Sets¶

Set¶

class sympy.sets.sets.Set(*args)[source]¶

The base class for any kind of set.

This is not meant to be used directly as a container of items. It does not behave like the builtin set; see FiniteSet for that.

Real intervals are represented by the Interval class and unions of sets by the Union class. The empty set is represented by the EmptySet class and available as a singleton as S.EmptySet.

Attributes

is_Complement
is_EmptySet
is_Intersection
is_UniversalSet

property boundary¶

The boundary or frontier of a set

A point x is on the boundary of a set S if

x is in the closure of S. I.e. Every neighborhood of x contains a point in S.
x is not in the interior of S. I.e. There does not exist an open set centered on x contained entirely within S.

There are the points on the outer rim of S. If S is open then these points need not actually be contained within S.

For example, the boundary of an interval is its start and end points. This is true regardless of whether or not the interval is open.

Examples

>>> from sympy import Interval
>>> Interval(0, 1).boundary
{0, 1}
>>> Interval(0, 1, True, False).boundary
{0, 1}

property closure¶

Property method which returns the closure of a set. The closure is defined as the union of the set itself and its boundary.

Examples

>>> from sympy import S, Interval
>>> S.Reals.closure
Reals
>>> Interval(0, 1).closure
Interval(0, 1)

complement(universe)[source]¶

The complement of ‘self’ w.r.t the given universe.

Examples

>>> from sympy import Interval, S
>>> Interval(0, 1).complement(S.Reals)
Union(Interval.open(-oo, 0), Interval.open(1, oo))

>>> Interval(0, 1).complement(S.UniversalSet)
UniversalSet() \ Interval(0, 1)

contains(other)[source]¶

Returns True if ‘other’ is contained in ‘self’ as an element.

As a shortcut it is possible to use the ‘in’ operator:

Examples

>>> from sympy import Interval
>>> Interval(0, 1).contains(0.5)
True
>>> 0.5 in Interval(0, 1)
True

property inf¶

The infimum of ‘self’

Examples

>>> from sympy import Interval, Union
>>> Interval(0, 1).inf
0
>>> Union(Interval(0, 1), Interval(2, 3)).inf
0

property interior¶

Property method which returns the interior of a set. The interior of a set S consists all points of S that do not belong to the boundary of S.

Examples

>>> from sympy import Interval
>>> Interval(0, 1).interior
Interval.open(0, 1)
>>> Interval(0, 1).boundary.interior
EmptySet()

intersect(other)[source]¶

Returns the intersection of ‘self’ and ‘other’.

>>> from sympy import Interval

>>> Interval(1, 3).intersect(Interval(1, 2))
Interval(1, 2)

>>> from sympy import imageset, Lambda, symbols, S
>>> n, m = symbols('n m')
>>> a = imageset(Lambda(n, 2*n), S.Integers)
>>> a.intersect(imageset(Lambda(m, 2*m + 1), S.Integers))
EmptySet()

intersection(other)[source]¶: Alias for intersect()

property is_closed¶

A property method to check whether a set is closed. A set is closed if it’s complement is an open set.

Examples

>>> from sympy import Interval
>>> Interval(0, 1).is_closed
True

is_disjoint(other)[source]¶

Returns True if ‘self’ and ‘other’ are disjoint

References

R458: http://en.wikipedia.org/wiki/Disjoint_sets

Examples

>>> from sympy import Interval
>>> Interval(0, 2).is_disjoint(Interval(1, 2))
False
>>> Interval(0, 2).is_disjoint(Interval(3, 4))
True

property is_open¶

Property method to check whether a set is open. A set is open if and only if it has an empty intersection with its boundary.

Examples

>>> from sympy import S
>>> S.Reals.is_open
True

is_proper_subset(other)[source]¶

Returns True if ‘self’ is a proper subset of ‘other’.

Examples

>>> from sympy import Interval
>>> Interval(0, 0.5).is_proper_subset(Interval(0, 1))
True
>>> Interval(0, 1).is_proper_subset(Interval(0, 1))
False

is_proper_superset(other)[source]¶

Returns True if ‘self’ is a proper superset of ‘other’.

Examples

>>> from sympy import Interval
>>> Interval(0, 1).is_proper_superset(Interval(0, 0.5))
True
>>> Interval(0, 1).is_proper_superset(Interval(0, 1))
False

is_subset(other)[source]¶

Returns True if ‘self’ is a subset of ‘other’.

