used in sorting n items?” This is taken as an estimate of the running
time. In the case of Merge Sort, we can express this quantity with a recurrence. Let
Tn be the maximum number of comparisons used while Merge Sorting a list of n
numbers. For now, assume that n is a power of 2. This ensures that the input can
be divided in half at every stage of the recursion.

If there is only one number in the list, then no comparisons are required, so
T1 D 0.

Otherwise, Tn includes comparisons used in sorting the first half (at most
Tn=2 ), in sorting the second half (also at most Tn=2 ), and in merging the two
halves. The number of comparisons in the merging step is at most n 1.
This is because at least one number is emitted after each comparison and one
more number is emitted at the end when one list becomes empty. Since n
items are emitted in all, there can be at most n 1 comparisons.
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948 Chapter 22 Recurrences

Therefore, the maximum number of comparisons needed to Merge Sort n items is
given by this recurrence:
T1 D 0
Tn D 2Tn=2 C n 1 (for n 2 and a power of 2):
This fully describes the number of comparisons, but not in a very useful way; a
closed-form expression would be much more helpful. To get that, we have to solve
the recurrence.

22.2.2 Solving the Recurrence
Let’s first try to solve the Merge Sort recurrence with the guess-and-verify tech-
nique. Here are the first few values:
T1 D 0
T2 D 2T1 C 2 1D1
T4 D 2T2 C 4 1D5
T8 D 2T4 C 8 1 D 17
T16 D 2T8 C 16 1 D 49:
We’re in trouble! Guessing the solution to this recurrence is hard because there is
no obvious pattern. So let’s try the plug-and-chug method instead.

Step 1: Plug and Chug Until a Pattern Appears
First, we expand the recurrence equation by alternately plugging and chugging until
a pattern appears.
Tn D 2Tn=2 C n 1
D 2.2Tn=4 C n=2 1/ C .n 1/ plug
D 4Tn=4 C .n 2/ C .n 1/ chug
D 4.2Tn=8 C n=4 1/ C .n 2/ C .n 1/ plug
D 8Tn=8 C .n 4/ C .n 2/ C .n 1/ chug
D 8.2Tn=16 C n=8 1/ C .n 4/ C .n 2/ C .n 1/ plug
D 16Tn=16 C .n 8/ C .n 4/ C .n 2/ C .n 1/ chug
A pattern is emerging. In particular, this formula seems holds:
Tn D 2k Tn=2k C .n 2k 1
/ C .n 2k 2
/ C C .n 20 /
D 2k Tn=2k C k n 2k 1
2k 2
20
D 2k Tn=2k C k n 2k C 1:
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22.2. Merge Sort 949

On the second line, we grouped the n terms and powers of 2. On the third, we
collapsed the geometric sum.

Step 2: Verify the Pattern
Next, we verify the pattern with one additional round of plug-and-chug. If we
guessed the wrong pattern, then this is where we’ll discover the mistake.

Tn D 2k Tn=2k C k n 2k C 1
D 2k .2Tn=2kC1 C n=2k 1/ C k n 2k C 1 plug
kC1 kC1
D2 Tn=2kC1 C .k C 1/n 2 C1 chug

The formula is unchanged except that k is replaced by k C 1. This amounts to the
induction step in a proof that the formula holds for all k 1.

Step 3: Write Tn Using Early Terms with Known Values
Finally, we express Tn using early terms whose values are known. Specifically, if
we let k D log n, then Tn=2k D T1 , which we know is 0:

Tn D 2k Tn=2k C k n 2k C 1
D 2log n Tn=2log n C n log n 2log n C 1
D nT1 C n log n nC1
D n log n n C 1:

We’re done! We have a closed-form expression for the maximum number of com-
parisons used in Merge Sorting a list of n numbers. In retrospect, it is easy to see
why guess-and-verify failed: this formula is fairly complicated.
As a check, we can confirm that this formula gives the same values that we
computed earlier:
n Tn n log n n C 1
1 0 1 log 1 1 C 1 D 0
2 1 2 log 2 2 C 1 D 1
4 5 4 log 4 4 C 1 D 5
8 17 8 log 8 8 C 1 D 17
16 49 16 log 16 16 C 1 D 49
As a double-check, we could write out an explicit induction proof. This would be
straightforward, because we already worked out the guts of the proof in step 2 of
the plug-and-chug procedure.
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950 Chapter 22 Recurrences

22.3 Linear Recurrences
So far we’ve solved recurrences with two techniques: guess-and-verify and plug-
and-chug. These methods require spotting a pattern in a sequence of numbers or
expressions. In this section and the next, we’ll give cookbook solutions for two
large classes of recurrences. These methods require no flash of insight; you just
follow the recipe and get the answer.

22.3.1 Climbing Stairs
How many different ways are there to climb n stairs, if you can either step up one
stair or hop up two? For example, there are five different ways to climb four stairs:
1. step, step, step, step
2. hop, hop
3. hop, step, step
4. step, hop step
5. step, step, hop
Working through this problem will demonstrate the major features of our first cook-
book method for solving recurrences. We’ll fill in the details of the general solution
afterward.

Finding a Recurrence
As special cases, there is 1 way to climb 0 stairs (do nothing) and 1 way to climb
1 stair (step up). In general, an ascent of n stairs consists of either a step followed
by an ascent of the remaining n 1 stairs or a hop followed by an ascent of n 2
stairs. So the total number of ways to climb n stairs is equal to the number of ways
to climb n 1 plus the number of ways to climb n 2. These observations define
a recurrence:

f .0/ D 1
f .1/ D 1
f .n/ D f .n 1/ C f .n 2/ for n 2:

Here, f .n/ denotes the number of ways to climb n stairs. Also, we’ve switched
from subscript notation to functional notation, from Tn to fn . Here the change is
cosmetic, but the expressiveness of functions will be useful later.
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22.3. Linear Recurrences 951

This is the Fibonaccifamous of all recurrence equations. Fibonacci numbers arise
in all sorts of applications and in nature. For example,
The recurrence was introduced by Fibonacci himself in thirteenth century to
model rabbit reproduction.

The sizes of the spiral patterns on the face of a sunflower grow in proportion
to Fibonacci numbers.

The input values on which Euclid’s gcd algorithm requires the greatest num-
ber of steps are consecutive Fibonacci numbers.

Solving the Recurrence
The Fibonacci recurrence belongs to the class of linear recurrences, which are es-
sentially all solvable with a technique that you can learn in an hour. This is some-
what amazing, since the Fibonacci recurrence remained unsolved for almost six
centuries!
In general, a homogeneous linear recurrence has the form

f .n/ D a1 f .n 1/ C a2 f .n 2/ C C ad f .n d/

where a1 ; a2 ; : : : ; ad and d are constants. The order of the recurrence is d . Com-
monly, the value of the function f is also specified at a few points; these are called
boundary conditions. For example, the Fibonacci recurrence has order d D 2 with
coefficients a1 D a2 D 1 and g.n/ D 0. The boundary conditions are f .0/ D 1
and f .1/ D 1. The word “homogeneous” sounds scary, but effectively means “the
simpler kind.” We’ll consider linear recurrences with a more complicated form
later.
Let’s try to solve the Fibonacci recurrence with the benefit centuries of hindsight.
In general, linear recurrences tend to have exponential solutions. So let’s guess that

f .n/ D x n

where x is a parameter introduced to improve our odds of making a correct guess.
We’ll figure out the best value for x later. To further improve our odds, let’s neglect
the boundary conditions f .0/ D 0 and f .1/ D 1 for now. Plugging this guess into
the recurrence f .n/ D f .n 1/ C f .n 2/ gives

xn D xn 1
C xn 2
:

Dividing both sides by x n 2 leaves a quadratic equation:

x 2 D x C 1:
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952 Chapter 22 Recurrences

Solving this equation gives two plausible values for the parameter x:
p
1˙ 5
xD :
2
This suggests that there are at least two different solutions to the recurrence, ne-
glecting the boundary conditions.
p !n p !n
1C 5 1 5
f .n/ D or f .n/ D
2 2

A charming features of homogeneous linear recurrences is that any linear com-
bination of solutions is another solution.

