# Sets¶

## Set¶

class diofant.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; 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.

boundary

The boundary or frontier of a set

A point x is on the boundary of a set S if

1. x is in the closure of S. I.e. Every neighborhood of x contains a point in S.
2. 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 diofant import Interval
>>> Interval(0, 1).boundary
{0, 1}
>>> Interval(0, 1, True, False).boundary
{0, 1}

complement(universe)[source]

The complement of ‘self’ w.r.t the given the universe.

Examples

>>> from diofant import Interval, S
>>> Interval(0, 1).complement(S.Reals)
(-oo, 0) U (1, oo)

>>> Interval(0, 1).complement(S.UniversalSet)
UniversalSet() \ [0, 1]

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 diofant import Interval
>>> Interval(0, 1).contains(0.5)
true
>>> 0.5 in Interval(0, 1)
True

inf

The infimum of ‘self’

Examples

>>> from diofant import Interval, Union
>>> Interval(0, 1).inf
0
>>> Union(Interval(0, 1), Interval(2, 3)).inf
0

intersect(other)[source]

Returns the intersection of ‘self’ and ‘other’.

>>> from diofant import Interval

>>> Interval(1, 3).intersect(Interval(1, 2))
[1, 2]

intersection(other)[source]

Alias for intersect()

is_disjoint(other)[source]

Returns True if ‘self’ and ‘other’ are disjoint

References

Examples

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

is_proper_subset(other)[source]

Returns True if ‘self’ is a proper subset of ‘other’.

Examples

>>> from diofant 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 diofant 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 diofant 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 diofant 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()

measure

The (Lebesgue) measure of ‘self’

Examples

>>> from diofant 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’.

References

Examples

>>> from diofant 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

sup

The supremum of ‘self’

Examples

>>> from diofant import Interval, Union
>>> Interval(0, 1).sup
1
>>> Union(Interval(0, 1), Interval(2, 3)).sup
3

union(other)[source]

Returns the union of ‘self’ and ‘other’.

Examples

As a shortcut it is possible to use the ‘+’ operator:

>>> from diofant 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]

diofant.sets.sets.imageset(*args)[source]

Image of 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 diofant import Interval, Symbol, imageset, sin, Lambda
>>> x = Symbol('x')

>>> imageset(x, 2*x, Interval(0, 2))
[0, 4]

>>> imageset(lambda x: 2*x, Interval(0, 2))
[0, 4]

>>> imageset(Lambda(x, sin(x)), Interval(-2, 1))
ImageSet(Lambda(x, sin(x)), [-2, 1])


## Interval¶

class diofant.sets.sets.Interval[source]

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

• 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

Examples

>>> from diofant import Symbol, Interval
>>> Interval(0, 1)
[0, 1]
>>> Interval(0, 1, False, True)
[0, 1)
>>> Interval.Ropen(0, 1)
[0, 1)
>>> Interval.Lopen(0, 1)
(0, 1]
>>> Interval.open(0, 1)
(0, 1)

>>> a = Symbol('a', extended_real=True)
>>> Interval(0, a)
[0, a]

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.

end

The right end point of ‘self’.

This property takes the same value as the ‘sup’ property.

Examples

>>> from diofant import Interval
>>> Interval(0, 1).end
1

is_left_unbounded

Return True if the left endpoint is negative infinity.

is_right_unbounded

Return True if the right endpoint is positive infinity.

left

The left end point of ‘self’.

This property takes the same value as the ‘inf’ property.

Examples

>>> from diofant import Interval
>>> Interval(0, 1).start
0

left_open

True if ‘self’ is left-open.

Examples

>>> from diofant 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.

right

The right end point of ‘self’.

This property takes the same value as the ‘sup’ property.

Examples

>>> from diofant import Interval
>>> Interval(0, 1).end
1

right_open

True if ‘self’ is right-open.

