11.8 Sets

Module: sage.sets.set

Sets

Author Log:

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

Module-level Functions

EnumeratedSet( X)

Return the enumerated set associated to .

The input object must be finite.

sage: EnumeratedSet([1,1,2,3])
{1, 2, 3}
sage: EnumeratedSet(ZZ)
Traceback (most recent call last):
...
ValueError: X (=Integer Ring) must be finite

Set( X)

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.

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

is_Set( x)

Returns true if is a SAGE Set (not to be confused with a Python 2.4 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

Class: Set_object

class Set_object

A set attached to an almost arbitrary object.

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))
\mathbf{Z}

Set_object( self, X)

Create a Set_object

This function is called by the Set function; users shouldn't call this directly.

sage: type(Set(QQ))
<class 'sage.sets.set.Set_object'>

Functions: cardinality, $\,$ difference, $\,$ intersection, $\,$ is_finite, $\,$ object, $\,$ subsets, $\,$ symmetric_difference, $\,$ union

cardinality( self)

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

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

difference( self, X)

Return the intersection of self and X.

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( self, X)

Return the intersection of self and X.

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_finite( self)

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( self)

Return underlying object.

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( self, [size=None])

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

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( self, X)

Returns the symmetric difference of self and X.

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

union( self, X)

Return the union of self and X.

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}

Special Functions: __add__, $\,$ __and__, $\,$ __cmp__, $\,$ __contains__, $\,$ __init__, $\,$ __iter__, $\,$ __or__, $\,$ __sub__, $\,$ _latex_, $\,$ _repr_

__add__( self, X)

Return the union of self and X.

sage: Set(RealField()) + Set(QQ^5)
 Set-theoretic union of Set of elements of Real Field with 53 bits of
precision and Set of elements of Vector space of dimension 5 over Rational
Field
sage: Set(GF(3)) + Set(GF(2))
{0, 1, 2, 0, 1}
sage: Set(GF(2)) + Set(GF(4,'a'))
{0, 1, a, a + 1}
sage: Set(GF(8,'b')) + Set(GF(4,'a'))
{0, 1, b, b + 1, b^2, b^2 + 1, b^2 + b, b^2 + b + 1, a, a + 1, 1, 0}

__and__( self, X)

Returns the intersection of self and X.

sage: Set([2,3]) \& Set([3,4])
{3}
sage: Set(ZZ) \& Set(QQ)
Set-theoretic intersection of Set of elements of Integer Ring and Set of
elements of Rational Field

__cmp__( self, right)

Compare self and right.

If right is not a Set compare types. If right is also a Set, returns comparison on the underlying objects.

Note: If is true this does not necessarily mean that is a subset of . Also, any two sets can be compared, which is a general Python philosophy.

sage: Set(ZZ) == Set(QQ)
False
sage: Set(ZZ) < Set(QQ)
True
sage: Primes() == Set(QQ)
False
sage: Primes() < Set(QQ)
True

sage: Set(QQ) == Primes()
False

__contains__( self, x)

Return True if is in self.

sage: X = Set(ZZ)
sage: 5 in X
True
sage: GF(7)(3) in X
True
sage: 2/1 in X
True
sage: 2/1 in ZZ
True
sage: 2/3 in X
False

Finite fields better illustrate the difference between __contains__ for objects and their underlying sets.

sage: X = Set(GF(7))
sage: X
{0, 1, 2, 3, 4, 5, 6}
sage: 5/3 in X
False
sage: 5/3 in GF(7)
False
sage: Set(GF(7)).union(Set(GF(5)))
{0, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 0}
sage: Set(GF(7)).intersection(Set(GF(5)))
{}

__iter__( self)

Iterate over the elements of this set.

sage: X = Set(ZZ)
sage: I = X.__iter__()
sage: I.next()
0
sage: I.next()
1
sage: I.next()
-1
sage: I.next()
2

__or__( self, X)

Return the union of self and X.

sage: Set([2,3]) | Set([3,4])
{2, 3, 4}
sage: Set(ZZ) | Set(QQ)
Set-theoretic union of Set of elements of Integer Ring and Set of elements
of Rational Field

__sub__( self, X)

Return the difference of self and X.

sage: X = Set(ZZ).difference(Primes())
sage: Y = Set(ZZ) - Primes()
sage: X == Y
True

_latex_( self)

Return latex representation of this set.

