Sets

AUTHORS:

• William Stein (2005) - first version
• William Stein (2006-02-16) - large number of documentation and examples; improved code
• Mike Hansen (2007-3-25) - added differences and symmetric differences; fixed operators
• Florent Hivert (2010-06-17) - Adapted to categories
• Nicolas M. Thiery (2011-03-15) - Added subset and superset methods
• Julian Rueth (2013-04-09) - Collected common code in Set_object_binary, fixed __hash__.

sage.sets.set.Set(X=frozenset([]))

Create the underlying set of X.

If X is a list, tuple, Python set, or X.is_finite() is True, this returns a wrapper around Python’s enumerated immutable frozenset type with extra functionality. Otherwise it returns a more formal wrapper.

If you need the functionality of mutable sets, use Python’s builtin set type.

EXAMPLES:

sage: X = Set(GF(9,'a'))
sage: X
{0, 1, 2, a, a + 1, a + 2, 2*a, 2*a + 1, 2*a + 2}
sage: type(X)
<class 'sage.sets.set.Set_object_enumerated_with_category'>
sage: Y = X.union(Set(QQ))
sage: Y
Set-theoretic union of {0, 1, 2, a, a + 1, a + 2, 2*a, 2*a + 1, 2*a + 2} and Set of elements of Rational Field
sage: type(Y)
<class 'sage.sets.set.Set_object_union_with_category'>

Usually sets can be used as dictionary keys.

sage: d={Set([2*I,1+I]):10}
sage: d                  # key is randomly ordered
{{I + 1, 2*I}: 10}
sage: d[Set([1+I,2*I])]
10
sage: d[Set((1+I,2*I))]
10

The original object is often forgotten.

sage: v = [1,2,3]
sage: X = Set(v)
sage: X
{1, 2, 3}
sage: v.append(5)
sage: X
{1, 2, 3}
sage: 5 in X
False

Set also accepts iterators, but be careful to only give finite sets.

sage: list(Set(iter([1, 2, 3, 4, 5])))
[1, 2, 3, 4, 5]

We can also create sets from different types:

sage: sorted(Set([Sequence([3,1], immutable=True), 5, QQ, Partition([3,1,1])]), key=str)
[5, Rational Field, [3, 1, 1], [3, 1]]

However each of the objects must be hashable:

sage: Set([QQ, [3, 1], 5])
Traceback (most recent call last):
...
TypeError: unhashable type: 'list'

TESTS:

sage: Set(Primes())
Set of all prime numbers: 2, 3, 5, 7, ...
sage: Set(Subsets([1,2,3])).cardinality()
8
sage: S = Set(iter([1,2,3])); S
{1, 2, 3}
sage: type(S)
<class 'sage.sets.set.Set_object_enumerated_with_category'>
sage: S = Set([])
sage: TestSuite(S).run()

Check that trac ticket #16090 is fixed:

sage: Set()
{}

class sage.sets.set.Set_object(X)

Bases: sage.structure.parent.Set_generic

A set attached to an almost arbitrary object.

EXAMPLES:

sage: K = GF(19)
sage: Set(K)
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}
sage: S = Set(K)

sage: latex(S)
\left\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18\right\}
sage: TestSuite(S).run()

sage: latex(Set(ZZ))
\Bold{Z}

TESTS:

See trac ticket trac ticket #14486:

sage: 0 == Set([1]), Set([1]) == 0
(False, False)
sage: 1 == Set([0]), Set([0]) == 1
(False, False)

an_element()

Returns the first element of self returned by __iter__()

If self is empty, the exception EmptySetError is raised instead.

This provides a generic implementation of the method _an_element_() for all parents in EnumeratedSets.

EXAMPLES:

sage: C = FiniteEnumeratedSets().example(); C
An example of a finite enumerated set: {1,2,3}
sage: C.an_element() # indirect doctest
1
sage: S = Set([])
sage: S.an_element()
Traceback (most recent call last):
...
EmptySetError

TESTS:

sage: super(Parent, C)._an_element_
Cached version of <function _an_element_from_iterator at ...>

cardinality()

Return the cardinality of this set, which is either an integer or Infinity.