Examples

>>> from sympy import Interval
>>> Interval(0, 0.5).is_subset(Interval(0, 1))
True
>>> Interval(0, 1).is_subset(Interval(0, 1, left_open=True))
False

is_superset(other)[source]¶

Returns True if ‘self’ is a superset of ‘other’.

Examples

>>> from sympy import Interval
>>> Interval(0, 0.5).is_superset(Interval(0, 1))
False
>>> Interval(0, 1).is_superset(Interval(0, 1, left_open=True))
True

isdisjoint(other)[source]¶: Alias for is_disjoint()

issubset(other)[source]¶: Alias for is_subset()

issuperset(other)[source]¶: Alias for is_superset()

property measure¶

The (Lebesgue) measure of ‘self’

Examples

>>> from sympy import Interval, Union
>>> Interval(0, 1).measure
1
>>> Union(Interval(0, 1), Interval(2, 3)).measure
2

powerset()[source]¶

Find the Power set of ‘self’.

References

R459: http://en.wikipedia.org/wiki/Power_set

Examples

>>> from sympy import FiniteSet, EmptySet
>>> A = EmptySet()
>>> A.powerset()
{EmptySet()}
>>> A = FiniteSet(1, 2)
>>> a, b, c = FiniteSet(1), FiniteSet(2), FiniteSet(1, 2)
>>> A.powerset() == FiniteSet(a, b, c, EmptySet())
True

property sup¶

The supremum of ‘self’

Examples

>>> from sympy import Interval, Union
>>> Interval(0, 1).sup
1
>>> Union(Interval(0, 1), Interval(2, 3)).sup
3

symmetric_difference(other)[source]¶

Returns symmetric difference of \(self\) and \(other\).

References

R460: https://en.wikipedia.org/wiki/Symmetric_difference

Examples

>>> from sympy import Interval, S
>>> Interval(1, 3).symmetric_difference(S.Reals)
Union(Interval.open(-oo, 1), Interval.open(3, oo))
>>> Interval(1, 10).symmetric_difference(S.Reals)
Union(Interval.open(-oo, 1), Interval.open(10, oo))

>>> from sympy import S, EmptySet
>>> S.Reals.symmetric_difference(EmptySet())
Reals

union(other)[source]¶

Returns the union of ‘self’ and ‘other’.

Examples

As a shortcut it is possible to use the ‘+’ operator:

>>> from sympy import Interval, FiniteSet
>>> Interval(0, 1).union(Interval(2, 3))
Union(Interval(0, 1), Interval(2, 3))
>>> Interval(0, 1) + Interval(2, 3)
Union(Interval(0, 1), Interval(2, 3))
>>> Interval(1, 2, True, True) + FiniteSet(2, 3)
Union(Interval.Lopen(1, 2), {3})

Similarly it is possible to use the ‘-‘ operator for set differences:

>>> Interval(0, 2) - Interval(0, 1)
Interval.Lopen(1, 2)
>>> Interval(1, 3) - FiniteSet(2)
Union(Interval.Ropen(1, 2), Interval.Lopen(2, 3))

sympy.sets.sets.imageset(*args)[source]¶

Return an image of the set under transformation f.

If this function can’t compute the image, it returns an unevaluated ImageSet object.

\[{ f(x) | x \in self }\]

See also

sympy.sets.fancysets.ImageSet

Examples

>>> from sympy import S, Interval, Symbol, imageset, sin, Lambda
>>> from sympy.abc import x, y

>>> imageset(x, 2*x, Interval(0, 2))
Interval(0, 4)

>>> imageset(lambda x: 2*x, Interval(0, 2))
Interval(0, 4)

>>> imageset(Lambda(x, sin(x)), Interval(-2, 1))
ImageSet(Lambda(x, sin(x)), Interval(-2, 1))

>>> imageset(sin, Interval(-2, 1))
ImageSet(Lambda(x, sin(x)), Interval(-2, 1))
>>> imageset(lambda y: x + y, Interval(-2, 1))
ImageSet(Lambda(_x, _x + x), Interval(-2, 1))

Expressions applied to the set of Integers are simplified to show as few negatives as possible and linear expressions are converted to a canonical form. If this is not desirable then the unevaluated ImageSet should be used.

