Module: sage.sets.set
Author Log:
Module-level Functions
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
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 set type
with extra functionality. Otherwise it returns a more formal
wrapper.
sage: X = Set(GF(9)) sage: X {a, 2*a + 1, a + 2, a + 1, 2*a + 2, 1, 0, 2, 2*a} sage: type(X) <class 'sage.sets.set.Set_object_enumerated'> sage: Y = X.union(Set(QQ)) sage: Y Set-theoretic union of Finite Field in a of size 3^2 and Rational Field sage: type(Y) <class 'sage.sets.set.Set_object_union'>
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_generic
Functions: category,
object
self) |
The category that this set belongs to, which is the category of all sets.
sage: Set(QQ).category() Category of sets
Class: Set_object
sage: K = GF(19) sage: Set(K) {11, 10, 13, 12, 15, 14, 17, 16, 18, 1, 0, 3, 2, 5, 4, 7, 6, 9, 8} sage: S = Set(K)
sage: latex(S) \left\{11, 10, 13, 12, 15, 14, 17, 16, 18, 1, 0, 3, 2, 5, 4, 7, 6, 9, 8\right\} sage: loads(S.dumps()) == S True
sage: latex(Set(ZZ)) \mbox{\bf{}Z}
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: intersection,
object,
order,
union
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
This is false since
does not have a canonical coercion
map to
.
sage: 2/1 in X False
This intersection is empty because canonical coercions are only defined if they are defined on the whole parent.
sage: X = Set(GF(9)).intersection(Set(GF(27))) sage: X {}
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, ...
self) |
Return the order of this set, which is either an integer or Infinity.
sage: Set(ZZ).order() Infinity sage: Primes().order() Infinity sage: Set(GF(5)).order() 5 sage: Set(GF(5^2)).order() 25
self, X) |
Return the union of self and X.
sage: Set(QQ).union(Set(ZZ)) Set-theoretic union of Rational Field and Integer Ring sage: Set(QQ) + Set(ZZ) Set-theoretic union of Rational Field and Integer Ring sage: X = Set(QQ).union(Set(GF(3))); X Set-theoretic union of Rational Field and Finite Field of size 3 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)) {1, 0, 1, 0, 3, 2, 5, 4, 6, 2}
Special Functions: __add__,
__cmp__,
__contains__,
__iter__,
_latex_,
_repr_
self, X) |
Return the union of self and X.
sage: Set(RealField()) + Set(QQ^5) Set-theoretic union of Real Field with 53 bits of precision and Vector space of dimension 5 over Rational Field sage: Set(GF(3)) + Set(GF(2)) {1, 0, 2, 1, 0} sage: Set(GF(2)) + Set(GF(4)) {1, 0, a, a + 1} sage: Set(GF(8)) + Set(GF(4)) {a, a^2 + 1, a + 1, a^2, 0, a + 1, 1, 0, 1, a, a^2 + a, a^2 + a + 1}
self, right) |
Compare self and right.
If right is not a Set always returns -1. 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
This illustrates that < is not the same as containment.
sage: Set(QQ) < Primes() True sage: Primes() < Set(QQ) True
self, x) |
Return True if
is in self.
This usually means that x can be canonically coerced into self. Note that "x in G" for most objects besides sets means that x can be coerced into G, not necessarily canonically. But for sets the coercion has to be canonical (for objects that support a notion of canonical coercion, i.e., an _coerce_ method).
sage: X = Set(ZZ) sage: 5 in X True sage: GF(7)(3) in X False sage: 2/1 in X False 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 {1, 0, 3, 2, 5, 4, 6} sage: 5/3 in X False sage: 5/3 in GF(7) True sage: Set(GF(7)).union(Set(GF(5))) {1, 0, 3, 1, 0, 3, 2, 5, 4, 6, 2, 4} sage: Set(GF(7)).intersection(Set(GF(5))) {}
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
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)) \mbox{\bf{}Q}
When the object is finite or a special set then the latex representation can be more interesting.