This is often the same as the latex representation of this object when the object is infinite.

sage: latex(Set(QQ))
\mathbf{Q}

When the object is finite or a special set then the latex representation can be more interesting.

sage: print latex(Primes())
\text{Set of all prime numbers: 2, 3, 5, 7, ...}
sage: print latex(Set([1,1,1,5,6]))
\left\{1, 5, 6\right\}

_repr_( self)

Print representation of this set.

sage: X = Set(ZZ)
sage: X
Set of elements of Integer Ring
sage: X.rename('{ integers }')
sage: X
{ integers }

Class: Set_object_difference

class Set_object_difference: Formal difference of two sets.

Set_object_difference( self, X, Y)

sage: S = Set(QQ)
sage: T = Set(ZZ)
sage: X = S.difference(T); X
Set-theoretic difference between Set of elements of Rational Field and Set
of elements of Integer Ring
sage: latex(X)
\mathbf{Q} - \mathbf{Z}

sage: loads(X.dumps()) == X
True

Functions: cardinality

cardinality( self)

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.

sage: X = Set(GF(13)).difference(Set(Primes()))
sage: X.cardinality()
8

Special Functions: __cmp__, $\,$ __contains__, $\,$ __init__, $\,$ __iter__, $\,$ _latex_, $\,$ _repr_

__cmp__( self, right)

Try to compare self and right.

Note: Comparison is basically not implemented, or rather it could say sets are not equal even though they are. I don't know how one could implement this for a generic intersection of sets in a meaningful manner. So be careful when using this.

sage: Y = Set(ZZ).difference(Set(QQ))
sage: Y == Set([])
False
sage: X = Set(QQ).difference(Set(ZZ))
sage: Y == X
False
sage: Z = X.difference(Set(ZZ))
sage: Z == X
False

This illustrates that equality testing for formal unions can be misleading in general.

sage: X == Set(QQ).difference(Set(ZZ))
True

__contains__( self, x)

Return true if self contains x.

Since self is a formal intersection of X and Y this function returns true if both X and Y contains x.

       sage: X = Set(QQ).difference(Set(ZZ))
       sage: 5 in X
       False
       sage: ComplexField().0 in X
       False
       sage: sqrt(2) in X     # since sqrt(2) is not a numerical approx
       False
sage: sqrt(RR(2)) in X # since sqrt(RR(2)) is a numerical approx
True
       sage: 5/2 in X
       True

__iter__( self)

Return iterator through elements of self.

Self is a formal difference of X and Y and this function is implemented by iterating through the elements of X and for each checking if it is not in Y, and if yielding it.

sage: X = Set(ZZ).difference(Primes())
sage: I = X.__iter__()
sage: I.next()
0
sage: I.next()
1
sage: I.next()
-1
sage: I.next()
-2
sage: I.next()
-3

_latex_( self)

Return latex representation of self.

sage: X = Set(QQ).difference(Set(ZZ))
sage: latex(X)
\mathbf{Q} - \mathbf{Z}

_repr_( self)

Return string representation of self.

sage: X = Set(QQ).difference(Set(ZZ)); X
Set-theoretic difference between Set of elements of Rational Field and Set
of elements of Integer Ring
sage: X.rename('Q - Z')
sage: X
Q - Z

Class: Set_object_enumerated

class Set_object_enumerated: A finite enumerated set.

Set_object_enumerated( self, X)

sage: S = EnumeratedSet(GF(19)); S
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}
sage: print latex(S)
\left\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18
ight\}
sage: loads(S.dumps()) == S
True

Functions: cardinality, $\,$ difference, $\,$ frozenset, $\,$ intersection, $\,$ set, $\,$ symmetric_difference, $\,$ union

cardinality( self)

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

difference( self, other)

Returns the set difference self-other.

