Multiset

In mathematics, a multiset 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.

The number of submultisets of size k in a set of size n is _{n + k − 1}C_k, 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 need not be 0 when k > n.)

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.