AUTHORS:
Return the enumerated set associated to 
.
The input object must be finite.
EXAMPLES:
sage: EnumeratedSet([1,1,2,3])
doctest:1: DeprecationWarning: EnumeratedSet is deprecated; use Set instead.
{1, 2, 3}
sage: EnumeratedSet(ZZ)
...
ValueError: X (=Integer Ring) must be finite
Create the underlying set of .
If 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'>
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'>
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]
TESTS:
sage: Set(Primes())
Set of all prime numbers: 2, 3, 5, 7, ...
sage: Set(Subsets([1,2,3])).cardinality()
8
sage: Set(iter([1,2,3]))
{1, 2, 3}
sage: type(_)
<class 'sage.sets.set.Set_object_enumerated'>
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: loads(S.dumps()) == S
True
sage: latex(Set(ZZ))
\Bold{Z}
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
Return the intersection of self and 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}
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
{}
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
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, ...
Return the Subset 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}]
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}
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}
Bases: sage.sets.set.Set_object
Formal difference of two sets.
This tries to return the cardinality of this formal intersection.
Note that this is not likely to work in very much generality, and may just hang if either set involved is infinite.
EXAMPLES:
sage: X = Set(GF(13)).difference(Set(Primes()))
sage: X.cardinality()
8
Bases: sage.sets.set.Set_object
A finite enumerated set.
EXAMPLES:
sage: Set([1,1]).cardinality()
1
Returns 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}
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
set([0, 1, c, c + 1, c^2, c^2 + 1, c^2 + c, c^2 + c + 1])
sage: hash(s)
...
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'>
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}
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'>
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 ...
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()
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'>
Returns the set difference self-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
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}
Bases: sage.sets.set.Set_object
Formal intersection of two sets.
This tries to return the cardinality of this formal intersection.
Note that this is not likely to work in very much generality, and may just hang if either set involved is infinite.
EXAMPLES:
sage: X = Set(GF(13)).intersection(Set(ZZ))
sage: X.cardinality()
13
Bases: sage.sets.set.Set_object
Formal symmetric difference of two sets.
This tries to return the cardinality of this formal symmetric difference.
Note that this is not likely to work in very much generality, and may just hang if either set involved is infinite.
EXAMPLES:
sage: X = Set(GF(13)).symmetric_difference(Set(range(5)))
sage: X.cardinality()
8
Bases: sage.sets.set.Set_object
A formal union of two sets.
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
Returns true if 
is a Sage Set (not to be confused with
a Python 2.4 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