Theorem 22.3.1. If f .n/ and g.n/ are both solutions to a homogeneous linear
recurrence, then h.n/ D sf .n/ C tg.n/ is also a solution for all s; t 2 R.

Proof.

h.n/ D sf .n/ C tg.n/
D s .a1 f .n 1/ C C ad f .n d // C t .a1 g.n 1/ C C ad g.n d //
D a1 .sf .n 1/ C tg.n 1// C C ad .sf .n d / C tg.n d //
D a1 h.n 1/ C C ad h.n d/

The first step uses the definition of the function h, and the second uses the fact that
f and g are solutions to the recurrence. In the last two steps, we rearrange terms
and use the definition of h again. Since the first expression is equal to the last, h is
also a solution to the recurrence.

The phenomenon described in this theorem—a linear combination of solutions is
another solution—also holds for many differential equations and physical systems.
In fact, linear recurrences are so similar to linear differential equations that you can
safely snooze through that topic in some future math class.
Returning to the Fibonacci recurrence, this theorem implies that
p !n p !n
1C 5 1 5
f .n/ D s Ct
2 2

is a solution for all real numbers s and t . The theorem expanded two solutions to
a whole spectrum of possibilities! Now, given all these options to choose from,
we can find one solution that satisfies the boundary conditions, f .0/ D 1 and
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22.3. Linear Recurrences 953

f .1/ D 1. Each boundary condition puts some constraints on the parameters s and
t. In particular, the first boundary condition implies that
p !0 p !0
1C 5 1 5
f .0/ D s Ct D s C t D 1:
2 2
Similarly, the second boundary condition implies that
p !1 p !1
1C 5 1 5
f .1/ D s Ct D 1:
2 2
Now we have two linear equations in two unknowns. The system is not degenerate,
so there is a unique solution:
p p
1 1C 5 1 1 5
sDp tD p :
5 2 5 2
These values of s and t identify a solution to the Fibonacci recurrence that also
satisfies the boundary conditions:
p p !n p p !n
1 1C 5 1C 5 1 1 5 1 5
f .n/ D p p
5 2 2 5 2 2
p !nC1 p !nC1
1 1C 5 1 1 5
Dp p :
5 2 5 2
It is easy to see why no one stumbled across this solution for almost six centuries.
All Fibonacci numbers are integers, but this expression is full of square roots of
five! Amazingly, the square roots always cancel out. This expression really does
give the Fibonacci numbers if we plug in n D 0; 1; 2, etc.
This closed form for Fibonacci numbers is known as Binet’s formula and has
some interesting corollaries.
p The first term tends to infinity because the base of
the exponential, .1 C 5/=2 D 1:618 : : : is greater than one. This value is often
denotedp and called the “golden ratio.” The second term tends to zero, because
.1 5/=2 D 0:618033988 : : : has absolute value less than 1. This implies that
the nth Fibonacci number is:
nC1
f .n/ D p C o.1/:
5
Remarkably, this expression involving irrational numbers is actually very close to
an integer for all large n—namely, a Fibonacci number! For example:
20
p D 6765:000029 f .19/:
5
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954 Chapter 22 Recurrences

This also implies that the ratio of consecutive Fibonacci numbers rapidly approaches
the golden ratio. For example:

f .20/ 10946
D D 1:618033998 : : : :
f .19/ 6765

22.3.2 Solving Homogeneous Linear Recurrences
The method we used to solve the Fibonacci recurrence can be extended to solve
any homogeneous linear recurrence; that is, a recurrence of the form

f .n/ D a1 f .n 1/ C a2 f .n 2/ C C ad f .n d/

where a1 ; a2 ; : : : ; ad and d are constants. Substituting the guess f .n/ D x n , as
with the Fibonacci recurrence, gives

x n D a1 x n 1
C a2 x n 2
C C ad x n d
:

Dividing by x n d gives

x d D a1 x d 1
C a2 x d 2
C C ad 1x C ad :

This is called the characteristic equation of the recurrence. The characteristic equa-
tion can be read off quickly since the coefficients of the equation are the same as
the coefficients of the recurrence.
The solutions to a linear recurrence are defined by the roots of the characteristic
equation. Neglecting boundary conditions for the moment:

If r is a nonrepeated root of the characteristic equation, then r n is a solution
to the recurrence.

If r is a repeated root with multiplicity k then r n , nr n , n2 r n , . . . , nk 1r n

are all solutions to the recurrence.

Theorem 22.3.1 implies that every linear combination of these solutions is also a
solution.
For example, suppose that the characteristic equation of a recurrence has roots s,
t and u twice. These four roots imply four distinct solutions:

f .n/ D s n f .n/ D t n f .n/ D un f .n/ D nun :

Furthermore, every linear combination

f .n/ D a s n C b t n C c un C d nun (22.1)
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22.3. Linear Recurrences 955

is also a solution.
All that remains is to select a solution consistent with the boundary conditions
by choosing the constants appropriately. Each boundary condition implies a linear
equation involving these constants. So we can determine the constants by solving
a system of linear equations. For example, suppose our boundary conditions were
f .0/ D 0, f .1/ D 1, f .2/ D 4 and f .3/ D 9. Then we would obtain four
equations in four unknowns:
f .0/ D 0 implies a s 0 C b t 0 C c u0 C d 0u0 D0
f .1/ D 1 implies a s 1 C b t 1 C c u1 C d 1u1 D1
f .2/ D 4 implies a s 2 C b t 2 C c u2 C d 2u2 D4
f .3/ D 9 implies a s 3 C b t 3 C c u3 C d 3u3 D9
This looks nasty, but remember that s, t and u are just constants. Solving this sys-
tem gives values for a, b, c and d that define a solution to the recurrence consistent
with the boundary conditions.

22.3.3 Solving General Linear Recurrences
We can now solve all linear homogeneous recurrences, which have the form

f .n/ D a1 f .n 1/ C a2 f .n 2/ C C ad f .n d /:

Many recurrences that arise in practice do not quite fit this mold. For example, the
Towers of Hanoi problem led to this recurrence:

f .1/ D 1
f .n/ D 2f .n 1/ C 1 (for n 2):

The problem is the extra C1; that is not allowed in a homogeneous linear recur-
rence. In general, adding an extra function g.n/ to the right side of a linear recur-
rence gives an inhomogeneous linear recurrence:

f .n/ D a1 f .n 1/ C a2 f .n 2/ C C ad f .n d / C g.n/:

Solving inhomogeneous linear recurrences is neither very different nor very dif-
ficult. We can divide the whole job into five steps:

1. Replace g.n/ by 0, leaving a homogeneous recurrence. As before, find roots
of the characteristic equation.
2. Write down the solution to the homogeneous recurrence, but do not yet use
the boundary conditions to determine coefficients. This is called the homo-
geneous solution.
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956 Chapter 22 Recurrences

3. Now restore g.n/ and find a single solution to the recurrence, ignoring bound-
ary conditions. This is called a particular solution. We’ll explain how to find
a particular solution shortly.

