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 |
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)
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
The infimum of ‘self’
>>> from sympy import Interval, Union
>>> Interval(0, 1).inf
0
>>> Union(Interval(0, 1), Interval(2, 3)).inf
0
Returns the intersection of ‘self’ and ‘other’.
>>> from sympy import Interval
>>> Interval(1, 3).intersect(Interval(1, 2))
[1, 2]
The (Lebesgue) measure of ‘self’
>>> from sympy import Interval, Union
>>> Interval(0, 1).measure
1
>>> Union(Interval(0, 1), Interval(2, 3)).measure
2
Give sort_key of infimum (if possible) else sort_key of the set.
Returns True if ‘other’ is a subset of ‘self’.
>>> from sympy import Interval
>>> Interval(0, 1).subset(Interval(0, 0.5))
True
>>> Interval(0, 1, left_open=True).subset(Interval(0, 1))
False
The supremum of ‘self’
>>> from sympy import Interval, Union
>>> Interval(0, 1).sup
1
>>> Union(Interval(0, 1), Interval(2, 3)).sup
3
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]
Represents a real interval as a Set.
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
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 |
Rewrite an interval in terms of inequalities and logic operators.
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
Return True if the left endpoint is negative infinity.
Return True if the right endpoint is positive infinity.
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
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
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
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
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
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 |
Rewrite a FiniteSet in terms of equalities and logic operators.
Represents a union of sets as a Set.
See also
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 |
Rewrite a Union in terms of equalities and logic operators.
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
Represents an intersection of sets as a Set.
See also
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 |
Rewrite an Intersection in terms of equalities and logic operators
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
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
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 |
Represents the empty set. The empty set is available as a singleton as S.EmptySet.
See also
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 |
Represents the set of all things. The universal set is available as a singleton as S.UniversalSet
See also
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 |
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(next(iterable))
1
>>> print(next(iterable))
2
>>> print(next(iterable))
3
>>> pprint(S.Naturals.intersect(Interval(0, 10)))
{1, 2, ..., 10}
Attributes
is_EmptySet | |
is_Intersection | |
is_UniversalSet |
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(next(iterable))
0
>>> print(next(iterable))
1
>>> print(next(iterable))
-1
>>> print(next(iterable))
2
>>> pprint(S.Integers.intersect(Interval(-4, 4)))
{-4, -3, ..., 4}
Attributes
is_EmptySet | |
is_Intersection | |
is_UniversalSet |
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_EmptySet | |
is_Intersection | |
is_UniversalSet |