Multiset

In mathematics, a multiset (sometimes also called 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.

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 union, 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 — "multiset coefficients"

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

<math>\left\langle \begin{matrix}n \\ k \end{matrix}\right\rangle = {n + k – 1 \choose n-1}={n+k-1 \choose k},<math>

where the expressions to the right of "=" are binomial coefficients, i.e., the number of such multisets is the same as the number of subsets of size k in a set of size n + k − 1. Unlike the situation with sets, this cardinality will not be 0 when k > n. One simple way to prove this involves representing multisets in the following way. First, consider the notation for multisets that would represent { a, a, a, a, a, a, b, b, c, c, c, d, d, d, d, d, d, d } (6 as, 2 bs, 3 cs, 7 ds) in this form:

<math>\bullet \bullet \bullet \bullet \bullet \bullet \mid \bullet \bullet \mid \bullet \bullet \bullet \mid \bullet \bullet \bullet \bullet \bullet \bullet \bullet <math>

This is a multiset of size 18 made of elements of a set of size 4. The number of characters including both dots and vertical lines used in this notation is 18 + 4 − 1. The number of vertical lines is 4 − 1. The number of multisets of size 18 is then the number of ways to arrange the 4 − 1 vertical lines among the 18 + 4 − 1 characters, and is thus the number of subsets of size 4 − 1 in a set of size 18 + 4 − 1. Equivalently, it is the number of ways to arrange the 18 dots among the 18 + 4 − 1 characters, which is the number of subsets of size 18 of a set of size 18 + 4 − 1. This is

<math>{18+4–1 \choose 4–1}={18+4–1 \choose 18},<math>

so that is the value of the multiset coefficient

<math>\left\langle\begin{matrix} 4 \\ 18 \end{matrix}\right\rangle.<math>

One may define a generalized binomial coefficient

<math>{n \choose k}={n(n-1)(n-2)\cdots(n-k+1) \over k!}<math>

in which n is not required to be a nonnegative integer, but may be negative or a non-integer, or a non-real complex number. (If k = 0, then the value of this coefficient is 1 because it is the product of no numbers.) Then the number of multisets of size k in a set of size n is

<math>\left\langle\begin{matrix} n \\ k \end{matrix}\right\rangle=(-1)^k{-n \choose k}.<math>

This fact led Gian-Carlo Rota to ask "Why are negative sets multisets?". He considered that question worthy of the attention of philosophers of mathematics.

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.