4. Add the homogeneous and particular solutions together to obtain the general
solution.

5. Now use the boundary conditions to determine constants by the usual method
of generating and solving a system of linear equations.

As an example, let’s consider a variation of the Towers of Hanoi problem. Sup-
pose that moving a disk takes time proportional to its size. Specifically, moving the
smallest disk takes 1 second, the next-smallest takes 2 seconds, and moving the nth
disk then requires n seconds instead of 1. So, in this variation, the time to complete
the job is given by a recurrence with a Cn term instead of a C1:

f .1/ D 1
f .n/ D 2f .n 1/ C n for n 2:

Clearly, this will take longer, but how much longer? Let’s solve the recurrence with
the method described above.
In Steps 1 and 2, dropping the Cn leaves the homogeneous recurrence f .n/ D
2f .n 1/. The characteristic equation is x D 2. So the homogeneous solution is
f .n/ D c2n .
In Step 3, we must find a solution to the full recurrence f .n/ D 2f .n 1/ C n,
without regard to the boundary condition. Let’s guess that there is a solution of the
form f .n/ D an C b for some constants a and b. Substituting this guess into the
recurrence gives

an C b D 2.a.n 1/ C b/ C n
0 D .a C 1/n C .b 2a/:

The second equation is a simplification of the first. The second equation holds for
all n if both a C 1 D 0 (which implies a D 1) and b 2a D 0 (which implies
that b D 2). So f .n/ D an C b D n 2 is a particular solution.
In the Step 4, we add the homogeneous and particular solutions to obtain the
general solution

f .n/ D c2n n 2:
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22.4. Divide-and-Conquer Recurrences 957

Finally, in step 5, we use the boundary condition f .1/ D 1 to determine the
value of the constant c:
f .1/ D 1 IMPLIES c21 1 2D1
IMPLIES c D 2:
Therefore, the function f .n/ D 2 2n
n 2 solves this variant of the Towers
of Hanoi recurrence. For comparison, the solution to the original Towers of Hanoi
problem was 2n 1. So if moving disks takes time proportional to their size, then
the monks will need about twice as much time to solve the whole puzzle.

22.3.4 How to Guess a Particular Solution
Finding a particular solution can be the hardest part of solving inhomogeneous
recurrences. This involves guessing, and you might guess wrong.1 However, some
rules of thumb make this job fairly easy most of the time.
Generally, look for a particular solution with the same form as the inhomo-
geneous term g.n/.
If g.n/ is a constant, then guess a particular solution f .n/ D c. If this doesn’t
work, try polynomials of progressively higher degree: f .n/ D bn C c, then
f .n/ D an2 C bn C c, etc.
More generally, if g.n/ is a polynomial, try a polynomial of the same degree,
then a polynomial of degree one higher, then two higher, etc. For example,
if g.n/ D 6n C 5, then try f .n/ D bn C c and then f .n/ D an2 C bn C c.
If g.n/ is an exponential, such as 3n , then first guess that f .n/ D c3n .
Failing that, try f .n/ D bn3n C c3n and then an2 3n C bn3n C c3n , etc.
The entire process is summarized on the following page.

22.4 Divide-and-Conquer Recurrences
We now have a recipe for solving general linear recurrences. But the Merge Sort
recurrence, which we encountered earlier, is not linear:
T .1/ D 0
T .n/ D 2T .n=2/ C n 1 (for n 2):
1 Chapter16 explains how to solve linear recurrences with generating functions—it’s a little more
complicated, but it does not require guessing.
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958 Chapter 22 Recurrences

Short Guide to Solving Linear Recurrences
A linear recurrence is an equation
f .n/ D a1 f .n 1/ C a2 f .n 2/ C C ad f .n d/ C g.n/
„ ƒ‚ … „ ƒ‚ …
homogeneous part inhomogeneous part

together with boundary conditions such as f .0/ D b0 , f .1/ D b1 , etc. Linear
recurrences are solved as follows:
1. Find the roots of the characteristic equation
x n D a1 x n 1
C a2 x n 2
C C ak 1x C ak :

2. Write down the homogeneous solution. Each root generates one term and
the homogeneous solution is their sum. A nonrepeated root r generates the
term cr n , where c is a constant to be determined later. A root r with multi-
plicity k generates the terms
d1 r n d2 nr n d3 n2 r n ::: dk nk 1 n
r
where d1 ; : : : dk are constants to be determined later.
3. Find a particular solution. This is a solution to the full recurrence that need
not be consistent with the boundary conditions. Use guess-and-verify. If
g.n/ is a constant or a polynomial, try a polynomial of the same degree, then
of one higher degree, then two higher. For example, if g.n/ D n, then try
f .n/ D bn C c and then an2 C bn C c. If g.n/ is an exponential, such as 3n ,
then first guess f .n/ D c3n . Failing that, try f .n/ D .bn C c/3n and then
.an2 C bn C c/3n , etc.
4. Form the general solution, which is the sum of the homogeneous solution
and the particular solution. Here is a typical general solution:
f .n/ D c2n C d. 1/n C C…
3nƒ‚
„ 1.
„ ƒ‚ …
homogeneous solution inhomogeneous solution

5. Substitute the boundary conditions into the general solution. Each boundary
condition gives a linear equation in the unknown constants. For example,
substituting f .1/ D 2 into the general solution above gives
2 D c 21 C d . 1/1 C 3 1 C 1
IMPLIES 2 D 2c d:
Determine the values of these constants by solving the resulting system of
linear equations.
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22.4. Divide-and-Conquer Recurrences 959

In particular, T .n/ is not a linear combination of a fixed number of immediately
preceding terms; rather, T .n/ is a function of T .n=2/, a term halfway back in the
sequence.
Merge Sort is an example of a divide-and-conquer algorithm: it divides the in-
put, “conquers” the pieces, and combines the results. Analysis of such algorithms
commonly leads to divide-and-conquer recurrences, which have this form:
k
X
T .n/ D ai T .bi n/ C g.n/
i D1

Here a1 ; : : : ak are positive constants, b1 ; : : : ; bk are constants between 0 and 1,
and g.n/ is a nonnegative function. For example, setting a1 D 2, b1 D 1=2 and
g.n/ D n 1 gives the Merge Sort recurrence.