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( self)

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

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)
Traceback (most recent call last):
...
TypeError: set objects are unhashable
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( self, other)

Return the intersection of self and other.

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

set( self)

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.

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'>

symmetric_difference( self, other)

Returns the set difference self-other.

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( self, other)

Return the union of self and other.

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}

Special Functions: __cmp__, $\,$ __init__, $\,$ __iter__, $\,$ __len__, $\,$ _latex_, $\,$ _repr_

__cmp__( self, other)

Compare the sets self and other.

sage: X = Set(GF(8,'c'))
sage: X == Set(GF(8,'c'))
True
sage: X == Set(GF(4,'a'))
False
sage: Set(QQ) == Set(ZZ)
False

__len__( self)

sage: len(Set([1,1]))
1

Class: Set_object_intersection

class Set_object_intersection: Formal intersection of two sets.

Set_object_intersection( self, X, Y)

sage: S = Set(QQ^2)
sage: T = Set(ZZ)
sage: X = S.intersection(T); X
Set-theoretic intersection of Set of elements of Vector space of dimension
2 over Rational Field and Set of elements of Integer Ring
sage: latex(X)
\mathbf{Q}^{2} \cap \mathbf{Z}

sage: loads(X.dumps()) == X
True

Functions: cardinality

cardinality( self)

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.

sage: X = Set(GF(13)).intersection(Set(ZZ))
sage: X.cardinality()
13

Special Functions: __cmp__, $\,$ __contains__, $\,$ __init__, $\,$ __iter__, $\,$ _latex_, $\,$ _repr_

__cmp__( self, right)

Try to compare self and right.

sage: Y = Set(ZZ).intersection(Set(QQ))
sage: X = Set(QQ).intersection(Set(ZZ))
sage: X == Y
True
sage: Y == X
True

This illustrates that equality testing for formal unions can be misleading in general.

sage: Set(ZZ).intersection(Set(QQ)) == Set(QQ)
False

__contains__( self, x)

Return true if self contains x.

Since self is a formal intersection of X and Y this function returns true if both X and Y contains x.

sage: X = Set(QQ).intersection(Set(RealField()))
sage: 5 in X
True
sage: ComplexField().0 in X
False

Floating-point numbers are rational.

sage: RR(sqrt(2)) in X
True

Real constants are not rational:

sage: pi in X
False

pi is not in RR either, since the comparison takes place in the symbolic ring.

sage: pi in RR
False

__iter__( self)

Return iterator through elements of self.

Self is a formal intersection of X and Y and this function is implemented by iterating through the elements of X and for each checking if it is in Y, and if yielding it.

sage: X = Set(ZZ).intersection(Primes())
sage: I = X.__iter__()
sage: I.next()
2

_latex_( self)

Return latex representation of self.

sage: X = Set(ZZ).intersection(Set(QQ))
sage: latex(X)
\mathbf{Z} \cap \mathbf{Q}

_repr_( self)

Return string representation of self.

sage: X = Set(ZZ).intersection(Set(QQ)); X
Set-theoretic intersection of Set of elements of Integer Ring and Set of
elements of Rational Field
sage: X.rename('Z /\ Q')
sage: X
Z /\ Q

Class: Set_object_symmetric_difference

class Set_object_symmetric_difference: Formal symmetric difference of two sets.

Set_object_symmetric_difference( self, X, Y)

sage: S = Set(QQ)
sage: T = Set(ZZ)
sage: X = S.symmetric_difference(T); X
Set-theoretic symmetric difference of Set of elements of Rational Field and
Set of elements of Integer Ring
sage: latex(X)
\mathbf{Q} \bigtriangleup \mathbf{Z}

sage: loads(X.dumps()) == X
True

Functions: cardinality

cardinality( self)

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.

sage: X = Set(GF(13)).symmetric_difference(Set(range(5)))
sage: X.cardinality()
8

Special Functions: __cmp__, $\,$ __contains__, $\,$ __init__, $\,$ __iter__, $\,$ _latex_, $\,$ _repr_

__cmp__( self, right)

Try to compare self and right.

