Editing Multiset (section)

==Definition==
A '''multiset''' may be formally defined as an [[ordered pair]] {{math|(''U'', ''m'')}} where {{mvar|U}} is a [[set (mathematics)|set]] called a [[Universe (mathematics)|''universe'']] or the ''underlying set'', and <math>m\colon U \to \mathbb{Z}_{\ge 0}</math> is a function from {{mvar|U}} to the [[nonnegative integer]]s. The value {{tmath|m(a)}} for an element {{tmath|a\in U}} is called the ''multiplicity'' of {{tmath|a}} in the multiset and intepreted as the number of occurences of {{tmath|a}} in the multiset.

The ''support'' of a multiset is the subset of {{tmath|U}} formed by the elements {{tmath|a\in U}} such that {{tmath|m(a)>0}}. A ''finite multiset'' is a multiset with a [[finite set|finite]] support. Most authors define ''multisets'' as finite multisets. This is the case in this article, where, unless otherwise stated, all multisets are finite multisets. 

Some authors{{who|date=February 2025}} define multisets with the additional constraint that {{tmath|m(a)>0}} for every {{tmath|a}}, or, equivalently, the support equals the underlying set.
Multisets with infinite multiplicities have also been studied;<ref>Cf., for instance, Richard Brualdi, ''Introductory Combinatorics'', Pearson.</ref> they are not considered in this article.
Some authors{{who|date=February 2025}} define a multiset in terms of a finite index set {{tmath|I}} and a function {{tmath|f \colon I \rightarrow U}} where the multiplicity of an element {{tmath|a \in U}} is {{tmath|{{!}}f^{-1}(a){{!}}}}, the number of elements of {{tmath|I}} that get mapped to {{tmath|a}} by {{tmath|f}}.

Multisets may be represented as sets, with some elements repeated. For example, the multiset with support {{tmath|\{a,b\} }} and multiplicity function such that {{tmath|1=m(a)=2, \; m(b)=1}} can be represented as {{math|{{mset|''a'', ''a'', ''b''}}}}. A more compact notation, in case of high multiplicities is {{tmath| \{(a, 2), (b, 1)\} }} for the same multiset.

If <math>A=\{a_1, \ldots, a_n\},</math>  a multiset with support included in {{tmath|A}}  is often represented as 
<math display=block>a_1^{m(a_1)} \cdots a_n^{m(a_n)},</math>
to which the computation rules of [[indeterminate (variable)|indeterminate]]s can be applied; that is, exponents 1 and factors with exponent 0 can be removed, and the multiset does not depend on the order of the factors. This allows extending the notation to infinite underlying sets as
<math display=block>\prod_{a\in U} a^{m(a)}.</math> 
An advantage of notation is that it allows using the notation without knowing the exact support. For example, the [[prime factor]]s of a [[natural number]] {{tmath|n}} form a multiset such that
<math display=block>n=\prod_{p \;\text {prime}} p^{m(p)}= 2^{m(2)} 3^{m(3)} 5^{m(5)}\cdots.</math> 

The finite subsets of a set {{tmath|U}} are exactly the multisets with underlying set {{tmath|U}}, such that {{tmath|m(a)\le 1}} for every {{tmath|a\in U}}.