22.4.1 The Akra-Bazzi Formula
The solution to virtually all divide and conquer solutions is given by the amazing
Akra-Bazzi formula. Quite simply, the asymptotic solution to the general divide-
and-conquer recurrence
k
X
T .n/ D ai T .bi n/ C g.n/
i D1

is n
g.u/
Z
p
T .n/ D ‚ n 1C du (22.2)
1 upC1
where p satisfies
k
p
X
ai bi D 1: (22.3)
i D1

A rarely-troublesome requirement is that the function g.n/ must not grow or
oscillate too quickly. Specifically, jg 0 .n/j must be bounded by some polynomial.
So, for example, the Akra-Bazzi formula is valid when g.n/ D x 2 log n, but not
when g.n/ D 2n .
Let’s solve the Merge Sort recurrence again, using the Akra-Bazzi formula in-
stead of plug-and-chug. First, we find the value p that satisfies

2 .1=2/p D 1:
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960 Chapter 22 Recurrences

Looks like p D 1 does the job. Then we compute the integral:
Z n
u 1
T .n/ D ‚ n 1 C du
1 u2
1 n

D ‚ n 1 C log u C
u 1

1
D ‚ n log n C
n
D ‚.n log n/:

The first step is integration and the second is simplification. We can drop the 1=n
term in the last step, because the log n term dominates. We’re done!
Let’s try a scary-looking recurrence:

T .n/ D 2T .n=2/ C .8=9/T .3n=4/ C n2 :

Here, a1 D 2, b1 D 1=2, a2 D 8=9 and b2 D 3=4. So we find the value p that
satisfies
2 .1=2/p C .8=9/.3=4/p D 1:
Equations of this form don’t always have closed-form solutions, so you may need
to approximate p numerically sometimes. But in this case the solution is simple:
p D 2. Then we integrate:
Z n 2
2 u
T .n/ D ‚ n 1 C 3
du
1 u
D ‚ n2 .1 C log n/

D ‚ n2 log n :

That was easy!

22.4.2 Two Technical Issues
Until now, we’ve swept a couple issues related to divide-and-conquer recurrences
under the rug. Let’s address those issues now.
First, the Akra-Bazzi formula makes no use of boundary conditions. To see why,
let’s go back to Merge Sort. During the plug-and-chug analysis, we found that

Tn D nT1 C n log n n C 1:

This expresses the nth term as a function of the first term, whose value is specified
in a boundary condition. But notice that Tn D ‚.n log n/ for every value of T1 .
The boundary condition doesn’t matter!
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22.4. Divide-and-Conquer Recurrences 961

This is the typical situation: the asymptotic solution to a divide-and-conquer
recurrence is independent of the boundary conditions. Intuitively, if the bottom-
level operation in a recursive algorithm takes, say, twice as long, then the overall
running time will at most double. This matters in practice, but the factor of 2 is
concealed by asymptotic notation. There are corner-case exceptions. For example,
the solution to T .n/ D 2T .n=2/ is either ‚.n/ or zero, depending on whether
T .1/ is zero. These cases are of little practical interest, so we won’t consider them
further.
There is a second nagging issue with divide-and-conquer recurrences that does
not arise with linear recurrences. Specifically, dividing a problem of size n may
create subproblems of non-integer size. For example, the Merge Sort recurrence
contains the term T .n=2/. So what if n is 15? How long does it take to sort seven-
and-a-half items? Previously, we dodged this issue by analyzing Merge Sort only
when the size of the input was a power of 2. But then we don’t know what happens
for an input of size, say, 100.
Of course, a practical implementation of Merge Sort would split the input ap-
proximately in half, sort the halves recursively, and merge the results. For example,
a list of 15 numbers would be split into lists of 7 and 8. More generally, a list of n
numbers would be split into approximate halves of size dn=2e and bn=2c. So the
maximum number of comparisons is actually given by this recurrence:
T .1/ D 0
T .n/ D T .dn=2e/ C T .bn=2c/ C n 1 (for n 2):
This may be rigorously correct, but the ceiling and floor operations make the recur-
rence hard to solve exactly.
Fortunately, the asymptotic solution to a divide and conquer recurrence is un-
affected by floors and ceilings. More precisely, the solution is not changed by
replacing a term T .bi n/ with either T .cei lbi n/ or T .bbi nc/. So leaving floors
and ceilings out of divide-and-conquer recurrences makes sense in many contexts;
those are complications that make no difference.

22.4.3 The Akra-Bazzi Theorem
The Akra-Bazzi formula together with our assertions about boundary conditions
and integrality all follow from the Akra-Bazzi Theorem, which is stated below.
Theorem 22.4.1 (Akra-Bazzi). Suppose that the function T W R ! R is nonnega-
tive and bounded for 0 x x0 and satisfies the recurrence
k
X
T .x/ D ai T .bi x C hi .x// C g.x/ for x > x0 ; (22.4)
i D1
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962 Chapter 22 Recurrences

where:

1. x0 is large enough so that T is well-defined,

2. a1 ; : : : ; ak are positive constants,

3. b1 ; : : : ; bk are constants between 0 and 1,

4. g.x/ is a nonnegative function such that jg 0 .x/j is bounded by a polynomial,

5. jhi .x/j D O.x= log2 x/.

Then x
g.u/
Z
p
T .x/ D ‚ x 1 C du
1 upC1
where p satisfies
k
p
X
ai bi D 1:
i D1

The Akra-Bazzi theorem can be proved using a complicated induction argument,
though we won’t do that here. But let’s at least go over the statement of the theorem.
All the recurrences we’ve considered were defined over the integers, and that is
the common case. But the Akra-Bazzi theorem applies more generally to functions
defined over the real numbers.
The Akra-Bazzi formula is lifted directed from the theorem statement, except
that the recurrence in the theorem includes extra functions, hi . These functions
extend the theorem to address floors, ceilings, and other small adjustments to the
sizes of subproblems. The trick is illustrated by this combination of parameters
lx m x
a1 D 1 b1 D 1=2 h1 .x/ D
j x2 k x2
a2 D 1 b2 D 1=2 h2 .x/ D
2 2
g.x/ D x 1

which corresponds the recurrence
x l x m x x j x k x
T .x/ D 1 T C C T C Cx 1
l x 2m 2j k
x
2 2 2 2
DT CT C x 1:
2 2
This is the rigorously correct Merge Sort recurrence valid for all input sizes,
complete with floor and ceiling operators. In this case, the functions h1 .x/ and
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22.4. Divide-and-Conquer Recurrences 963

h2 .x/ are both at most 1, which is easily O.x= log2 x/ as required by the theorem
statement. These functions hi do not affect—or even appear in—the asymptotic
solution to the recurrence. This justifies our earlier claim that applying floor and
ceiling operators to the size of a subproblem does not alter the asymptotic solution
to a divide-and-conquer recurrence.

22.4.4 The Master Theorem
There is a special case of the Akra-Bazzi formula known as the Master Theorem
that handles some of the recurrences that commonly arise in computer science. It
is called the Master Theorem because it was proved long before Akra and Bazzi
arrived on the scene and, for many years, it was the final word on solving divide-
and-conquer recurrences. We include the Master Theorem here because it is still
widely referenced in algorithms courses and you can use it without having to know
anything about integration.