Note: Comparison is basically not implemented, or rather it could say sets are not equal even though they are. I don't know how one could implement this for a generic symmetric difference of sets in a meaningful manner. So be careful when using this.

sage: Y = Set(ZZ).symmetric_difference(Set(QQ))
sage: X = Set(QQ).symmetric_difference(Set(ZZ))
sage: X == Y
True
sage: Y == X
True

__contains__( self, x)

Return true if self contains x.

Since self is the formal symmetric difference of X and Y this function returns true if either X or Y (but now both) contains x.

       sage: X = Set(QQ).symmetric_difference(Primes())
       sage: 4 in X
       True
       sage: ComplexField().0 in X
       False
       sage: sqrt(2) in X      # since sqrt(2) is currently symbolic
       False
sage: sqrt(RR(2)) in X # since sqrt(RR(2)) is currently approximated
True
       sage: pi in X
       False
       sage: 5/2 in X
       True
       sage: 3 in X
       False

__iter__( self)

Return iterator through elements of self.

Self is the formal symmetric difference of X and Y. This function is implemented by first iterating through the elements of X and yielding it if it is not in Y. Then it will iterate throw all the elements of Y and yielding it if it is not in X.

sage: X = Set(ZZ).symmetric_difference(Primes())
sage: I = X.__iter__()
sage: I.next()
0
sage: I.next()
1
sage: I.next()
-1
sage: I.next()
-2
sage: I.next()
-3

_latex_( self)

Return latex representation of self.

sage: X = Set(ZZ).symmetric_difference(Set(QQ))
sage: latex(X)
\mathbf{Z} \bigtriangleup \mathbf{Q}

_repr_( self)

Return string representation of self.

sage: X = Set(ZZ).symmetric_difference(Set(QQ)); X
Set-theoretic symmetric difference of Set of elements of Integer Ring and
Set of elements of Rational Field
sage: X.rename('Z symdif Q')
sage: X
Z symdif Q

Class: Set_object_union

class Set_object_union: A formal union of two sets.

Set_object_union( self, X, Y)

sage: S = Set(QQ^2)
sage: T = Set(ZZ)
sage: X = S.union(T); X
Set-theoretic union of Set of elements of Vector space of dimension 2 over
Rational Field and Set of elements of Integer Ring

sage: latex(X)
\mathbf{Q}^{2} \cup \mathbf{Z}

sage: loads(X.dumps()) == X
True

Functions: cardinality

cardinality( self)

Return the cardinality of this set.

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

Special Functions: __cmp__, $\,$ __contains__, $\,$ __init__, $\,$ __iter__, $\,$ _latex_, $\,$ _repr_

__cmp__( self, right)

Try to compare self and right.

Note: Comparison is basically not implemented, or rather it could say sets are not equal even though they are. I don't know how one could implement this for a generic union of sets in a meaningful manner. So be careful when using this.

sage: Y = Set(ZZ^2).union(Set(ZZ^3))
sage: X = Set(ZZ^3).union(Set(ZZ^2))
sage: X == Y
True
sage: Y == X
True

This illustrates that equality testing for formal unions can be misleading in general.

sage: Set(ZZ).union(Set(QQ)) == Set(QQ)
False

__contains__( self, x)

Returns True if x is an element of self.

sage: X = Set(GF(3)).union(Set(GF(2)))
sage: GF(5)(1) in X
False
sage: GF(3)(2) in X
True
sage: GF(2)(0) in X
True
sage: GF(5)(0) in X
False

__iter__( self)

Return iterator over the elements of self.

sage: [x for x in Set(GF(3)).union(Set(GF(2)))]
[0, 1, 2, 0, 1]

_latex_( self)

Return latex representation of self.

sage: latex(Set(ZZ).union(Set(GF(5))))
\mathbf{Z} \cup \left\{0, 1, 2, 3, 4\right\}

_repr_( self)

Return string representation of self.

sage: Set(ZZ).union(Set(GF(5)))
Set-theoretic union of Set of elements of Integer Ring and {0, 1, 2, 3, 4}

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