sage: print latex(Primes()) \mbox{\rm Set of all prime numbers: 2, 3, 5, 7, ...} sage: print latex(Set([1,1,1,5,6])) \left\{1, 5, 6\right\}
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_enumerated
self, X) |
sage: S = EnumeratedSet(GF(19)); S {11, 10, 13, 12, 15, 14, 17, 16, 18, 1, 0, 3, 2, 5, 4, 7, 6, 9, 8} sage: print latex(S) \left\{11, 10, 13, 12, 15, 14, 17, 16, 18, 1, 0, 3, 2, 5, 4, 7, 6, 9, 8 ight\} sage: loads(S.dumps()) == S True
Functions: intersection,
order,
set,
union
self, other) |
Return the intersection of self and other.
sage: X = Set(GF(8)) sage: Y = Set([GF(8).0, 1, 2, 3]) sage: X.intersection(Y) {a}
self) |
sage: Set([1,1]).order() 1
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)) sage: X {a, a^2 + 1, a + 1, a^2, 1, 0, a^2 + a, a^2 + a + 1} sage: X.set() set([a, a^2 + 1, a + 1, a^2, 1, 0, a^2 + a, a^2 + a + 1]) sage: type(X.set()) <type 'set'> sage: type(X) <class 'sage.sets.set.Set_object_enumerated'>
self, other) |
Return the union of self and other.
sage: X = Set(GF(8)) sage: Y = Set([GF(8).0, 1, 2, 3]) sage: X {a, a^2 + 1, a + 1, a^2, 1, 0, a^2 + a, a^2 + a + 1} sage: Y {a, 1, 2, 3} sage: X.union(Y) {a, a^2 + 1, 2, 3, a + 1, a^2, 1, 1, 0, a^2 + a, a^2 + a + 1}
Special Functions: __cmp__,
__iter__,
__len__,
_latex_,
_repr_
self, other) |
Compare the sets self and other.
sage: X = Set(GF(8)) sage: X == Set(GF(8)) True sage: X == Set(GF(4)) False sage: Set(QQ) == Set(ZZ) False
self) |
sage: len(Set([1,1])) 1
Class: 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 Vector space of dimension 2 over Rational Field and Integer Ring sage: latex(X) \mbox{\bf{}Q}^{2} \cap \mbox{\bf{}Z}
sage: loads(X.dumps()) == X True
Functions: order
self) |
This tries to return the order 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.order() 0
Special Functions: __cmp__,
__contains__,
__iter__,
_latex_,
_repr_
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).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
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 sage: sqrt(2) in X False sage: pi in X False sage: pi in RR True
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
self) |
Return latex representation of self.
sage: X = Set(ZZ).intersection(Set(QQ)) sage: latex(X) \mbox{\bf{}Z} \cap \mbox{\bf{}Q}
self) |
Return string representation of self.
sage: X = Set(ZZ).intersection(Set(QQ)); X Set-theoretic intersection of Integer Ring and Rational Field sage: X.rename('Z /\ Q') sage: X Z /\ Q
Class: 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 Vector space of dimension 2 over Rational Field and Integer Ring
sage: latex(X) \mbox{\bf{}Q}^{2} \cup \mbox{\bf{}Z}
sage: loads(X.dumps()) == X True
Functions: order
self) |
Return the order of this set.
sage: X = Set(GF(3)).union(Set(GF(2))) sage: X {1, 0, 2, 1, 0} sage: X.order() 5
sage: X = Set(GF(3)).union(Set(ZZ)) sage: X.order() Infinity
Special Functions: __cmp__,
__contains__,
__iter__,
_latex_,
_repr_
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
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
self) |
Return iterator over the elements of self.
sage: [x for x in Set(GF(3)).union(Set(GF(2)))] [1, 0, 2, 1, 0]
self) |
Return latex representation of self.
sage: latex(Set(ZZ).union(Set(GF(5)))) \mbox{\bf{}Z} \cup \left\{1, 0, 3, 2, 4\right\}
self) |
Return string representation of self.
sage: Set(ZZ).union(Set(GF(5))) Set-theoretic union of Integer Ring and Finite Field of size 5
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