>>> imageset(x, -2*x + 5, S.Integers)
ImageSet(Lambda(x, 2*x + 1), Integers)

Elementary Sets¶

Interval¶

class sympy.sets.sets.Interval(start, end, left_open=False, right_open=False)[source]¶

Represents a real interval as a Set.

Usage:

Returns an interval with end points “start” and “end”.

For left_open=True (default left_open is False) the interval will be open on the left. Similarly, for right_open=True the interval will be open on the right.

Notes

Only real end points are supported
Interval(a, b) with a > b will return the empty set
Use the evalf() method to turn an Interval into an mpmath ‘mpi’ interval instance

References

R461: http://en.wikipedia.org/wiki/Interval_%28mathematics%29

Examples

>>> from sympy import Symbol, Interval
>>> Interval(0, 1)
Interval(0, 1)
>>> Interval.Ropen(0, 1)
Interval.Ropen(0, 1)
>>> Interval.Ropen(0, 1)
Interval.Ropen(0, 1)
>>> Interval.Lopen(0, 1)
Interval.Lopen(0, 1)
>>> Interval.open(0, 1)
Interval.open(0, 1)

>>> a = Symbol('a', real=True)
>>> Interval(0, a)
Interval(0, a)

Attributes

is_Complement
is_EmptySet
is_Intersection
is_UniversalSet

classmethod Lopen(a, b)[source]¶: Return an interval not including the left boundary.

classmethod Ropen(a, b)[source]¶: Return an interval not including the right boundary.

as_relational(x)[source]¶: Rewrite an interval in terms of inequalities and logic operators.

property end¶

The right end point of ‘self’.

This property takes the same value as the ‘sup’ property.

Examples

>>> from sympy import Interval
>>> Interval(0, 1).end
1

property is_left_unbounded¶: Return True if the left endpoint is negative infinity.

property is_right_unbounded¶: Return True if the right endpoint is positive infinity.

property left¶

The left end point of ‘self’.

This property takes the same value as the ‘inf’ property.

Examples

>>> from sympy import Interval
>>> Interval(0, 1).start
0

property left_open¶

True if ‘self’ is left-open.

Examples

>>> from sympy import Interval
>>> Interval(0, 1, left_open=True).left_open
True
>>> Interval(0, 1, left_open=False).left_open
False

classmethod open(a, b)[source]¶: Return an interval including neither boundary.

property right¶

The right end point of ‘self’.

This property takes the same value as the ‘sup’ property.

Examples

>>> from sympy import Interval
>>> Interval(0, 1).end
1

property right_open¶

True if ‘self’ is right-open.

Examples

>>> from sympy import Interval
>>> Interval(0, 1, right_open=True).right_open
True
>>> Interval(0, 1, right_open=False).right_open
False

property start¶

The left end point of ‘self’.

This property takes the same value as the ‘inf’ property.

Examples

>>> from sympy import Interval
>>> Interval(0, 1).start
0

FiniteSet¶

class sympy.sets.sets.FiniteSet(*args, **kwargs)[source]¶

Represents a finite set of discrete numbers

References

R462: http://en.wikipedia.org/wiki/Finite_set

Examples

>>> from sympy import FiniteSet
>>> FiniteSet(1, 2, 3, 4)
{1, 2, 3, 4}
>>> 3 in FiniteSet(1, 2, 3, 4)
True

>>> members = [1, 2, 3, 4]
>>> f = FiniteSet(*members)
>>> f
{1, 2, 3, 4}
>>> f - FiniteSet(2)
{1, 3, 4}
>>> f + FiniteSet(2, 5)
{1, 2, 3, 4, 5}

Attributes

is_Complement
is_EmptySet
is_Intersection
is_UniversalSet

as_relational(symbol)[source]¶: Rewrite a FiniteSet in terms of equalities and logic operators.

Compound Sets¶

Union¶

class sympy.sets.sets.Union(*args, **kwargs)[source]¶

Represents a union of sets as a Set.

