Multiset

In mathematics, a multiset (also a bag) differs from a set in that each member has a multiplicity, which is a cardinal number indicating (loosely speaking) how many times it is a member, or perhaps how many memberships it has in the multiset. For example, in the multiset { a, a, b, b, b, c }, the multiplicities of the members a, b, and c are respectively 2, 3, and 1.

One of the most natural and simple examples is the multiset of prime factors of a number. Another is the multiset of solutions of an algebraic equation. Everyone learns in secondary school that a quadratic equation has two solutions, but in some cases they are both the same number. Thus the multiset of solutions of the equation could be { 3, 5 }, or it could be { 4, 4 }. In the latter case it has a solution of multiplicity 2.

Table of contents

1 Formal definition 2 Operations 3 Counting 4 Free commutative monoids

Formal definition

Formally multisets can be defined, within set theory, as partial functions that map elements to positive natural numbers. So in terms of sets

• the multiset written as {a, b, b} is defined as {(a, 1), (b, 2)},
• likewise {a, a, b} is defined as {(a, 2), (b, 1)}, and
• the multiset meaning of {a, b} is defined as {(a, 1), (b, 1)}.

Operations

The usual set operations such as set union, set intersection and cartesian product can be easily generalized for multisets.

• the union of A and B can be defined as the function F on the union of the domain of A and B such that F(x) = A(x) + B(x)
• the intersection of A and B can be defined as the function F on the intersection of the domain of A and B such that F(x) = MIN(A(x), B(x))
• the cartesian product of A and B can be defined as the function F on the cartesian product of the domains of A and B such that F((x,y)) = A(x).B(y)

Counting

The number of submultisets of size k in a set of size n is the binomial coefficient

i.e., it is the same as the number of subsets of size k in a set of size n + k − 1. (Note that, unlike the situation with sets, this cardinality will not be 0 when k > n.)

Free commutative monoids

There is a connection with the free object concept: the free commutative monoid on a set X can be taken to be the set of finite multisets with elements drawn from X, with the obvious addition operation.