Sets

Basic Sets

Set

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

The base class for any kind of set.

Explanation

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.

property boundary

The boundary or frontier of a set.

Explanation

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)
Complement(UniversalSet, Interval(0, 1))

contains(other)[source]

Returns a SymPy value indicating whether other is contained in self: true if it is, false if it isn’t, else an unevaluated Contains expression (or, as in the case of ConditionSet and a union of FiniteSet/Intervals, an expression indicating the conditions for containment).

Examples

>>> from sympy import Interval, S
>>> from sympy.abc import x

>>> Interval(0, 1).contains(0.5)
True

As a shortcut it is possible to use the in operator, but that will raise an error unless an affirmative true or false is not obtained.

>>> Interval(0, 1).contains(x)
(0 <= x) & (x <= 1)
>>> x in Interval(0, 1)
Traceback (most recent call last):
...
TypeError: did not evaluate to a bool: None

The result of ‘in’ is a bool, not a SymPy value

>>> 1 in Interval(0, 2)
True
>>> _ is S.true
False

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’.

Examples

>>> 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.

Explanation

A set is closed if its complement is an open set. The closedness of a subset of the reals is determined with respect to R and its standard topology.

Examples

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

is_disjoint(other)[source]

Returns True if self and other are disjoint.

Examples

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

References

R739: https://en.wikipedia.org/wiki/Disjoint_sets

property is_open

Property method to check whether a set is open.

Explanation

A set is open if and only if it has an empty intersection with its boundary. In particular, a subset A of the reals is open if and only if each one of its points is contained in an open interval that is a subset of A.

Examples

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

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 kind

The kind of a Set

Explanation

Any Set will have kind SetKind which is parametrised by the kind of the elements of the set. For example most sets are sets of numbers and will have kind SetKind(NumberKind). If elements of sets are different in kind than their kind will SetKind(UndefinedKind). See sympy.core.kind.Kind for an explanation of the kind system.

Examples

>>> from sympy import Interval, Matrix, FiniteSet, EmptySet, ProductSet, PowerSet

>>> FiniteSet(Matrix([1, 2])).kind
SetKind(MatrixKind(NumberKind))

>>> Interval(1, 2).kind
SetKind(NumberKind)

>>> EmptySet.kind
SetKind()

A sympy.sets.powerset.PowerSet is a set of sets:

>>> PowerSet({1, 2, 3}).kind
SetKind(SetKind(NumberKind))

A ProductSet represents the set of tuples of elements of other sets. Its kind is sympy.core.containers.TupleKind parametrised by the kinds of the elements of those sets:

>>> p = ProductSet(FiniteSet(1, 2), FiniteSet(3, 4))
>>> list(p)
[(1, 3), (2, 3), (1, 4), (2, 4)]
>>> p.kind
SetKind(TupleKind(NumberKind, NumberKind))

When all elements of the set do not have same kind, the kind will be returned as SetKind(UndefinedKind):

>>> FiniteSet(0, Matrix([1, 2])).kind
SetKind(UndefinedKind)

The kind of the elements of a set are given by the element_kind attribute of SetKind:

>>> Interval(1, 2).kind.element_kind
NumberKind

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.

Examples

>>> from sympy import EmptySet, FiniteSet, Interval

A power set of an empty set:

>>> A = EmptySet
>>> A.powerset()
{EmptySet}

A power set of a finite set:

>>> A = FiniteSet(1, 2)
>>> a, b, c = FiniteSet(1), FiniteSet(2), FiniteSet(1, 2)
>>> A.powerset() == FiniteSet(a, b, c, EmptySet)
True

A power set of an interval:

>>> Interval(1, 2).powerset()
PowerSet(Interval(1, 2))

References

R740: https://en.wikipedia.org/wiki/Power_set

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.

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

References

R741: https://en.wikipedia.org/wiki/Symmetric_difference

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({3}, Interval.Lopen(1, 2))

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.

Explanation

If this function cannot compute the image, it returns an unevaluated ImageSet object.

\[\{ f(x) \mid x \in \mathrm{self} \}\]

Examples

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

>>> 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(y, x + y), 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)

See also

sympy.sets.fancysets.ImageSet

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.

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)

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

R742: https://en.wikipedia.org/wiki/Interval_%28mathematics%29

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 the interval.

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_open

True if interval 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_open

True if interval 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 the interval.

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.

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}

References

R743: https://en.wikipedia.org/wiki/Finite_set

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.

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)

See also

Intersection

References

R744: https://en.wikipedia.org/wiki/Union_%28set_theory%29

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.