See also

Intersection

References

R463: http://en.wikipedia.org/wiki/Union_%28set_theory%29

Examples

>>> from sympy import Union, Interval
>>> Union(Interval(1, 2), Interval(3, 4))
Union(Interval(1, 2), Interval(3, 4))

The Union constructor will always try to merge overlapping intervals, if possible. For example:

>>> Union(Interval(1, 2), Interval(2, 3))
Interval(1, 3)

Attributes

is_Complement
is_EmptySet
is_Intersection
is_UniversalSet

as_relational(symbol)[source]¶: Rewrite a Union in terms of equalities and logic operators.

Intersection¶

class sympy.sets.sets.Intersection(*args, **kwargs)[source]¶

Represents an intersection of sets as a Set.

See also

Union

References

R464: http://en.wikipedia.org/wiki/Intersection_%28set_theory%29

Examples

>>> from sympy import Intersection, Interval
>>> Intersection(Interval(1, 3), Interval(2, 4))
Interval(2, 3)

We often use the .intersect method

>>> Interval(1,3).intersect(Interval(2,4))
Interval(2, 3)

Attributes

is_Complement
is_EmptySet
is_UniversalSet

as_relational(symbol)[source]¶: Rewrite an Intersection in terms of equalities and logic operators

ProductSet¶

class sympy.sets.sets.ProductSet(*sets, **assumptions)[source]¶

Represents a Cartesian Product of Sets.

Returns a Cartesian product given several sets as either an iterable or individual arguments.

Can use ‘*’ operator on any sets for convenient shorthand.

Notes

Passes most operations down to the argument sets
Flattens Products of ProductSets

References

R465: http://en.wikipedia.org/wiki/Cartesian_product

Examples

>>> from sympy import Interval, FiniteSet, ProductSet
>>> I = Interval(0, 5); S = FiniteSet(1, 2, 3)
>>> ProductSet(I, S)
Interval(0, 5) x {1, 2, 3}

>>> (2, 2) in ProductSet(I, S)
True

>>> Interval(0, 1) * Interval(0, 1) # The unit square
Interval(0, 1) x Interval(0, 1)

>>> coin = FiniteSet('H', 'T')
>>> set(coin**2)
{(H, H), (H, T), (T, H), (T, T)}

Attributes

is_Complement
is_EmptySet
is_Intersection
is_UniversalSet

property is_iterable¶

A property method which tests whether a set is iterable or not. Returns True if set is iterable, otherwise returns False.

Examples

>>> from sympy import FiniteSet, Interval, ProductSet
>>> I = Interval(0, 1)
>>> A = FiniteSet(1, 2, 3, 4, 5)
>>> I.is_iterable
False
>>> A.is_iterable
True

Complement¶

class sympy.sets.sets.Complement(a, b, evaluate=True)[source]¶

Represents the set difference or relative complement of a set with another set.

\(A - B = \{x \in A| x \\notin B\}\)

See also

Intersection, Union

References

R466: http://mathworld.wolfram.com/ComplementSet.html

Examples

>>> from sympy import Complement, FiniteSet
>>> Complement(FiniteSet(0, 1, 2), FiniteSet(1))
{0, 2}

Attributes

is_EmptySet
is_Intersection
is_UniversalSet

static reduce(A, B)[source]¶: Simplify a Complement.

Singleton Sets¶

EmptySet¶

class sympy.sets.sets.EmptySet(*args, **kwargs)[source]¶

Represents the empty set. The empty set is available as a singleton as S.EmptySet.

See also

UniversalSet

References

R467: http://en.wikipedia.org/wiki/Empty_set

Examples

>>> from sympy import S, Interval
>>> S.EmptySet
EmptySet()

>>> Interval(1, 2).intersect(S.EmptySet)
EmptySet()

Attributes

is_Complement
is_Intersection
is_UniversalSet

UniversalSet¶

class sympy.sets.sets.UniversalSet(*args, **kwargs)[source]¶

Represents the set of all things. The universal set is available as a singleton as S.UniversalSet

See also

EmptySet

References

R468: http://en.wikipedia.org/wiki/Universal_set

Examples

>>> from sympy import S, Interval
>>> S.UniversalSet
UniversalSet()

>>> Interval(1, 2).intersect(S.UniversalSet)
Interval(1, 2)

Attributes

is_Complement
is_EmptySet
is_Intersection

Special Sets¶

Naturals¶

class sympy.sets.fancysets.Naturals(*args, **kwargs)[source]¶

Represents the natural numbers (or counting numbers) which are all positive integers starting from 1. This set is also available as the Singleton, S.Naturals.

