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Informal Definition

A multiset is an extension of the concept of a set.

While a set can contain only one occurrence of any given element, a multiset may contain multiple occurrences of the same element.

Note that by this definition, a set is also classified as a multiset.


A multiset is a pair $\struct {S, \mu}$ where:

$S$ is a set
$\mu: S \to \N_{>0}$ is a mapping to the strictly positive natural numbers

For $s \in S$ the natural number $\map \mu s$ is called the multiplicity of $s$.

Note that the multiplicities of elements is finite: we do not allow infinitely many occurrences of the same element, though the set $S$ itself may be finite, countably infinite or uncountably infinite.


Care must be taken to define equality of multisets such that no restriction is placed on the ordering of elements.

Let $\struct {S, \mu}$ and $\struct {T, \nu}$ be multisets.

We say that $\struct {S, \mu}$ and $\struct {T, \nu}$ are equal if and only if there exists a bijection $\sigma: S \to T$ such that $\mu = \nu \circ \sigma$.

That is, $\map \mu s = \map \nu {\map \sigma s}$ for all $s \in S$.


To distinguish multisets from sets, sometimes multisets are written with double braces, for example:

$\set {\set {1, 2, 3, 4} }$

Beware the fact that such a notation is also used to mean a set containing a set, so be sure you know what notation is being used.


While $\set {a, b, c, d}$ is a set, $\set {\set {a, b, c, c, d} }$ is a multiset.