Theorem 22.4.2 (Master Theorem). Let T be a recurrence of the form
n
T .n/ D aT C g.n/:
b

Case 1: If g.n/ D O nlogb .a/ for some constant > 0, then

T .n/ D ‚ nlogb .a/ :

Case 2: If g.n/ D ‚ nlogb .a/ logk .n/ for some constant k 0, then

T .n/ D ‚ nlogb .a/ logkC1 .n/ :

Case 3: If g.n/ D nlogb .a/C for some constant > 0 and ag.n=b/ < cg.n/
for some constant c < 1 and sufficiently large n, then

T .n/ D ‚.g.n//:

The Master Theorem can be proved by induction on n or, more easily, as a corol-
lary of Theorem 22.4.1. We will not include the details here.
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964 Chapter 22 Recurrences

22.5 A Feel for Recurrences
We’ve guessed and verified, plugged and chugged, found roots, computed integrals,
and solved linear systems and exponential equations. Now let’s step back and look
for some rules of thumb. What kinds of recurrences have what sorts of solutions?
Here are some recurrences we solved earlier:
Recurrence Solution
Towers of Hanoi Tn D 2Tn 1 C 1 Tn 2n
Merge Sort Tn D 2Tn=2 C n 1 Tn n log n
Hanoi variation Tn D 2Tn 1 C n Tn 2 2n p
Fibonacci Tn D Tn 1 C Tn 2 Tn .1:618 : : : /nC1 = 5

Notice that the recurrence equations for Towers of Hanoi and Merge Sort are some-
what similar, but the solutions are radically different. Merge Sorting n D 64 items
takes a few hundred comparisons, while moving n D 64 disks takes more than
1019 steps!
Each recurrence has one strength and one weakness. In the Towers of Hanoi,
we broke a problem of size n into two subproblem of size n 1 (which is large),
but needed only 1 additional step (which is small). In Merge Sort, we divided the
problem of size n into two subproblems of size n=2 (which is small), but needed
.n 1/ additional steps (which is large). Yet, Merge Sort is faster by a mile!
This suggests that generating smaller subproblems is far more important to al-
gorithmic speed than reducing the additional steps per recursive call. For example,
shifting to the variation of Towers of Hanoi increased the last term from C1 to Cn,
but the solution only doubled. And one of the two subproblems in the Fibonacci
recurrence is just slightly smaller than in Towers of Hanoi (size n 2 instead of
n 1). Yet the solution is exponentially smaller! More generally, linear recurrences
(which have big subproblems) typically have exponential solutions, while divide-
and-conquer recurrences (which have small subproblems) usually have solutions
bounded above by a polynomial.
All the examples listed above break a problem of size n into two smaller prob-
lems. How does the number of subproblems affect the solution? For example,
suppose we increased the number of subproblems in Towers of Hanoi from 2 to 3,
giving this recurrence:
Tn D 3Tn 1 C 1
This increases the root of the characteristic equation from 2 to 3, which raises the
solution exponentially, from ‚.2n / to ‚.3n /.
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22.5. A Feel for Recurrences 965

Divide-and-conquer recurrences are also sensitive to the number of subproblems.
For example, for this generalization of the Merge Sort recurrence:
T1 D 0
Tn D aTn=2 C n 1:
the Akra-Bazzi formula gives:
8
<‚.n/
ˆ for a < 2
Tn D ‚.n log n/ for a D 2
ˆ
‚.nlog a / for a > 2:
:

So the solution takes on three completely different forms as a goes from 1.99
to 2.01!
How do boundary conditions affect the solution to a recurrence? We’ve seen
that they are almost irrelevant for divide-and-conquer recurrences. For linear re-
currences, the solution is usually dominated by an exponential whose base is de-
termined by the number and size of subproblems. Boundary conditions matter
greatly only when they give the dominant term a zero coefficient, which changes
the asymptotic solution.
So now we have a rule of thumb! The performance of a recursive procedure is
usually dictated by the size and number of subproblems, rather than the amount
of work per recursive call or time spent at the base of the recursion. In particular,
if subproblems are smaller than the original by an additive factor, the solution is
most often exponential. But if the subproblems are only a fraction the size of the
original, then the solution is typically bounded by a polynomial.

Problems for Section 22.4
Homework Problems
Problem 22.1.
The running time of an algorithm A is described by the recurrence T .n/ D 7T .n=2/C
n2 . A competing algorithm A0 has a running time of T 0 .n/ D aT 0 .n=4/ C n2 . For
what values of a is A0 asymptotically faster than A?

Problem 22.2.
Use the Akra-Bazzi formula to find ‚./ asymptotic bounds for the following divide-
and-conquer recurrences. For each recurrence, T .1/ D 1 and T .n/ D ‚.1/ for all
“mcs” — 2017/3/10 — 22:22 — page 966 — #974

966 Chapter 22 Recurrences

constant n. State the value of p you get for each recurrence (which can be left in the
form of logs). Also, state the values of the ai ; bi ; and hi .n/ for each recurrence.

1. T .n/ D 3T .bn=3c/ C n.

2. T .n/ D 4T .bn=3c/ C n2 .

3. T .n/ D 3T .bn=4c/ C n.

4. T .n/ D T .bn=4c/ C T .bn=3c/ C n.

5. T .n/ D T .dn=4e/ C T .b3n=4c/ C n.
p
6. T .n/ D 2T .bn=4c/ C n.
p
7. T .n/ D 2T .bn=4c C 1/ C n.
p ˘
8. T .n/ D 2T . n=4 C n / C 1.
l m
9. T .n/ D 3T . n1=3 / C log3 n. (For this problem, T .2/ D 1.)
p j k
10. T .n/ D eT . n1=e / C ln n.

Class Problems
Problem 22.3.
We have devised an error-tolerant version of MergeSort. We call our exciting new
algorithm OverSort.
Here is how the new algorithm works. The input is a list of n distinct numbers.
If the list contains a single number, then there is nothing to do. If the list contains
two numbers, then we sort them with a single comparison. If the list contains more
than two numbers, then we perform the following sequence of steps.

We make a list containing the first 23 n numbers and sort it recursively.

We make a list containing the last 23 n numbers and sort it recursively.

We make a list containing the first 13 n numbers and the last 13 n numbers and
sort it recursively.

We merge the first and second lists, throwing out duplicates.

We merge this combined list with the third list, again throwing out duplicates.
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22.5. A Feel for Recurrences 967

The final, merged list is the output. What’s great is that because multiple copies
of each number are maintained, even if the sorter occasionally forgets about a num-
ber, OverSort can still output a complete, sorted list.
(a) Let T .n/ be the maximum number of comparisons that OverSort could use
to sort a list of n distinct numbers, assuming the sorter never forgets a number and
n is a power of 3. What is T .3/? Write a recurrence relation for T .n/. (Hint:
Merging a list of j distinct numbers and a list of k distinct numbers, and throwing
out duplicates of numbers that appear in both lists, requires j C k d comparisons,
when d > 0 is the number of duplicates.)

(b) Now we’re going to apply the Akra-Bazzi Theorem to find a ‚ bound on
T .n/. Begin by identifying the following constants and functions in the Akra-Bazzi
recurrence (22.4):

The constant k.
The constants ai .
The constants bi .
The functions hi .
The function g.
The constant p. You can leave p in terms of logarithms, but you’ll need a
rough estimate of its value later on.

(d) Does the condition jhi .x/j D O.x= log2 x/ hold?

(e) Determine a ‚ bound on T .n/ by integration.