See also

Naturals0: non-negative integers (i.e. includes 0, too)
Integers: also includes negative integers

Examples

>>> from sympy import S, Interval, pprint
>>> 5 in S.Naturals
True
>>> iterable = iter(S.Naturals)
>>> next(iterable)
1
>>> next(iterable)
2
>>> next(iterable)
3
>>> pprint(S.Naturals.intersect(Interval(0, 10)))
{1, 2, ..., 10}

Attributes

is_Complement
is_EmptySet
is_Intersection
is_UniversalSet

Naturals0¶

class sympy.sets.fancysets.Naturals0(*args, **kwargs)[source]¶

Represents the whole numbers which are all the non-negative integers, inclusive of zero.

See also

Naturals: positive integers; does not include 0
Integers: also includes the negative integers

Attributes

is_Complement
is_EmptySet
is_Intersection
is_UniversalSet

Integers¶

class sympy.sets.fancysets.Integers(*args, **kwargs)[source]¶

Represents all integers: positive, negative and zero. This set is also available as the Singleton, S.Integers.

See also

Naturals0: non-negative integers
Integers: positive and negative integers and zero

Examples

>>> from sympy import S, Interval, pprint
>>> 5 in S.Naturals
True
>>> iterable = iter(S.Integers)
>>> next(iterable)
0
>>> next(iterable)
1
>>> next(iterable)
-1
>>> next(iterable)
2

>>> pprint(S.Integers.intersect(Interval(-4, 4)))
{-4, -3, ..., 4}

Attributes

is_Complement
is_EmptySet
is_Intersection
is_UniversalSet

ImageSet¶

class sympy.sets.fancysets.ImageSet(flambda, *sets)[source]¶

Image of a set under a mathematical function. The transformation must be given as a Lambda function which has as many arguments as the elements of the set upon which it operates, e.g. 1 argument when acting on the set of integers or 2 arguments when acting on a complex region.

This function is not normally called directly, but is called from \(imageset\).

See also

sympy.sets.sets.imageset

Examples

>>> from sympy import Symbol, S, pi, Dummy, Lambda
>>> from sympy.sets.sets import FiniteSet, Interval
>>> from sympy.sets.fancysets import ImageSet

>>> x = Symbol('x')
>>> N = S.Naturals
>>> squares = ImageSet(Lambda(x, x**2), N) # {x**2 for x in N}
>>> 4 in squares
True
>>> 5 in squares
False

>>> FiniteSet(0, 1, 2, 3, 4, 5, 6, 7, 9, 10).intersect(squares)
{1, 4, 9}

>>> square_iterable = iter(squares)
>>> for i in range(4):
...     next(square_iterable)
1
4
9
16

If you want to get value for \(x\) = 2, 1/2 etc. (Please check whether the \(x\) value is in \(base_set\) or not before passing it as args)

>>> squares.lamda(2)
4
>>> squares.lamda(S(1)/2)
1/4

>>> n = Dummy('n')
>>> solutions = ImageSet(Lambda(n, n*pi), S.Integers) # solutions of sin(x) = 0
>>> dom = Interval(-1, 1)
>>> dom.intersect(solutions)
{0}

Attributes

is_Complement
is_EmptySet
is_Intersection
is_UniversalSet

Range¶

class sympy.sets.fancysets.Range(*args)[source]¶

Represents a range of integers. Can be called as Range(stop), Range(start, stop), or Range(start, stop, step); when stop is not given it defaults to 1.

\(Range(stop)\) is the same as \(Range(0, stop, 1)\) and the stop value (juse as for Python ranges) is not included in the Range values.

>>> from sympy import Range
>>> list(Range(3))
[0, 1, 2]

The step can also be negative:

>>> list(Range(10, 0, -2))
[10, 8, 6, 4, 2]

The stop value is made canonical so equivalent ranges always have the same args:

>>> Range(0, 10, 3)
Range(0, 12, 3)

Infinite ranges are allowed. If the starting point is infinite, then the final value is stop - step. To iterate such a range, it needs to be reversed:

>>> from sympy import oo
>>> r = Range(-oo, 1)
>>> r[-1]
0
>>> next(iter(r))
Traceback (most recent call last):
...
ValueError: Cannot iterate over Range with infinite start
>>> next(iter(r.reversed))
0

Although Range is a set (and supports the normal set operations) it maintains the order of the elements and can be used in contexts where \(range\) would be used.