Examples

>>> from diofant import Interval
>>> Interval(0, 1, right_open=True).right_open
true
>>> Interval(0, 1, right_open=False).right_open
false

start

The left end point of ‘self’.

This property takes the same value as the ‘inf’ property.

Examples

>>> from diofant import Interval
>>> Interval(0, 1).start
0


## FiniteSet¶

class diofant.sets.sets.FiniteSet[source]

Represents a finite set of discrete numbers

References

Examples

>>> from diofant import FiniteSet
>>> FiniteSet(1, 2, 3, 4)
{1, 2, 3, 4}
>>> 3 in FiniteSet(1, 2, 3, 4)
True

as_relational(symbol)[source]

Rewrite a FiniteSet in terms of equalities and logic operators.

## Union¶

class diofant.sets.sets.Union[source]

Represents a union of sets as a Set.

References

Examples

>>> from diofant 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]

as_relational(symbol)[source]

Rewrite a Union in terms of equalities and logic operators.

static reduce(args)[source]

Simplify a Union using known rules

Then we iterate through all pairs and ask the constituent sets if they can simplify themselves with any other constituent

## Intersection¶

class diofant.sets.sets.Intersection[source]

Represents an intersection of sets as a Set.

References

Examples

>>> from diofant 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]

as_relational(symbol)[source]

Rewrite an Intersection in terms of equalities and logic operators

static reduce(args)[source]

Simplify an intersection using known rules

We first start with global rules like ‘if any empty sets return empty set’ and ‘distribute any unions’

Then we iterate through all pairs and ask the constituent sets if they can simplify themselves with any other constituent

## ProductSet¶

class diofant.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.

Notes

• Passes most operations down to the argument sets
• Flattens Products of ProductSets

References

Examples

>>> from diofant import Interval, FiniteSet, ProductSet, Symbol
>>> 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]

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


## Complement¶

class diofant.sets.sets.Complement[source]

Represents relative complement of a set with another set.

$$A - B = \{x \in A| x \notin B\}$$

References

Examples

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

static reduce(A, B)[source]

Simplify a Complement.

## EmptySet¶

class diofant.sets.sets.EmptySet[source]

Represents the empty set.

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

References

Examples

>>> from diofant import S, Interval
>>> S.EmptySet
EmptySet()

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


## UniversalSet¶

class diofant.sets.sets.UniversalSet[source]

Represents the set of all things.

The universal set is available as a singleton as S.UniversalSet

References

Examples

>>> from diofant import S, Interval
>>> S.UniversalSet
UniversalSet()

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


## Naturals¶

class diofant.sets.fancysets.Naturals[source]

The set of natural numbers.

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.

Naturals0
non-negative integers
Integers
also includes negative integers

Examples

>>> from diofant 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¶

class diofant.sets.fancysets.Naturals0[source]

The set of natural numbers, starting from 0.

Represents the whole numbers which are all the non-negative integers, inclusive of zero.

Naturals
positive integers
Integers
also includes the negative integers

## Integers¶

class diofant.sets.fancysets.Integers[source]

The set of all integers.

Represents all integers: positive, negative and zero. This set is also available as the Singleton, S.Integers.

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

Examples

>>> from diofant 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}


## Rationals¶

class diofant.sets.fancysets.Rationals[source]

The set of all rationals.

## ImageSet¶

class diofant.sets.fancysets.ImageSet[source]

Image of a set under a mathematical function.

Examples

>>> from diofant 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


## Range¶

class diofant.sets.fancysets.Range[source]

Represents a range of integers.

Examples

>>> from diofant import Range
>>> list(Range(5)) # 0 to 5
[0, 1, 2, 3, 4]
>>> list(Range(10, 15)) # 10 to 15
[10, 11, 12, 13, 14]
>>> list(Range(10, 20, 2)) # 10 to 20 in steps of 2
[10, 12, 14, 16, 18]
>>> list(Range(20, 10, -2)) # 20 to 10 backward in steps of 2
[12, 14, 16, 18, 20]