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 boundary or frontier of a set
A point x is on the boundary of a set S if
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.
>>> from sympy import Interval
>>> Interval(0, 1).boundary
{0, 1}
>>> Interval(0, 1, True, False).boundary
{0, 1}
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
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 |
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
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
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 |
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 |
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 |
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 (or counting numbers) which are all positive integers starting from 1. This set is also available as the Singleton, S.Naturals.
See also
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_EmptySet | |
is_Intersection | |
is_UniversalSet |
Represents all integers: positive, negative and zero. This set is also available as the Singleton, S.Integers.
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_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 |