EXAMPLES:

sage: Set(ZZ).cardinality()
+Infinity
sage: Primes().cardinality()
+Infinity
sage: Set(GF(5)).cardinality()
5
sage: Set(GF(5^2,'a')).cardinality()
25

difference(X)

Return the set difference self - X.

EXAMPLES:

sage: X = Set(ZZ).difference(Primes())
sage: 4 in X
True
sage: 3 in X
False

sage: 4/1 in X
True

sage: X = Set(GF(9,'b')).difference(Set(GF(27,'c')))
sage: X
{0, 1, 2, b, b + 1, b + 2, 2*b, 2*b + 1, 2*b + 2}

sage: X = Set(GF(9,'b')).difference(Set(GF(27,'b')))
sage: X
{0, 1, 2, b, b + 1, b + 2, 2*b, 2*b + 1, 2*b + 2}

intersection(X)

Return the intersection of self and X.

EXAMPLES:

sage: X = Set(ZZ).intersection(Primes())
sage: 4 in X
False
sage: 3 in X
True

sage: 2/1 in X
True

sage: X = Set(GF(9,'b')).intersection(Set(GF(27,'c')))
sage: X
{}

sage: X = Set(GF(9,'b')).intersection(Set(GF(27,'b')))
sage: X
{}

is_empty()

Return boolean representing emptiness of the set.

OUTPUT:

True if the set is empty, false if otherwise.

EXAMPLES:

sage: Set([]).is_empty()
True
sage: Set([0]).is_empty()
False
sage: Set([1..100]).is_empty()
False
sage: Set(SymmetricGroup(2).list()).is_empty()
False
sage: Set(ZZ).is_empty()
False

TESTS:

sage: Set([]).is_empty()
True
sage: Set([1,2,3]).is_empty()
False
sage: Set([1..100]).is_empty()
False
sage: Set(DihedralGroup(4).list()).is_empty()
False
sage: Set(QQ).is_empty()
False

is_finite()

Return True if self is finite.

EXAMPLES:

sage: Set(QQ).is_finite()
False
sage: Set(GF(250037)).is_finite()
True
sage: Set(Integers(2^1000000)).is_finite()
True
sage: Set([1,'a',ZZ]).is_finite()
True

object()

Return underlying object.

EXAMPLES:

sage: X = Set(QQ)
sage: X.object()
Rational Field
sage: X = Primes()
sage: X.object()
Set of all prime numbers: 2, 3, 5, 7, ...

subsets(size=None)

Return the Subsets object representing the subsets of a set. If size is specified, return the subsets of that size.

EXAMPLES:

sage: X = Set([1,2,3])
sage: list(X.subsets())
[{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}]
sage: list(X.subsets(2))
[{1, 2}, {1, 3}, {2, 3}]

symmetric_difference(X)

Returns the symmetric difference of self and X.

EXAMPLES:

sage: X = Set([1,2,3]).symmetric_difference(Set([3,4]))
sage: X
{1, 2, 4}

union(X)

Return the union of self and X.

EXAMPLES:

sage: Set(QQ).union(Set(ZZ))
Set-theoretic union of Set of elements of Rational Field and Set of elements of Integer Ring
sage: Set(QQ) + Set(ZZ)
Set-theoretic union of Set of elements of Rational Field and Set of elements of Integer Ring
sage: X = Set(QQ).union(Set(GF(3))); X
Set-theoretic union of Set of elements of Rational Field and {0, 1, 2}
sage: 2/3 in X
True
sage: GF(3)(2) in X
True
sage: GF(5)(2) in X
False
sage: Set(GF(7)) + Set(GF(3))
{0, 1, 2, 3, 4, 5, 6, 1, 2, 0}

class sage.sets.set.Set_object_binary(X, Y, op, latex_op)

Bases: sage.sets.set.Set_object

An abstract common base class for sets defined by a binary operation (ex. Set_object_union, Set_object_intersection, Set_object_difference, and Set_object_symmetric_difference).