>>> from sympy import Interval
>>> Range(0, 10, 2).intersect(Interval(3, 7))
Range(4, 8, 2)
>>> list(_)
[4, 6]

Athough slicing of a Range will always return a Range – possibly empty – an empty set will be returned from any intersection that is empty:

>>> Range(3)[:0]
Range(0, 0, 1)
>>> Range(3).intersect(Interval(4, oo))
EmptySet()
>>> Range(3).intersect(Range(4, oo))
EmptySet()

Attributes

is_Complement
is_EmptySet
is_Intersection
is_UniversalSet

property reversed¶

Return an equivalent Range in the opposite order.

Examples

>>> from sympy import Range
>>> Range(10).reversed
Range(9, -1, -1)

ComplexRegion¶

class sympy.sets.fancysets.ComplexRegion(sets, polar=False)[source]¶

Represents the Set of all Complex Numbers. It can represent a region of Complex Plane in both the standard forms Polar and Rectangular coordinates.

Polar Form Input is in the form of the ProductSet or Union of ProductSets of the intervals of r and theta, & use the flag polar=True.

Z = {z in C | z = r*[cos(theta) + I*sin(theta)], r in [r], theta in [theta]}

Rectangular Form Input is in the form of the ProductSet or Union of ProductSets of interval of x and y the of the Complex numbers in a Plane. Default input type is in rectangular form.

Z = {z in C | z = x + I*y, x in [Re(z)], y in [Im(z)]}

See also

Reals

Examples

>>> from sympy.sets.fancysets import ComplexRegion
>>> from sympy.sets import Interval
>>> from sympy import S, I, Union
>>> a = Interval(2, 3)
>>> b = Interval(4, 6)
>>> c = Interval(1, 8)
>>> c1 = ComplexRegion(a*b)  # Rectangular Form
>>> c1
ComplexRegion(Interval(2, 3) x Interval(4, 6), False)

c1 represents the rectangular region in complex plane surrounded by the coordinates (2, 4), (3, 4), (3, 6) and (2, 6), of the four vertices.

>>> c2 = ComplexRegion(Union(a*b, b*c))
>>> c2
ComplexRegion(Union(Interval(2, 3) x Interval(4, 6), Interval(4, 6) x Interval(1, 8)), False)

c2 represents the Union of two rectangular regions in complex plane. One of them surrounded by the coordinates of c1 and other surrounded by the coordinates (4, 1), (6, 1), (6, 8) and (4, 8).

>>> 2.5 + 4.5*I in c1
True
>>> 2.5 + 6.5*I in c1
False

>>> r = Interval(0, 1)
>>> theta = Interval(0, 2*S.Pi)
>>> c2 = ComplexRegion(r*theta, polar=True)  # Polar Form
>>> c2  # unit Disk
ComplexRegion(Interval(0, 1) x Interval.Ropen(0, 2*pi), True)

c2 represents the region in complex plane inside the Unit Disk centered at the origin.

>>> 0.5 + 0.5*I in c2
True
>>> 1 + 2*I in c2
False

>>> unit_disk = ComplexRegion(Interval(0, 1)*Interval(0, 2*S.Pi), polar=True)
>>> upper_half_unit_disk = ComplexRegion(Interval(0, 1)*Interval(0, S.Pi), polar=True)
>>> intersection = unit_disk.intersect(upper_half_unit_disk)
>>> intersection
ComplexRegion(Interval(0, 1) x Interval(0, pi), True)
>>> intersection == upper_half_unit_disk
True

Attributes

is_Complement
is_EmptySet
is_Intersection
is_UniversalSet

property a_interval¶

Return the union of intervals of \(x\) when, self is in rectangular form, or the union of intervals of \(r\) when self is in polar form.

Examples

>>> from sympy import Interval, ComplexRegion, Union
>>> a = Interval(2, 3)
>>> b = Interval(4, 5)
>>> c = Interval(1, 7)
>>> C1 = ComplexRegion(a*b)
>>> C1.a_interval
Interval(2, 3)
>>> C2 = ComplexRegion(Union(a*b, b*c))
>>> C2.a_interval
Union(Interval(2, 3), Interval(4, 5))

property b_interval¶

Return the union of intervals of \(y\) when, self is in rectangular form, or the union of intervals of \(theta\) when self is in polar form.