Exam Problems
Problem 22.4.
Use the Akra-Bazzi formula to find ‚./ asymptotic bounds for the following recur-
rences. For each recurrence T .0/ D 1 and n 2 N.
(a) T .n/ D 2T .bn=4c/ C T .bn=3c/ C n
p ˘
(b) T .n/ D 4T n=2 C n C n2

(c) A society of devil-worshipers meets every week in a catacomb to initiate new
members. Members who have been in the society for two or more weeks initiate
four new members each and members who have been in the society for only one
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968 Chapter 22 Recurrences

week initiate one new member each. On week 0 there is one devil-worshiper. There
are two devil-worshipers on week 1.
Write a recurrence relation for the number of members D.n/ in the society on the
nth week.
You do NOT need to solve the recurrence. Be sure to include the base cases.
“mcs” — 2017/3/10 — 22:22 — page 969 — #977
“mcs” — 2017/3/10 — 22:22 — page 970 — #978
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971

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“mcs” — 2017/3/10 — 22:22 — page 976 — #984

976 Glossary of Symbols

Glossary of Symbols
symbol meaning
WWD is defined to be
end of proof symbol
¤ not equal
^ and, AND
_ or, OR
! implies, if . . . , then , IMPLIES
! state transition
:P; P not P , NOT.p/
! iff, equivalent, IFF
˚ xor, exclusive-or, XOR
9 exists
8 for all
2 is a member of, is in
is a (possibly =) subset of
6 is not a (possibly =) subset of
is a proper (not =) subset of
6 is not a proper (not =) subset of
[
S set union
i 2I Si union of sets Si where i ranges over set I of indices
\
T set intersection
i 2I Si intersection of sets Si where i ranges over set I of indices
; the empty set, f g
A complement of set A
set difference
pow.A/ powerset of set, A
AB Cartesian product of sets A and B
Sn Cartesian product of n copies of set S
Z integers
N; Z0 nonnegative integers
ZC ; NC positive integers
Z negative integers
Q rational numbers
R real numbers
C complex numbers
brc the floor of r: the greatest integer r
dre the ceiling of r: the least integer r
jrj the absolute value of a real number r
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977 Glossary of Symbols

symbol meaning
R.X / image of set X under binary relation R
R 1 inverse of binary relation R
R 1 .X / inverse image of set X under relation R
surj A surj B iff 9f W A ! B: f is a surjective function
inj A inj B iff 9R W A ! B: R is an injective total relation
bij A bij B iff 9f W A ! B: f is a bijection
Œ 1 in injective property of a relation
Œ 1 in surjective property of a relation
Œ 1 out function property of a relation
Œ 1 out total property of a relation
ŒD 1 out; D 1 in bijection relation
ı relational composition operator
the empty string/list
A the finite strings over alphabet A
A! the infinite strings over alphabet A
rev.s/ the reversal of string s
st concatenation of strings s; t; append.s; t /
#c .s/ number of occurrences of character c in string s
mjn integer m divides integer n; m is a factor of n
gcd greatest common divisor
log the base 2 logarithm, log2
ln the natural logarithm, loge
lcm least common multiple
.k::n/ fi 2 Z j k < i < ng
Œk::n/ fi 2 Z j k i < ng
.k::n fi 2 Z j k < i ng
Œk::n
P fi 2 Z j k i ng
Q i 2I ri sum of numbers ri where i ranges over set I of indices
i 2I ri product of numbers ri where i ranges over set I of indices
qcnt.n; d / quotient of n divided by d
rem.n; d / remainder of n divided by d
.mod n/ congruence modulo n
6 not congruent
Zn the ring of integers modulo n
Cn ; n addition and multiplication operations in Zn
Zn the set of numbers in Œ0; n/ relatively prime to n
.n/ Euler’s totient function WWDjZn j
hu ! vi directed edge from vertex u to vertex v
IdA identity relation on set A: aIdA a0 iff a D a0
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978 Glossary of Symbols

symbol meaning
R path relation of relation R; reflexive transitive closure of R
RC positive path relation of R; transitive closure of R
xg
fb merge of walk f with end vertex x
and walk g with start vertex x
fbg merge of walk f and walk g
where f’s end vertex equals g’s start vertex
hu—vi undirected edge connecting vertices u ¤ v
E.G/ the edges of graph G
V .G/ the vertices of graph G
Cn the length-n undirected cycle
Ln the length-n line graph
Kn the n-vertex complete graph
Hn the n-dimensional hypercube
L.G/ the “left” vertices of bipartite graph G
R.G/ the “right” vertices of bipartite graph G
Kn;m the complete bipartite graph with n left and m right vertices
.G/ chromatic number of simple Pgraph G
n
Hn the nth Harmonic number i D1 1= i
asymptotic equality
nŠ n factorial WWDn .n 1/ 2 1
n
m WWDnŠ=mŠ..n m/Š; the binomial coefficient
o./ asymptotic notation “little oh”
O./ asymptotic notation “big oh”
‚./ asymptotic notation “Theta”
./ asymptotic notation “big Omega”
!./ asymptotic notation “little omega”
PrŒA
probability of event A
Pr A j B conditional probability of A given B
S sample space
IA indicator variable for event A
PDF probability density function
CDF cumulative distribution function
ExŒR expectation of random variable R
ExŒR j A conditional expectation of R given event A
Ex2 ŒR abbreviation for .ExŒR/2
VarŒR variance of R
Var2 R

the square of the variance of R
R standard deviation of R
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Index
Cn , 459, 473 little omega, 586
IE , indicator for event E, 808 asymptotic relations, 594
K3;3 , 525 average, 819, 861
K5 , 525 average degree, 515
big omega, 585 axiom, 4, 9
‚./, 583 axiomatic method, 9
bij, 108 equivalence axioms, see equiva-
.G/, 468 lence (logic)
Ex2 ŒR, 869 ZFC axioms, 9, 269
inj, 108 axiom of choice, 270, 271, 286
, 580 axiom of extensionality, 98, 269
surj, 108 axiom of pairing, 269
k-edge connected, 476 foundation axiom, 270
k-way independent, 777, 810 infinity axiom, 269
r-permutation, 652 power set axiom, 269
IQ, 862, 868 replacement axiom, 269
icr , 482 subset axiom, 269
union axiom, 269
absolute value, 24, 300
Ackermann function, 217 Banach-Tarski Theorem, 271
acyclic, see also directed graph, 385 base case, see induction, see recur-
adjacency matrix, 381 sive data type
walk counting matrix, 382 Bayes’ Rule, 766
Adleman, Leonard, 334 Bayesian, 767
Akra-Bazzi formula, 959 Bayes Theorem, 779
Akra-Bazzi Theorem, 961, 967 Beneš, Václav, 440
annuity, 555, 556 Bernoulli distribution, 813
antichain, 392 Bernoulli variable, 869
antisymmetric relation, 396, 402 Bernoulli variables, 808
a posteriori, 766 Bezout’s lemma, 305
asymmetric relation, 395, 402, 424 biased, 913
asymptotic notation, 580 big O, see asymptotic notation
asymptotically smaller, o, little o, bijection, see binary relation
580 Binary GCD, 343
asymptotic equality, 574 binary relation, 103, 163, 384
big O, 581 bijection, 105, 254
big Omega, 585 image, 106
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980 INDEX