INPUT:

• X, Y – sets, the operands to op
• op – a string describing the binary operation
• latex_op – a string used for rendering this object in LaTeX

EXAMPLES:

sage: X = Set(QQ^2)
sage: Y = Set(ZZ)
sage: from sage.sets.set import Set_object_binary
sage: S = Set_object_binary(X, Y, "union", "\\cup"); S
Set-theoretic union of Set of elements of Vector space of dimension 2
 over Rational Field and Set of elements of Integer Ring

class sage.sets.set.Set_object_difference(X, Y)

Bases: sage.sets.set.Set_object_binary

Formal difference of two sets.

is_finite()

Return whether this set is finite.

EXAMPLES:

sage: X = Set(range(10))
sage: Y = Set(range(-10,5))
sage: Z = Set(QQ)
sage: X.difference(Y).is_finite()
True
sage: X.difference(Z).is_finite()
True
sage: Z.difference(X).is_finite()
False
sage: Z.difference(Set(ZZ)).is_finite()
Traceback (most recent call last):
...
NotImplementedError

class sage.sets.set.Set_object_enumerated(X)

Bases: sage.sets.set.Set_object

A finite enumerated set.

cardinality()

Return the cardinality of self.

EXAMPLES:

sage: Set([1,1]).cardinality()
1

difference(other)

Return the set difference self - other.

EXAMPLES:

sage: X = Set([1,2,3,4])
sage: Y = Set([1,2])
sage: X.difference(Y)
{3, 4}
sage: Z = Set(ZZ)
sage: W = Set([2.5, 4, 5, 6])
sage: W.difference(Z)
{2.50000000000000}

frozenset()

Return the Python frozenset object associated to this set, which is an immutable set (hence hashable).

EXAMPLES:

sage: X = Set(GF(8,'c'))
sage: X
{0, 1, c, c + 1, c^2, c^2 + 1, c^2 + c, c^2 + c + 1}
sage: s = X.set(); s
{0, 1, c, c + 1, c^2, c^2 + 1, c^2 + c, c^2 + c + 1}
sage: hash(s)
Traceback (most recent call last):
...
TypeError: unhashable type: 'set'
sage: s = X.frozenset(); s
frozenset({0, 1, c, c + 1, c^2, c^2 + 1, c^2 + c, c^2 + c + 1})
sage: hash(s)
-1390224788            # 32-bit
 561411537695332972    # 64-bit
sage: type(s)
<type 'frozenset'>

intersection(other)

Return the intersection of self and other.

EXAMPLES:

sage: X = Set(GF(8,'c'))
sage: Y = Set([GF(8,'c').0, 1, 2, 3])
sage: X.intersection(Y)
{1, c}

is_finite()

Return True as this is a finite set.

EXAMPLES:

sage: Set(GF(19)).is_finite()
True

issubset(other)

Return whether self is a subset of other.

INPUT:

• other – a finite Set

EXAMPLES:

sage: X = Set([1,3,5])
sage: Y = Set([0,1,2,3,5,7])
sage: X.issubset(Y)
True
sage: Y.issubset(X)
False
sage: X.issubset(X)
True

TESTS:

sage: len([Z for Z in Y.subsets() if Z.issubset(X)])
8

issuperset(other)

Return whether self is a superset of other.

INPUT:

• other – a finite Set

EXAMPLES:

sage: X = Set([1,3,5])
sage: Y = Set([0,1,2,3,5])
sage: X.issuperset(Y)
False
sage: Y.issuperset(X)
True
sage: X.issuperset(X)
True

TESTS:

sage: len([Z for Z in Y.subsets() if Z.issuperset(X)])
4

list()

Return the elements of self, as a list.

EXAMPLES:

sage: X = Set(GF(8,'c'))
sage: X
{0, 1, c, c + 1, c^2, c^2 + 1, c^2 + c, c^2 + c + 1}
sage: X.list()
[0, 1, c, c + 1, c^2, c^2 + 1, c^2 + c, c^2 + c + 1]
sage: type(X.list())
<type 'list'>

Todo

FIXME: What should be the order of the result? That of self.object()? Or the order given by set(self.object())? Note that __getitem__() is currently implemented in term of this list method, which is really inefficient ...

set()

Return the Python set object associated to this set.