Examples

>>> from sympy import Interval, ComplexRegion, Union
>>> a = Interval(2, 3)
>>> b = Interval(4, 5)
>>> c = Interval(1, 7)
>>> C1 = ComplexRegion(a*b)
>>> C1.b_interval
Interval(4, 5)
>>> C2 = ComplexRegion(Union(a*b, b*c))
>>> C2.b_interval
Interval(1, 7)

classmethod from_real(sets)[source]¶

Converts given subset of real numbers to a complex region.

Examples

>>> from sympy import Interval, ComplexRegion
>>> unit = Interval(0,1)
>>> ComplexRegion.from_real(unit)
ComplexRegion(Interval(0, 1) x {0}, False)

property polar¶

Returns True if self is in polar form.

Examples

>>> from sympy import Interval, ComplexRegion, Union, S
>>> a = Interval(2, 3)
>>> b = Interval(4, 5)
>>> theta = Interval(0, 2*S.Pi)
>>> C1 = ComplexRegion(a*b)
>>> C1.polar
False
>>> C2 = ComplexRegion(a*theta, polar=True)
>>> C2.polar
True

property psets¶

Return a tuple of sets (ProductSets) input of the self.

Examples

>>> from sympy import Interval, ComplexRegion, Union
>>> a = Interval(2, 3)
>>> b = Interval(4, 5)
>>> c = Interval(1, 7)
>>> C1 = ComplexRegion(a*b)
>>> C1.psets
(Interval(2, 3) x Interval(4, 5),)
>>> C2 = ComplexRegion(Union(a*b, b*c))
>>> C2.psets
(Interval(2, 3) x Interval(4, 5), Interval(4, 5) x Interval(1, 7))

property sets¶

Return raw input sets to the self.

Examples

>>> from sympy import Interval, ComplexRegion, Union
>>> a = Interval(2, 3)
>>> b = Interval(4, 5)
>>> c = Interval(1, 7)
>>> C1 = ComplexRegion(a*b)
>>> C1.sets
Interval(2, 3) x Interval(4, 5)
>>> C2 = ComplexRegion(Union(a*b, b*c))
>>> C2.sets
Union(Interval(2, 3) x Interval(4, 5), Interval(4, 5) x Interval(1, 7))

sympy.sets.fancysets.normalize_theta_set(theta)[source]¶

Normalize a Real Set \(theta\) in the Interval [0, 2*pi). It returns a normalized value of theta in the Set. For Interval, a maximum of one cycle [0, 2*pi], is returned i.e. for theta equal to [0, 10*pi], returned normalized value would be [0, 2*pi). As of now intervals with end points as non-multiples of \(pi\) is not supported.

Raises

NotImplementedError

The algorithms for Normalizing theta Set are not yet implemented.

ValueError

The input is not valid, i.e. the input is not a real set.

RuntimeError

It is a bug, please report to the github issue tracker.

Examples

>>> from sympy.sets.fancysets import normalize_theta_set
>>> from sympy import Interval, FiniteSet, pi
>>> normalize_theta_set(Interval(9*pi/2, 5*pi))
Interval(pi/2, pi)
>>> normalize_theta_set(Interval(-3*pi/2, pi/2))
Interval.Ropen(0, 2*pi)
>>> normalize_theta_set(Interval(-pi/2, pi/2))
Union(Interval(0, pi/2), Interval.Ropen(3*pi/2, 2*pi))
>>> normalize_theta_set(Interval(-4*pi, 3*pi))
Interval.Ropen(0, 2*pi)
>>> normalize_theta_set(Interval(-3*pi/2, -pi/2))
Interval(pi/2, 3*pi/2)
>>> normalize_theta_set(FiniteSet(0, pi, 3*pi))
{0, pi}

Sets¶

Set¶

Elementary Sets¶

Interval¶

FiniteSet¶

Compound Sets¶

Union¶

Intersection¶

ProductSet¶

Complement¶

Singleton Sets¶

EmptySet¶

UniversalSet¶

Special Sets¶

Naturals¶

Naturals0¶

Integers¶

ImageSet¶

Range¶

ComplexRegion¶

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