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
; seeFiniteSet
for that.Real intervals are represented by the
Interval
class and unions of sets by theUnion
class. The empty set is represented by theEmptySet
class and available as a singleton asS.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 FiniteSet(0, 1) >>> Interval(0, 1, True, False).boundary FiniteSet(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 inself
:true
if it is,false
if it isn’t, else an unevaluatedContains
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
andother
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
- 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 ofother
.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 ofother
.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 ofother
.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 ofother
.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 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() FiniteSet(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
- 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
andother
.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
- union(other)[source]¶
Returns the union of
self
andother
.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(FiniteSet(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 can’t 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
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
- property end¶
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
- 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
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
- property right_open¶
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
- property start¶
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(*args, **kwargs)[source]¶
Represents a finite set of discrete numbers.
Examples
>>> from sympy import FiniteSet >>> FiniteSet(1, 2, 3, 4) FiniteSet(1, 2, 3, 4) >>> 3 in FiniteSet(1, 2, 3, 4) True
>>> members = [1, 2, 3, 4] >>> f = FiniteSet(*members) >>> f FiniteSet(1, 2, 3, 4) >>> f - FiniteSet(2) FiniteSet(1, 3, 4) >>> f + FiniteSet(2, 5) FiniteSet(1, 2, 3, 4, 5)
References
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
References
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
References
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), FiniteSet(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
- 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)) FiniteSet(0, 2)
See also
References
- 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)) FiniteSet(1, 2, 4, 5)
See also
References
Singleton Sets¶
EmptySet¶
UniversalSet¶
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.
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}
Naturals0¶
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}
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
Complexes¶
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.sets.sets import FiniteSet, Interval >>> from sympy.sets.fancysets import ImageSet
>>> 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) FiniteSet(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) FiniteSet(0)
See also
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 (juse 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 ofIntegers
). If the starting point is infinite, then the final value isstop - 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}
- 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, & use the flag polar=True.
Z = {z in C | z = r*[cos(theta) + I*sin(theta)], r in [r], theta in [theta]}
Rectangular Form Input is in the form of the ProductSet or Union of ProductSets of interval of x and y the of the Complex numbers in a Plane. Default input type is in rectangular form.
Z = {z in C | z = x + I*y, x in [Re(z)], y in [Im(z)]}
Examples
>>> from sympy.sets.fancysets import ComplexRegion >>> from sympy.sets import Interval >>> from sympy import S, I, Union >>> a = Interval(2, 3) >>> b = Interval(4, 6) >>> c = Interval(1, 8) >>> 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.
>>> 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
See also
- 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), FiniteSet(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 C | z = x + I*y, x in [Re(z)], y in [Im(z)]}
Examples
>>> from sympy.sets.fancysets import ComplexRegion >>> from sympy.sets.sets import Interval >>> from sympy import I >>> region = ComplexRegion(Interval(1, 3) * Interval(4, 6)) >>> 2 + 5*I in region True >>> 5*I in region False
See also
- class sympy.sets.fancysets.PolarComplexRegion(sets)[source]¶
Set representing a polar region of the complex plane.
Z = {z in C | z = r*[cos(theta) + I*sin(theta)], r in [r], theta in [theta]}
Examples
>>> from sympy.sets.fancysets import ComplexRegion, Interval >>> from sympy import 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
See also
- 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)) FiniteSet(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.sets.powerset import PowerSet >>> from sympy import S, FiniteSet
A power set of a finite set:
>>> PowerSet(FiniteSet(1, 2, 3)) PowerSet(FiniteSet(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(FiniteSet(1), FiniteSet(1, 2), FiniteSet(1, 3), FiniteSet(1, 2, 3), FiniteSet(2), FiniteSet(2, 3), FiniteSet(3), EmptySet)
References
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 | condition(x) is True for 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, FiniteSet(y, z))
To check if
pi
is inc
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 >>> from sympy.sets.contains import 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