injection, 105 Chinese Appetizer problem, 863
product of relations, 398 Chinese Remainder Theorem, 359
properties, 393, 395, 401, 424 chromatic number, 468
relation on a set, 103 closed forms, 555
surjection, 105, 254 closed walk, 471
total, 105 CNF, see conjunctive form
binary trees, 229 codomain, 101, 103
Binet’s formula, 158, 694, 953 collateralized debt obligation, 909
binomial distribution, 813, 817, 872 Collatz conjecture, 216
Binomial Theorem, 620 collusion, 827, 829
binomial, 620 colorable, 468
binomial coefficient, 621 coloring, 444, 468
bin packing, 879 solid, 484
bipartite graph, 461, 465, 498, 538 combinatorial proof, 643, 681, 682
degree-constrained, 465 communcation net
birthday principle, 741 2-dimensional array, 447
Book Stacking Problem, 568 latency, 435
Boole’s inequality, 743 communication net, 375, 433
Boole, George, 48 2-dimensional array, 437
Boolean variable, 48 2-layer array, 447
Borel-Cantelli Lemma, 910 Beneš net, 440
bottleneck, 465 butterfly net, 438, 450
boundary conditions, 951 complete binary tree, 433
bridge, 534 congestion, 436
Brin, Sergey, 375, 923 for min-latency, 449, 450
buildup error, 478 diameter, 434
busy, 849, 850 latency
for min-congestion, 449, 450
Cantor, Georg, 254 Reasoner net, 450
cardinality, 107 routing, 434
Cartesian product, 100, 114 commutative ring, 323
CDO, 909 ring of formal power series, 701
ceiling, 976 ring of intergers modulo n, 323
chain, 389 complement, see st97
characteristic equation, 954 Complement Rule, 743
Chebyshev’s Theorem, 865, 877 complete bipartite graph, 525
Chebyshev Bound, 908 complete graph, 525
Chebyshev bound, 895 composite, see prime
one-sided, 898 composition
Chernoff Bound, 880
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981 INDEX

of functions, 103 cycle of a graph, 473
of relations, 384
Comprehension Axiom, 269 DAG, 295, see directed graph
concatenation, 209, 380 de Bruijn, Nicolaas, 404
concatentation, 242 degree d linear recurrence, 698
conditional expectation, 822 degree-constrained, 465, 633, 666
confidence, 905 De Morgan’s Laws, see also equiva-
confidence level, 878 lence (logic), 59
congestion, see communication net De Morgan’s Laws for Quantifiers, 66
congruence, see also modular arithemtic, derived variables, 175
318 deviation from the mean, 861
conjunctive form, 57, 81 diagonal argument, 262
conjunctive normal form, 58, 61 diameter, 513, 517
connected, 384, 473, 476 Die Hard, 300
k-edge, 476 Difference Rule, 743
k-vertex, 476 digraph, see directed graph
edge, 476 Dilworth’s lemma, 392
connected components, 474 directed graph, 375, 377
continuous faces, 529 cycle, 379
Continuum Hypothesis, 271 degree, 377
contrapositive, 13, 55 directed acyclic graph, 385
converse, 55 shortest path, 383
convex function, 886 walk
Convolution, 687 trail, 406
convolution, 687 walk counting matrix, 382
Convolution Counting Principle, 704 walk relation, 384, 393
corollary, 9 directed graphs, 295
countable, 257, 272, 275 discrete faces, 532
countably infinite, 257 disjunctive form, 57, 81
counter-model, 68 disjunctive normal form, 57, 61
coupon collector problem, 834 Distributive Law
cover, 464 sets, 99
covering edge, 420 divide-and-conquer, 959
critical path, 391, 415 divisibility, 297, 377, 396
cumulative distribution function, 811 Division Rule (counting), 613
cut edge, 476 Division Theorem, 299
cycle, see directed graph, simple graph, DNF, see disjunctive form
468, 471 domain, 101, 103
of length n, 459 domain of discourse, 66, 672
Dongles, 534
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982 INDEX

dot product, 898 exponential backoff, 817
Double or nothing, 736 exponential growth, 62
double summations, 577 extends F , 483
drawing, 525
face-down four-card trick, 667
edge factorial, 215, 556, 575
directed graph, 377 factoring, 309
simple graph, 454 fair, 825
edge connected, 476 Fast Exponentiation, 173
edge cover, 464 feasible spouse, 182
ellipsis, 31 Fermat, 7
empty graph, 468 Fermat’s Last Theorem, 7
empty relation, 396, 419, 422, 424, Fermat’s Little Theorem, 332
425, 429 Fibonacci numbers, 137
empty string, 88 floor, 976
endpoints, 454 Floyd, Robert W., 169, 212
equivalence (logic), 53, 58 flush, 669
axioms, 58 formal power series, 701
equivalence relation, 319, 399, 402 Four Color Theorem, 6
equivalence class, 400, 430 Four Step Method, 749
Euclid, 8, 298 four-step method, 784
Euclid’s algorithm, 303 Frege, Gotlob, 267
Euler, 6 frequentist, 767
formula, 536 function, 101, 105
Euler’s function, 329 partial, 102
Euler’s constant, 572 total, 102
Euler’s formula, 543 Fundamental Theorem of Arithmetic,
Euler’s Theorem, 329, 354 see Unique Factorization The-
for Zn , 331 orem
Euler’s theorem, 358, 366
Euler tour, 405 Gale, 183
event, 727, 742 Gauss, 311, 317, 318
events, 807 gcd, see greatest common divisor
exclusive-or, 49 general binomial density function, 818
existential quantification, 64 Generalized Pigeonhole Principle, 628
expectation, 819 Generalized Product Rule, 610
expected absolute deviation, 849, 897, generating function, 703, 710
897 Generating Functions, 683
expected return, 825 generating functions, 554
expected value, 722, 819, 820, 861 geometric distribution, 825, 825
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983 INDEX

geometric series, 683 identity relation, 385, 396, 424, 430
geometric sum, 555 image, 102, 106, 465
Gödel, Kurt, 271 inverse image, 106
going broke, 913 implication, 11, 13, 50
Goldbach, 7 false hypothesis, 50
Goldbach’s Conjecture, see also prime, Inclusion-Exclusion, 636, 638
64, 310 inclusion-exclusion for probabilities,
golden ratio, 304, 342 743
good count, 240, 712, 713 Inclusion-Exclusion Rule, 636
Google, 913 incompleteness theorem, 271
googol, 44 independence, 772
graph independent, 871
bipartite, 461 independent random variables, 809
coloring problem, 468 indicator random variable, 808
diameter, 434 indicator variable, 820, 869, 894
matching, 464 indicator variables, 810
perfect, 464 indirect proof, see proof by contra-
valid coloring, 468 diction
width, 501 induction, 128, 143, 226
graph coloring, 468 induciton hypothesis, 130
gray edge, 484 inductive step, 130
Greatest Common Divisor, 309 structural induction, 207, 209
greatest common divisor, 232, 297, inference rules, 10
302 infinity, see countable, Mapping Rules,
guess-and-verify, 941 set, set theory
inhomogeneous linear recurrence, 955
half-adder, 71 injection, see binary relation
Hall’s Matching Theorem, 462 Integral Method, 589
Hall’s Theorem, 465, 666 intended profit, 913
Hall’s theorem, 498 interest rate, 588
Halting Problem, 263 intersection, 97
Hamiltonian Cycle Problem, 509 invariant, 163, 165, 167, 301
Handshaking Lemma, 457 Invariant Principle, 168
Hardy, G. H., 297 inverse (multiplicative), 325
harmonic number, 571 irreducible, 348
harmonic numbers, 578 irreflexive relation, 394, 401, 424
Hat-Check problem, 863 isomorphic, 397, 423, 548
homogeneous linear recurrence, 951
homogeneous solution, 955 k-combination, see Sbset Split Theo-
hypercube, 510 rem620
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984 INDEX