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)

See also

Union

References

R745: https://en.wikipedia.org/wiki/Intersection_%28set_theory%29

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.

Explanation

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

Can use * operator on any sets for convenient shorthand.

Examples

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

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

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

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

The Cartesian product is not commutative or associative e.g.:

>>> I*S == S*I
False
>>> (I*I)*I == I*(I*I)
False

Notes

Passes most operations down to the argument sets

References

R746: https://en.wikipedia.org/wiki/Cartesian_product

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
>>> 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.

\[A - B = \{x \in A \mid x \notin B\}\]

Examples

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

See also

Intersection, Union

References

R747: http://mathworld.wolfram.com/ComplementSet.html

as_relational(symbol)[source]: Rewrite a complement in terms of equalities and logic operators

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

SymmetricDifference

class sympy.sets.sets.SymmetricDifference(a, b, evaluate=True)[source]

Represents the set of elements which are in either of the sets and not in their intersection.

Examples

>>> from sympy import SymmetricDifference, FiniteSet
>>> SymmetricDifference(FiniteSet(1, 2, 3), FiniteSet(3, 4, 5))
{1, 2, 4, 5}

See also

Complement, Union

References

R748: https://en.wikipedia.org/wiki/Symmetric_difference

as_relational(symbol)[source]: Rewrite a symmetric_difference in terms of equalities and logic operators

DisjointUnion

class sympy.sets.sets.DisjointUnion(*sets)[source]

Represents the disjoint union (also known as the external disjoint union) of a finite number of sets.

Examples

>>> from sympy import DisjointUnion, FiniteSet, Interval, Union, Symbol
>>> A = FiniteSet(1, 2, 3)
>>> B = Interval(0, 5)
>>> DisjointUnion(A, B)
DisjointUnion({1, 2, 3}, Interval(0, 5))
>>> DisjointUnion(A, B).rewrite(Union)
Union(ProductSet({1, 2, 3}, {0}), ProductSet(Interval(0, 5), {1}))
>>> C = FiniteSet(Symbol('x'), Symbol('y'), Symbol('z'))
>>> DisjointUnion(C, C)
DisjointUnion({x, y, z}, {x, y, z})
>>> DisjointUnion(C, C).rewrite(Union)
ProductSet({x, y, z}, {0, 1})

References

https://en.wikipedia.org/wiki/Disjoint_union

Singleton Sets

EmptySet

class sympy.sets.sets.EmptySet[source]

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

Examples

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

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

See also

UniversalSet

References

R749: https://en.wikipedia.org/wiki/Empty_set

UniversalSet

class sympy.sets.sets.UniversalSet[source]

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

Examples

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

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

See also

EmptySet

References

R750: https://en.wikipedia.org/wiki/Universal_set

Special Sets

Rationals

class sympy.sets.fancysets.Rationals[source]

Represents the rational numbers. This set is also available as the singleton S.Rationals.

Examples

>>> from sympy import S
>>> S.Half in S.Rationals
True
>>> iterable = iter(S.Rationals)
>>> [next(iterable) for i in range(12)]
[0, 1, -1, 1/2, 2, -1/2, -2, 1/3, 3, -1/3, -3, 2/3]

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.

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}

See also

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

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

Integers

class sympy.sets.fancysets.Integers[source]

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}

See also

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

Reals

class sympy.sets.fancysets.Reals[source]

Represents all real numbers from negative infinity to positive infinity, including all integer, rational and irrational numbers. This set is also available as the singleton S.Reals.

Examples

>>> from sympy import S, Rational, pi, I
>>> 5 in S.Reals
True
>>> Rational(-1, 2) in S.Reals
True
>>> pi in S.Reals
True
>>> 3*I in S.Reals
False
>>> S.Reals.contains(pi)
True

See also

ComplexRegion

Complexes

class sympy.sets.fancysets.Complexes[source]

The Set of all complex numbers

Examples

>>> from sympy import S, I
>>> S.Complexes
Complexes
>>> 1 + I in S.Complexes
True

See also

Reals, ComplexRegion

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.

Examples

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

>>> 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}

See also

sympy.sets.sets.imageset

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 step is not given it defaults to 1.

