Sets

Set

class sympy.sets.sets.Set[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  
boundary[source]

The boundary or frontier of a set

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

  1. x is in the closure of S. I.e. Every neighborhood of x contains a point in S.
  2. 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}
complement(universe)[source]

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

Examples

>>> from sympy import Interval, S
>>> Interval(0, 1).complement(S.Reals)
(-oo, 0) U (1, oo)
>>> Interval(0, 1).complement(S.UniversalSet)
UniversalSet() \ [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
inf[source]

The infimum of ‘self’

Examples

>>> from sympy import Interval, Union
>>> Interval(0, 1).inf
0
>>> Union(Interval(0, 1), Interval(2, 3)).inf
0
intersect(other)[source]

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

>>> from sympy import Interval
>>> Interval(1, 3).intersect(Interval(1, 2))
[1, 2]
intersection(other)[source]

Alias for intersect()

is_disjoint(other)[source]

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

References

[R346]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
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()

measure[source]

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

[R347]http://en.wikipedia.org/wiki/Power_set

Examples

>>> from sympy import FiniteSet, EmptySet
>>> A = EmptySet()
>>> A.powerset()
{EmptySet()}
>>> A = FiniteSet(1, 2)
>>> A.powerset() == FiniteSet(FiniteSet(1), FiniteSet(2), FiniteSet(1, 2), EmptySet())
True
subset(*args, **kwargs)[source]

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

sup[source]

The supremum of ‘self’

Examples

>>> from sympy import Interval, Union
>>> Interval(0, 1).sup
1
>>> Union(Interval(0, 1), Interval(2, 3)).sup
3
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))
[0, 1] U [2, 3]
>>> Interval(0, 1) + Interval(2, 3)
[0, 1] U [2, 3]
>>> Interval(1, 2, True, True) + FiniteSet(2, 3)
(1, 2] U {3}

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

>>> Interval(0, 2) - Interval(0, 1)
(1, 2]
>>> Interval(1, 3) - FiniteSet(2)
[1, 2) U (2, 3]
sympy.sets.sets.imageset(*args)[source]

Image of set under transformation f.

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

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

Examples

>>> from sympy import Interval, Symbol, imageset, sin, Lambda
>>> x = Symbol('x')
>>> imageset(x, 2*x, Interval(0, 2))
[0, 4]
>>> imageset(lambda x: 2*x, Interval(0, 2))
[0, 4]
>>> imageset(Lambda(x, sin(x)), Interval(-2, 1))
ImageSet(Lambda(x, sin(x)), [-2, 1])

Elementary Sets

Interval

class sympy.sets.sets.Interval[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

[R348]http://en.wikipedia.org/wiki/Interval_%28mathematics%29

Examples

>>> from sympy import Symbol, Interval
>>> Interval(0, 1)
[0, 1]
>>> Interval(0, 1, False, True)
[0, 1)
>>> a = Symbol('a', real=True)
>>> Interval(0, a)
[0, a]

Attributes

is_Complement  
is_EmptySet  
is_Intersection  
is_UniversalSet  
as_relational(symbol)[source]

Rewrite an interval in terms of inequalities and logic operators.

end[source]

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
is_left_unbounded[source]

Return True if the left endpoint is negative infinity.

is_right_unbounded[source]

Return True if the right endpoint is positive infinity.

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
left_open[source]

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
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
right_open[source]

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
start[source]

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[source]

Represents a finite set of discrete numbers

References

[R349]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

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[source]

Represents a union of sets as a Set.

See also

Intersection

References

[R350]http://en.wikipedia.org/wiki/Union_%28set_theory%29

Examples

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

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

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

static reduce(args)[source]

Simplify a Union using known rules

We first start with global rules like ‘Merge all FiniteSets’

Then we iterate through all pairs and ask the constituent sets if they can simplify themselves with any other constituent

Intersection

class sympy.sets.sets.Intersection[source]

Represents an intersection of sets as a Set.

See also

Union

References

[R351]http://en.wikipedia.org/wiki/Intersection_%28set_theory%29

Examples

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

We often use the .intersect method

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

Attributes

is_Complement  
is_EmptySet  
is_UniversalSet  
as_relational(symbol)[source]

Rewrite an Intersection in terms of equalities and logic operators

static reduce(args)[source]

Simplify an intersection using known rules

We first start with global rules like ‘if any empty sets return empty set’ and ‘distribute any unions’

Then we iterate through all pairs and ask the constituent sets if they can simplify themselves with any other constituent

ProductSet

class sympy.sets.sets.ProductSet[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

[R352]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)
[0, 5] x {1, 2, 3}
>>> (2, 2) in ProductSet(I, S)
True
>>> Interval(0, 1) * Interval(0, 1) # The unit square
[0, 1] x [0, 1]
>>> coin = FiniteSet('H', 'T')
>>> set(coin**2)
set([(H, H), (H, T), (T, H), (T, T)])

Attributes

is_Complement  
is_EmptySet  
is_Intersection  
is_UniversalSet  

Complement

class sympy.sets.sets.Complement[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

http://mathworld.wolfram.com/SetComplement.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[source]

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

See also

UniversalSet

References

[R353]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[source]

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

See also

EmptySet

References

[R354]http://en.wikipedia.org/wiki/Universal_set

Examples

>>> from sympy import S, Interval
>>> S.UniversalSet
UniversalSet()
>>> Interval(1, 2).intersect(S.UniversalSet)
[1, 2]

Attributes

is_Complement  
is_EmptySet  
is_Intersection  

Special Sets

Naturals

class sympy.sets.fancysets.Naturals[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[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[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[source]

Image of a set under a mathematical function

Examples

>>> from sympy import Symbol, S, ImageSet, FiniteSet, Lambda
>>> 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

Attributes

is_Complement  
is_EmptySet  
is_Intersection  
is_UniversalSet  

Table Of Contents

Previous topic

Series Expansions

Next topic

Simplify

This Page