Python has a notion of finite set, and often Sage sets have an associated Python set. This function returns that set.

EXAMPLES:

sage: X = Set(GF(8,'c'))
sage: X
{0, 1, c, c + 1, c^2, c^2 + 1, c^2 + c, c^2 + c + 1}
sage: X.set()
{0, 1, c, c + 1, c^2, c^2 + 1, c^2 + c, c^2 + c + 1}
sage: type(X.set())
<type 'set'>
sage: type(X)
<class 'sage.sets.set.Set_object_enumerated_with_category'>

symmetric_difference(other)

Return the symmetric difference of self and other.

EXAMPLES:

sage: X = Set([1,2,3,4])
sage: Y = Set([1,2])
sage: X.symmetric_difference(Y)
{3, 4}
sage: Z = Set(ZZ)
sage: W = Set([2.5, 4, 5, 6])
sage: U = W.symmetric_difference(Z)
sage: 2.5 in U
True
sage: 4 in U
False
sage: V = Z.symmetric_difference(W)
sage: V == U
True
sage: 2.5 in V
True
sage: 6 in V
False

union(other)

Return the union of self and other.

EXAMPLES:

sage: X = Set(GF(8,'c'))
sage: Y = Set([GF(8,'c').0, 1, 2, 3])
sage: X
{0, 1, c, c + 1, c^2, c^2 + 1, c^2 + c, c^2 + c + 1}
sage: Y
{1, c, 3, 2}
sage: X.union(Y)
{0, 1, c, c + 1, c^2, c^2 + 1, c^2 + c, c^2 + c + 1, 2, 3}

class sage.sets.set.Set_object_intersection(X, Y)

Bases: sage.sets.set.Set_object_binary

Formal intersection of two sets.

is_finite()

Return whether this set is finite.

EXAMPLES:

sage: X = Set(IntegerRange(100))
sage: Y = Set(ZZ)
sage: X.intersection(Y).is_finite()
True
sage: Y.intersection(X).is_finite()
True
sage: Y.intersection(Set(QQ)).is_finite()
Traceback (most recent call last):
...
NotImplementedError

class sage.sets.set.Set_object_symmetric_difference(X, Y)

Bases: sage.sets.set.Set_object_binary

Formal symmetric difference of two sets.

is_finite()

Return whether this set is finite.

EXAMPLES:

sage: X = Set(range(10))
sage: Y = Set(range(-10,5))
sage: Z = Set(QQ)
sage: X.symmetric_difference(Y).is_finite()
True
sage: X.symmetric_difference(Z).is_finite()
False
sage: Z.symmetric_difference(X).is_finite()
False
sage: Z.symmetric_difference(Set(ZZ)).is_finite()
Traceback (most recent call last):
...
NotImplementedError

class sage.sets.set.Set_object_union(X, Y)

Bases: sage.sets.set.Set_object_binary

A formal union of two sets.

cardinality()

Return the cardinality of this set.

EXAMPLES:

sage: X = Set(GF(3)).union(Set(GF(2)))
sage: X
{0, 1, 2, 0, 1}
sage: X.cardinality()
5

sage: X = Set(GF(3)).union(Set(ZZ))
sage: X.cardinality()
+Infinity

is_finite()

Return whether this set is finite.

EXAMPLES:

sage: X = Set(range(10))
sage: Y = Set(range(-10,0))
sage: Z = Set(Primes())
sage: X.union(Y).is_finite()
True
sage: X.union(Z).is_finite()
False

sage.sets.set.is_Set(x)

Returns True if x is a Sage Set_object (not to be confused with a Python set).

EXAMPLES:

sage: from sage.sets.set import is_Set
sage: is_Set([1,2,3])
False
sage: is_Set(set([1,2,3]))
False
sage: is_Set(Set([1,2,3]))
True
sage: is_Set(Set(QQ))
True
sage: is_Set(Primes())
True