Range(stop) is the same as Range(0, stop, 1) and the stop value (just 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. oo and -oo are never included in the set (Range is always a subset of Integers). 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):
...
TypeError: 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]

Although 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

Range will accept symbolic arguments but has very limited support for doing anything other than displaying the Range:

>>> from sympy import Symbol, pprint
>>> from sympy.abc import i, j, k
>>> Range(i, j, k).start
i
>>> Range(i, j, k).inf
Traceback (most recent call last):
...
ValueError: invalid method for symbolic range

Better success will be had when using integer symbols:

>>> n = Symbol('n', integer=True)
>>> r = Range(n, n + 20, 3)
>>> r.inf
n
>>> pprint(r)
{n, n + 3, ..., n + 18}

as_relational(x)[source]: Rewrite a Range in terms of equalities and logic operators.

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, and use the flag polar=True.

\[Z = \{z \in \mathbb{C} \mid z = r\times (\cos(\theta) + I\sin(\theta)), r \in [\texttt{r}], \theta \in [\texttt{theta}]\}\]
Rectangular Form Input is in the form of the ProductSet or Union of ProductSets of interval of x and y, the real and imaginary parts of the Complex numbers in a plane. Default input type is in rectangular form.

\[Z = \{z \in \mathbb{C} \mid z = x + Iy, x \in [\operatorname{re}(z)], y \in [\operatorname{im}(z)]\}\]

Examples

>>> from sympy import ComplexRegion, Interval, S, I, Union
>>> a = Interval(2, 3)
>>> b = Interval(4, 6)
>>> c1 = ComplexRegion(a*b)  # Rectangular Form
>>> c1
CartesianComplexRegion(ProductSet(Interval(2, 3), Interval(4, 6)))

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.

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

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
PolarComplexRegion(ProductSet(Interval(0, 1), Interval.Ropen(0, 2*pi)))

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
PolarComplexRegion(ProductSet(Interval(0, 1), Interval(0, pi)))
>>> intersection == upper_half_unit_disk
True

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)
CartesianComplexRegion(ProductSet(Interval(0, 1), {0}))

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
(ProductSet(Interval(2, 3), Interval(4, 5)),)
>>> C2 = ComplexRegion(Union(a*b, b*c))
>>> C2.psets
(ProductSet(Interval(2, 3), Interval(4, 5)), ProductSet(Interval(4, 5), 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
ProductSet(Interval(2, 3), Interval(4, 5))
>>> C2 = ComplexRegion(Union(a*b, b*c))
>>> C2.sets
Union(ProductSet(Interval(2, 3), Interval(4, 5)), ProductSet(Interval(4, 5), Interval(1, 7)))

class sympy.sets.fancysets.CartesianComplexRegion(sets)[source]

Set representing a square region of the complex plane.

\[Z = \{z \in \mathbb{C} \mid z = x + Iy, x \in [\operatorname{re}(z)], y \in [\operatorname{im}(z)]\}\]

Examples

>>> from sympy import ComplexRegion, I, Interval
>>> region = ComplexRegion(Interval(1, 3) * Interval(4, 6))
>>> 2 + 5*I in region
True
>>> 5*I in region
False

class sympy.sets.fancysets.PolarComplexRegion(sets)[source]

Set representing a polar region of the complex plane.

\[Z = \{z \in \mathbb{C} \mid z = r\times (\cos(\theta) + I\sin(\theta)), r \in [\texttt{r}], \theta \in [\texttt{theta}]\}\]

Examples

>>> from sympy import ComplexRegion, Interval, oo, pi, I
>>> rset = Interval(0, oo)
>>> thetaset = Interval(0, pi)
>>> upper_half_plane = ComplexRegion(rset * thetaset, polar=True)
>>> 1 + I in upper_half_plane
True
>>> 1 - I in upper_half_plane
False

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}

Power sets

PowerSet

class sympy.sets.powerset.PowerSet(arg, evaluate=None)[source]

A symbolic object representing a power set.

Parameters

arg : Set

The set to take power of.

evaluate : bool

The flag to control evaluation.

If the evaluation is disabled for finite sets, it can take advantage of using subset test as a membership test.

Notes

Power set \(\mathcal{P}(S)\) is defined as a set containing all the subsets of \(S\).

If the set \(S\) is a finite set, its power set would have \(2^{\left| S \right|}\) elements, where \(\left| S \right|\) denotes the cardinality of \(S\).

