# Sets¶

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

>>> 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

>>> 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

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

inf

The infimum of ‘self’

Examples

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

intersection(other)[source]

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

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

is_disjoint(other)[source]

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

Examples

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


References

• https//en.wikipedia.org/wiki/Disjoint_sets
is_open

Test if a set is open.

A set is open if it has an empty intersection with its boundary.

Examples

>>> S.Reals.is_open
True

is_proper_subset(other)[source]

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

Examples

>>> 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

>>> 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

>>> 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

>>> 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

>>> Interval(0, 1).measure
1
>>> Union(Interval(0, 1), Interval(2, 3)).measure
2

powerset()[source]

Find the Power set of ‘self’.

Examples

>>> 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

• https//en.wikipedia.org/wiki/Power_set
sup

The supremum of ‘self’

Examples

>>> 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:

>>> 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

>>> 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])


## Elementary Sets¶

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.

Examples

>>> 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', real=True)
>>> Interval(0, a)
[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

• 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.

end

The right end point of ‘self’.

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

Examples

>>> 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

>>> Interval(0, 1).start
0

left_open

True if ‘self’ is left-open.

Examples

>>> 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

>>> Interval(0, 1).end
1

right_open

True if ‘self’ is right-open.

Examples

>>> 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

>>> Interval(0, 1).start
0

class diofant.sets.sets.FiniteSet[source]

Represents a finite set of discrete numbers

Examples

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


References

• https//en.wikipedia.org/wiki/Finite_set
as_relational(symbol)[source]

Rewrite a FiniteSet in terms of equalities and logic operators.

## Compound Sets¶

class diofant.sets.sets.Union[source]

Represents a union of sets as a Set.

Examples

>>> 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]


References

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

class diofant.sets.sets.Intersection[source]

Represents an intersection of sets as a Set.

Examples

>>> Intersection(Interval(1, 3), Interval(2, 4))
[2, 3]


We often use the .intersect method

>>> Interval(1, 3).intersection(Interval(2, 4))
[2, 3]


References

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

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.

Examples

>>> 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)}


Notes

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

References

• https//en.wikipedia.org/wiki/Cartesian_product
class diofant.sets.sets.Complement[source]

Represents relative complement of a set with another set.

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

Examples

>>> Complement(FiniteSet(0, 1, 2), FiniteSet(1))
{0, 2}


References

static reduce(A, B)[source]

Simplify a Complement.

## Singleton Sets¶

class diofant.sets.sets.EmptySet[source]

Represents the empty set.

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

Examples

>>> S.EmptySet
EmptySet()

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


References

• https//en.wikipedia.org/wiki/Empty_set
class diofant.sets.sets.UniversalSet[source]

Represents the set of all things.

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

Examples

>>> S.UniversalSet
UniversalSet()

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


References

• https//en.wikipedia.org/wiki/Universal_set

## Special Sets¶

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.

Examples

>>> 5 in S.Naturals
True
>>> iterable = iter(S.Naturals)
>>> next(iterable)
1
>>> next(iterable)
2
>>> next(iterable)
3
>>> S.Naturals.intersection(Interval(0, 10))
Range(1, 11, 1)


Naturals0
non-negative integers
Integers
also includes negative integers
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
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.

Examples

>>> 5 in S.Naturals
True
>>> iterable = iter(S.Integers)
>>> next(iterable)
0
>>> next(iterable)
1
>>> next(iterable)
-1
>>> next(iterable)
2

>>> S.Integers.intersection(Interval(-4, 4))
Range(-4, 5, 1)


Naturals0
non-negative integers
Integers
positive and negative integers and zero
class diofant.sets.fancysets.Rationals[source]

The set of all rationals.

class diofant.sets.fancysets.ImageSet[source]

Image of a set under a mathematical function.

Examples

>>> squares = ImageSet(Lambda(x, x**2), S.Naturals)
>>> 4 in squares
True
>>> 5 in squares
False

>>> FiniteSet(0, 1, 2, 3, 4, 5, 6, 7, 9, 10).intersection(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.One/2)
1/4

class diofant.sets.fancysets.Range[source]

Represents a range of integers.

Examples

>>> list(Range(5))
[0, 1, 2, 3, 4]
>>> list(Range(10, 15))
[10, 11, 12, 13, 14]
>>> list(Range(10, 20, 2))
[10, 12, 14, 16, 18]
>>> list(Range(20, 10, -2))
[12, 14, 16, 18, 20]