k-to-1 function, 613 merging vertices, 544
Meyer, Albert R, 169
latency, see communcation net minimal, 428
Latin square, 496 minimal (graphing), 388
lattice basis reduction, 630 minimum (graphing), 388
Law of Large Numbers, 877 minimum weight spanning tree, 482
leaf, 478 minor, 544
lemma, 9 modular arithmetic, 318, 320
length-n cycle, 459 congruence, 318
length-n walk relation, 384 modulo n, see modular arithmetic
length of a walk, 471 modus ponens, 10
Let’s Make a Deal, 748 Monty Hall Problem, 723
limit superior, 581 Multinomial Theorem, 681
linear combination (integers), 299 multinomial coefficient, 618
Linearity of Expectation, 830, 831 multiplicative inverse, 325, 701
linear relation, 402 Murphy’s Law, 887, 908
literal, 852 mutual independence, 871
little o, see asymptotic notation mutually independent, 774, 810, 872,
load balancing, 879, 883 881
lower bound, 34 mutually recursive, 710
Mapping Rules, 627 neighbors, 465, 491
for finite sets, 109 nonconstructive proof, 630
bijection, 109, 605 nondecreasing, 564
for infinite sets, 254 nonincreasing, 564
Markov’s bound, 898 non-unique factorization, 348
Markov’s Theorem, 862, 890 norm, 300, 348, 898
Markov Bound, 908 number theory, 297
Markov bound, 885
matching, 462, 464 O (big O), see asymptotic notation
matching birthdays, 872 o (little o), see asymptotic notation
matching condition, 463 odds, 779
matrix multiplication, 582 optimal spouse, 182
maximum dilation, 936 order, 951
maximum element, 389 ordinals, 256, 290
max-value, 250 outcome, 725, 742
mean, 819 outside face, 529
mean square deviation, 865
Menger, 477 Page, Larry, 375, 923
Merge Sort, 946, 966 page rank, 925, 927
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985 INDEX

pair, 288 Prime Number Theorem, 310, 330
pairwise independence, 871 relatively prime, 326
pairwise independent, 777, 872, 873 Prime Factorization Theorem, see Unique
Pairwise Independent Additivity, 872 Factorization Theorem
Pairwise Independent Sampling, 876, probability density function, 811
904 probability density function,, 810
partial correctness, 172 probability function, 742, 790
partial fractions, 691 probability of an event, 742
partial order, 393, 423 probability space, 742
strict, 394, 402 Product Rule, 761
weak, 395, 402 product rule, 804
particular solution, 956 Product Rule (counting), 607
partition, 391 Product Rule (generating functions),
Pascal’s Triangle Identity, 643 686
path, 843 proof, 9, 17
perfect graph, 464 proof by contradiction, 16
perfect number, 298, 338 proposition, 4, 5
permutation, 550, 612 propositional variable, 48
Perturbation Method, 557 public key cryptography, 334
perturbation method, 684 Pulverizer, 305, 339, 340
pessimal spouse, 182 pure first-order formula, 268
Pick-4, 882 P vs. NP, 63
planar drawing, 525 Pythagoreans, 541
planar embedding, 531, 532, 548
planar graph, 529 quicksort, 817
planar graphs, 471 quotient, 300
planar subgraph, 539
randomized algorithm, 817
plug-and-chug, 941
random sample, 906
pointwise, 102
random sampling, 904
Polyhedra, 541
random variable, 807
polyhedron, 541
random variables, 808
polynomial time, 62, 309, 461
random walk, 843, 926
population size, 877
Random Walks, 913
power, 778
range, 102
predicate, 8
reachable, 167
pre-MST, 483
reciprocal, 701
preserved invariant, see invariant, 165
recognizes, 281
preserved under isomorphism, 460
recurrence, 941
prime, 5, 28, 309
recursive data type, 207
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986 INDEX

ambiguity, 214 infinite set, 253
reflexive relation, 394, 401 multiset, 95
register allocation, 499 power set, 97, 259
regular, 493 set builder notation, 98
Regular graphs, 466 subset, 96, 398, 423
Regularity Axiom, 270 set difference, 97
regular polyhedron, 541 set theory, 268
relation, see binary relation Shamir, Adi, 334
relaxed, 849, 850 Shapley, 183
remainder, 300 significance, 778
remainder arithmetic, see modular arith- simple graph, 453, 454
metic complete graph, 457
Riemann Hypothesis, 330 degree, 454
ring, see commutative ring empty graph, 458
ripple-carry, 72, 148 simple graphs, 295
Rivest, Ronald, 334 Simpson’s Paradox, 770
root mean square, 867 sink, 932
routing, see communication net smallest counterexample, 31
RSA, 297, 334, 370 solid coloring, 484
Rubin, Herman, 884 spanning subgraph, 481
ruined, 913 spanning tree, 481
Russel, Bertrand, 267 square modulo N , 371
Russell’s Paradox, 267, 270 Square Multiple Rule, 870
Russell, Bertrand, 271 square root, 360
square root of s modulo N , 371
sample space, 725, 742 stable distributions, 932
sampling, 904 stable matching, 178
satisfiability, 56 standard deviation, 14, 866, 868, 871
SAT, 62, 336 star graph, 469
SAT-solver, 62 state machine, 33, 163, 301, 304
satisfiable, 852 execution, 167
Schröder-Bernstein, 255, 275 stationary distribution, 927
self-loop, 455 Stirling’s Formula, 575
self-loops, 379 strict, 254
sequence, 100 strictly decreasing, 175, 564
empty sequence, 100 strictly increasing, 564
set, 95 strict subset, 96
combining sets, 96 strong induction, 136, 143
complement, 97 strongly connected, 936
covering, 464
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987 INDEX

structure, 295 universal quantification, 64
Subset Split Rule, 618 unlucky, 849, 850
summation notation, 31, 215 upper bound, 34
Sum Rule, 743
Sum Rule (counting), 608 valid coloring, 468
surjection, see binary relation validity (logic), 56, 62, 67
symmetric, 295, 936 value of an annuity, 558
symmetric relation, 401, 453 variance, 865, 875, 894
vertex
tails, 817 directed graph, 377
tails of the distribution, 817 simple graph, 454
termination (state machine), 172 vertex connected, 476
The Bookkeeper Rule, 618
theorem, 9 walk, see directed graph, simple graph,
topological sort, 387 509
total expectation, 823 walk in a simple graph, 471
totient function, 329 Weak Law of Large Numbers, 877,
tournament digraph, 408, 410, 426, 904
752, 846 weakly connected, 406
Towers of Hanoi, 709, 943 weakly decreasing, 158, 175, 311, 564
transitive, 738 weakly increasing, 175, 564
transitive relation, 393, 402, 424 weight of a graph, 482
tree diagram, 725, 784 well founded, 287, 428
Triangle Identity, 643 Well Ordering Principle, 29, 143, 156,
truth table, 48 172
Turing, 314, 328 well ordered set, 33
Turing’s code, 323, 328 winnings, 825
Twin Prime Conjecture, 309 wrap, 713
type-checking, see Halting Problem
Zermelo-Fraenkel Set Theory, 268, 271
unbiased binomial distribution, 817, ZFC, see Zermello-Fraenkel Set The-
849 ory
unbiased game, 913
unbounded Gambler’s ruin, 922
uncountable, 257, 280, 285
uniform, 736, 744, 813
uniform distribution, 813
union, 96
Union Bound, 744
Unique Factorization Theorem, 32, 313