Subsections

• Boolean Functions and Operators
• Binary Set Operators
• Other Set Operations

As explained in the Introduction, when elements are taken out of a set their parent will be the universe of the set (or, if the universe is itself a set, the universe of the universe, etc.); in particular, the set itself is not the parent. Hence equality testing on set elements is in fact equality testing between two elements of certain algebraic structures, and the sets are irrelevant. We only list the (in)equality operator for convenience here.

Element membership testing is of critical importance for all types of sets.

Testing whether or not R is a subset of S can be done if R is an enumerated or indexed set and S is any set; hence (in)equality testing is only possible between sets that are not formal sets.

IsNull(R) : SetEnum -> BoolElt

IsNull(R) : SetIndx -> BoolElt

IsNull(R) : SetMulti -> BoolElt

Returns true if and only if the enumerated, indexed, or multi- set R is empty and does not have its universe defined.

IsEmpty(R) : SetIndx -> BoolElt

IsEmpty(R) : SetMulti -> BoolElt

Returns true if and only if the enumerated, indexed or multi- set R is empty.

Given an element x of a set R with universe U and an element y of a set S with universe V, where a common overstructure W can be found with U subset W supset V (see the Introduction for details on overstructures), return true if and only if x and y are equal as elements of W.

Given an element x of a set R with universe U and an element y of a set S with universe V, where a common overstructure W can be found with U subset W supset V (see the Introduction for details on overstructures), return true if and only if x and y are distinct as elements of W.

True if and only if the element x is a member of the set R. If x is not an element of the universe U of R, it is attempted to coerce x into U; if this fails, an error occurs.

True if and only if the element x is not a member of the set R. If x is not an element of the parent structure U of R, it is attempted to coerce x into U; if this fails, an error occurs.

Return the multiplicity in multiset S of element x. If x is not in S, zero is returned.

R subset S : SetIndx, Set -> BoolElt

R subset S : SetMulti, Set -> BoolElt

True if the enumerated, indexed or multi- set R is a subset of the set S, false otherwise. For multisets, if an element x of R has multiplicity n in R, the multiplicity of x in S must be at least n. Coercion of the elements of R into S is attempted if necessary, and an error occurs if this fails.

R notsubset S : SetIndx, Set -> BoolElt

R notsubset S : SetMulti, Set -> BoolElt

True if the enumerated, indexed, or multi- set R is a not a subset of the set S, false otherwise. Coercion of the elements of R into S is attempted if necessary, and an error occurs if this fails.

True if and only if R and S are identical sets, where R and S are enumerated, indexed or multi- sets For indexed sets, the index function is irrelevant for deciding equality. For multisets, matching multiplicities must also be equal. Coercion of the elements of R into S is attempted if necessary, and an error occurs if this fails.

True if and only if R and S are distinct sets, where R and S are enumerated indexed, or multi- sets. For indexed sets, the index function is irrelevant for deciding equality. For multisets, matching multiplicities must also be equal. Coercion of the elements of R into S is attempted if necessary, and an error occurs if this fails.

IsDisjoint(R, S) : SetIndx, SetIndx -> BoolElt

IsDisjoint(R, S) : SetMulti, SetMulti -> BoolElt

True iff the enumerated, indexed or multi- sets R and S are disjoint. Coercion of the elements of R into S is attempted if necessary, and an error occurs if this fails.

For each of the following operators, R and S are sets of the same type. If R and S are both formal sets, then an error will occur unless both have been constructed with the same carrier structure F in the definition. If R and S are both enumerated, indexed, or multi- sets, then an error occurs unless the universes of R and S are compatible, as defined in the Introduction to this Part.

Note that

Q := { ! x in R !}

R join S : SetIndx, SetIndx -> SetIndx

R join S : SetMulti, SetMulti -> SetMulti

R join S : SetForm, SetForm -> SetForm

Union of the sets R and S (see above for the restrictions on R and S). For multisets, matching multiplicities are added in the union.

R meet S : SetIndx, SetIndx -> SetIndx

R meet S : SetMulti, SetMulti -> SetMulti

R meet S : SetForm, SetForm -> SetForm

Intersection of the sets R and S (see above for the restrictions on R and S). For multisets, the minimum of matching multiplicities is stored in the intersection.

R diff S : SetForm, SetForm -> SetForm

Difference of the sets R and S, i.e., the set consisting of those elements of R which are not members of S (see above for the restrictions on R and S).

R sdiff S : SetForm, SetForm -> SetForm

Symmetric difference of the sets R and S, i.e., the set consisting of those elements which are members of either R or S but not both (see above for the restrictions on R and S).

> R := { 1, 2, 3 };
> S := { 1, 1/2, 1/3 };
> R join S;
{ 1/3, 1/2, 1, 2, 3 }
> R meet S;
{ 1 }
> R diff S;
{ 2, 3 }
> S diff R;
{ 1/3, 1/2 }
> R sdiff S;
{ 1/3, 1/2, 2, 3 }

The set of all subsets of S.

Subsets(S, k) : SetEnum, RngIntElt -> SetEnum

The set of subsets of S of size k. If k is larger than the cardinality of S then the result will be empty.

The set of multisets consisting of k not necessarily distinct elements of S.

The set of sequences of length k with elements from S.

The set of permutations (stored as sequences) of the elements of S.

Permutations(S, k) : SetEnum, RngIntElt -> SetEnum;

The set of permutations (stored as sequences) of each of the subsets of S of cardinality k.