Sets

Set

class sympy.core.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_EmptySet  
is_Intersection  
is_UniversalSet  
complement[source]

The complement of ‘self’.

As a shortcut it is possible to use the ‘~’ or ‘-‘ operators:

>>> from sympy import Interval
>>> Interval(0, 1).complement
(-oo, 0) U (1, oo)
>>> ~Interval(0, 1)
(-oo, 0) U (1, oo)
>>> -Interval(0, 1)
(-oo, 0) U (1, oo)
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:

>>> from sympy import Interval
>>> Interval(0, 1).contains(0.5)
True
>>> 0.5 in Interval(0, 1)
True
inf[source]

The infimum of ‘self’

>>> 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]
measure[source]

The (Lebesgue) measure of ‘self’

>>> from sympy import Interval, Union
>>> Interval(0, 1).measure
1
>>> Union(Interval(0, 1), Interval(2, 3)).measure
2
sort_key(order=None)[source]

Give sort_key of infimum (if possible) else sort_key of the set.

subset(other)[source]

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

>>> from sympy import Interval
>>> Interval(0, 1).contains(0)
True
>>> Interval(0, 1, left_open=True).contains(0)
False
sup[source]

The supremum of ‘self’

>>> 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’.

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]

Elementary Sets

Interval

class sympy.core.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

<http://en.wikipedia.org/wiki/Interval_(mathematics)>

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_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.

>>> 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.

>>> from sympy import Interval
>>> Interval(0, 1).start
0
left_open[source]

True if ‘self’ is left-open.

>>> 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.

>>> from sympy import Interval
>>> Interval(0, 1).end
1
right_open[source]

True if ‘self’ is right-open.

>>> 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.

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

FiniteSet

class sympy.core.sets.FiniteSet[source]

Represents a finite set of discrete numbers

References

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_EmptySet  
is_Intersection  
is_UniversalSet  
as_relational(symbol)[source]

Rewrite a FiniteSet in terms of equalities and logic operators.

Compound Sets

Union

class sympy.core.sets.Union[source]

Represents a union of sets as a Set.

See also

Intersection

References

<http://en.wikipedia.org/wiki/Union_(set_theory)>

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_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.core.sets.Intersection[source]

Represents an intersection of sets as a Set.

See also

Union

References

<http://en.wikipedia.org/wiki/Intersection_(set_theory)>

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_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.core.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

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_EmptySet  
is_Intersection  
is_UniversalSet  

Singleton Sets

EmptySet

class sympy.core.sets.EmptySet[source]

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

See also

UniversalSet

References

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_Intersection  
is_UniversalSet  

UniversalSet

class sympy.core.sets.UniversalSet[source]

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

See also

EmptySet

References

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_EmptySet  
is_Intersection  

Special Sets

Naturals

class sympy.sets.fancysets.Naturals[source]

Represents the Natural Numbers. The Naturals are available as a singleton as S.Naturals

Examples

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

Attributes

is_EmptySet  
is_Intersection  
is_UniversalSet  

Integers

class sympy.sets.fancysets.Integers[source]

Represents the Integers. The Integers are available as a singleton as S.Integers

Examples

>>> from sympy import S, Interval, pprint
>>> 5 in S.Naturals
True
>>> iterable = iter(S.Integers)
>>> print iterable.next()
0
>>> print iterable.next()
1
>>> print iterable.next()
-1
>>> print iterable.next()
2
>>> pprint(S.Integers.intersect(Interval(-4, 4)))
{-4, -3, ..., 4}

Attributes

is_EmptySet  
is_Intersection  
is_UniversalSet  

TransformationSet

class sympy.sets.fancysets.TransformationSet[source]

A set that is a transformation of another through some algebraic expression

Examples

>>> from sympy import Symbol, S, TransformationSet, FiniteSet, Lambda
>>> x = Symbol('x')
>>> N = S.Naturals
>>> squares = TransformationSet(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):
...     square_iterable.next()
1
4
9
16

Attributes

is_EmptySet  
is_Intersection  
is_UniversalSet  

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