Sets¶
Set¶
-
class
sympy.sets.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; seeFiniteSetfor that.Real intervals are represented by the
Intervalclass and unions of sets by theUnionclass. The empty set is represented by theEmptySetclass and available as a singleton asS.EmptySet.-
boundary¶ The boundary or frontier of a set
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}
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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)
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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) UniversalSet() \ Interval(0, 1)
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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:
Examples
>>> from sympy import Interval >>> Interval(0, 1).contains(0.5) True >>> 0.5 in Interval(0, 1) True
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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
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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()
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intersect(other)[source]¶ Returns the intersection of ‘self’ and ‘other’.
>>> 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()
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intersection(other)[source]¶ Alias for
intersect()
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is_closed¶ A property method to check whether a set is closed. A set is closed if it’s complement is an open set.
Examples
>>> from sympy import Interval >>> Interval(0, 1).is_closed True
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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
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is_open¶ Property method to check whether a set is open. A set is open if and only if it has an empty intersection with its boundary.
Examples
>>> from sympy import S >>> S.Reals.is_open True
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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
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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
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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
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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
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isdisjoint(other)[source]¶ Alias for
is_disjoint()
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issubset(other)[source]¶ Alias for
is_subset()
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issuperset(other)[source]¶ Alias for
is_superset()
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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
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powerset()[source]¶ Find the Power set of ‘self’.
Examples
>>> from sympy import FiniteSet, EmptySet >>> A = EmptySet() >>> A.powerset() {EmptySet()} >>> A = FiniteSet(1, 2) >>> a, b, c = FiniteSet(1), FiniteSet(2), FiniteSet(1, 2) >>> A.powerset() == FiniteSet(a, b, c, EmptySet()) True
References
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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
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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
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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(Interval.Lopen(1, 2), {3})
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))
-
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sympy.sets.sets.imageset(*args)[source]¶ Return an image of the set under transformation
f.If this function can’t compute the image, it returns an unevaluated ImageSet object.
\[{ f(x) | x \in self }\]Examples
>>> from sympy import S, Interval, Symbol, imageset, sin, Lambda >>> from sympy.abc import x, y
>>> 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(_x, _x + x), 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[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
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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
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is_left_unbounded¶ Return
Trueif the left endpoint is negative infinity.
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is_right_unbounded¶ Return
Trueif the right endpoint is positive infinity.
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left¶ 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
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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
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right¶ 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
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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
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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[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
Compound Sets¶
Union¶
-
class
sympy.sets.sets.Union[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[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[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.
Examples
>>> from sympy import Interval, FiniteSet, ProductSet >>> I = Interval(0, 5); S = FiniteSet(1, 2, 3) >>> ProductSet(I, S) Interval(0, 5) x {1, 2, 3}
>>> (2, 2) in ProductSet(I, S) True
>>> Interval(0, 1) * Interval(0, 1) # The unit square Interval(0, 1) x Interval(0, 1)
>>> coin = FiniteSet('H', 'T') >>> set(coin**2) {(H, H), (H, T), (T, H), (T, T)}
Notes
Passes most operations down to the argument sets
Flattens Products of ProductSets
References
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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, ProductSet >>> 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[source]¶ Represents the set difference or relative complement of a set with another set.
\(A - B = \{x \in A| x \\notin B\}\)
Examples
>>> from sympy import Complement, FiniteSet >>> Complement(FiniteSet(0, 1, 2), FiniteSet(1)) {0, 2}
See also
References
-
static
reduce(A, B)[source]¶ Simplify a
Complement.
-
static
Singleton Sets¶
EmptySet¶
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
References
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}
ImageSet¶
-
class
sympy.sets.fancysets.ImageSet[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) {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
Range¶
-
class
sympy.sets.fancysets.Range[source]¶ Represents a range of integers. Can be called as Range(stop), Range(start, stop), or Range(start, stop, step); when stop 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.
ooand-ooare never included in the set (Rangeis 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): ... ValueError: 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()
-
reversed¶ Return an equivalent Range in the opposite order.
Examples
>>> from sympy import Range >>> Range(10).reversed Range(9, -1, -1)
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ComplexRegion¶
-
class
sympy.sets.fancysets.ComplexRegion[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 ComplexRegion(Interval(2, 3) x Interval(4, 6), False)
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 ComplexRegion(Union(Interval(2, 3) x Interval(4, 6), Interval(4, 6) x Interval(1, 8)), False)
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 ComplexRegion(Interval(0, 1) x Interval.Ropen(0, 2*pi), True)
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 ComplexRegion(Interval(0, 1) x Interval(0, pi), True) >>> intersection == upper_half_unit_disk True
See also
Reals-
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))
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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)
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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) ComplexRegion(Interval(0, 1) x {0}, False)
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polar¶ Returns True if self is in polar form.
Examples
>>> from sympy import Interval, ComplexRegion, Union, S >>> a = Interval(2, 3) >>> b = Interval(4, 5) >>> theta = Interval(0, 2*S.Pi) >>> C1 = ComplexRegion(a*b) >>> C1.polar False >>> C2 = ComplexRegion(a*theta, polar=True) >>> C2.polar True
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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 (Interval(2, 3) x Interval(4, 5),) >>> C2 = ComplexRegion(Union(a*b, b*c)) >>> C2.psets (Interval(2, 3) x Interval(4, 5), Interval(4, 5) x Interval(1, 7))
-
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 Interval(2, 3) x Interval(4, 5) >>> C2 = ComplexRegion(Union(a*b, b*c)) >>> C2.sets Union(Interval(2, 3) x Interval(4, 5), Interval(4, 5) x Interval(1, 7))
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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}