Examples

>>> from sympy import PowerSet, S, FiniteSet

A power set of a finite set:

>>> PowerSet(FiniteSet(1, 2, 3))
PowerSet({1, 2, 3})

A power set of an empty set:

>>> PowerSet(S.EmptySet)
PowerSet(EmptySet)
>>> PowerSet(PowerSet(S.EmptySet))
PowerSet(PowerSet(EmptySet))

A power set of an infinite set:

>>> PowerSet(S.Reals)
PowerSet(Reals)

Evaluating the power set of a finite set to its explicit form:

>>> PowerSet(FiniteSet(1, 2, 3)).rewrite(FiniteSet)
FiniteSet(EmptySet, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3})

References

R751: https://en.wikipedia.org/wiki/Power_set
R752: https://en.wikipedia.org/wiki/Axiom_of_power_set

Iteration over sets

For set unions, \(\{a, b\} \cup \{x, y\}\) can be treated as \(\{a, b, x, y\}\) for iteration regardless of the distinctiveness of the elements, however, for set intersections, assuming that \(\{a, b\} \cap \{x, y\}\) is \(\varnothing\) or \(\{a, b \}\) would not always be valid, since some of \(a\), \(b\), \(x\) or \(y\) may or may not be the elements of the intersection.

Iterating over the elements of a set involving intersection, complement, or symmetric difference yields (possibly duplicate) elements of the set provided that all elements are known to be the elements of the set. If any element cannot be determined to be a member of a set then the iteration gives TypeError. This happens in the same cases where x in y would give an error.

There are some reasons to implement like this, even if it breaks the consistency with how the python set iterator works. We keep in mind that sympy set comprehension like FiniteSet(*s) from a existing sympy sets could be a common usage. And this approach would make FiniteSet(*s) to be consistent with any symbolic set processing methods like FiniteSet(*simplify(s)).

Condition Sets

ConditionSet

class sympy.sets.conditionset.ConditionSet(sym, condition, base_set=UniversalSet)[source]

Set of elements which satisfies a given condition.

\[\{x \mid \textrm{condition}(x) = \texttt{True}, x \in S\}\]

Examples

>>> from sympy import Symbol, S, ConditionSet, pi, Eq, sin, Interval
>>> from sympy.abc import x, y, z

>>> sin_sols = ConditionSet(x, Eq(sin(x), 0), Interval(0, 2*pi))
>>> 2*pi in sin_sols
True
>>> pi/2 in sin_sols
False
>>> 3*pi in sin_sols
False
>>> 5 in ConditionSet(x, x**2 > 4, S.Reals)
True

If the value is not in the base set, the result is false:

>>> 5 in ConditionSet(x, x**2 > 4, Interval(2, 4))
False

Notes

Symbols with assumptions should be avoided or else the condition may evaluate without consideration of the set:

>>> n = Symbol('n', negative=True)
>>> cond = (n > 0); cond
False
>>> ConditionSet(n, cond, S.Integers)
EmptySet

Only free symbols can be changed by using \(subs\):

>>> c = ConditionSet(x, x < 1, {x, z})
>>> c.subs(x, y)
ConditionSet(x, x < 1, {y, z})

To check if pi is in c use:

>>> pi in c
False

If no base set is specified, the universal set is implied:

>>> ConditionSet(x, x < 1).base_set
UniversalSet

Only symbols or symbol-like expressions can be used:

>>> ConditionSet(x + 1, x + 1 < 1, S.Integers)
Traceback (most recent call last):
...
ValueError: non-symbol dummy not recognized in condition

When the base set is a ConditionSet, the symbols will be unified if possible with preference for the outermost symbols:

>>> ConditionSet(x, x < y, ConditionSet(z, z + y < 2, S.Integers))
ConditionSet(x, (x < y) & (x + y < 2), Integers)

Relations on sets

class sympy.sets.conditionset.Contains(x, s)[source]

Asserts that x is an element of the set S.

Examples

>>> from sympy import Symbol, Integer, S, Contains
>>> Contains(Integer(2), S.Integers)
True
>>> Contains(Integer(-2), S.Naturals)
False
>>> i = Symbol('i', integer=True)
>>> Contains(i, S.Naturals)
Contains(i, Naturals)

References

R753: https://en.wikipedia.org/wiki/Element_%28mathematics%29

SetKind

class sympy.sets.conditionset.SetKind(element_kind=None)[source]

SetKind is kind for all Sets

Every instance of Set will have kind SetKind parametrised by the kind of the elements of the Set. The kind of the elements might be NumberKind, or TupleKind or something else. When not all elements have the same kind then the kind of the elements will be given as UndefinedKind.

Parameters: element_kind: Kind (optional)

The kind of the elements of the set. In a well defined set all elements will have the same kind. Otherwise the kind should sympy.core.kind.UndefinedKind. The element_kind argument is optional but should only be omitted in the case of EmptySet whose kind is simply SetKind()

Examples

>>> from sympy import Interval
>>> Interval(1, 2).kind
SetKind(NumberKind)
>>> Interval(1,2).kind.element_kind
NumberKind