A multiset is a set for which repeated elements are considered.

Note that the standard definition of a set also allows repeated elements, but these are not treated as repeated elements. For example, as a set is actually equal to . However, as a multiset, is not simplifiable further.

A definition that makes clear the distinction between set and multiset follows:

Multiset 1   A multiset is a pair , where is a set, and is a function mapping to the cardinal numbers greater than zero. For any , is called the multiplicity of .

Using this definition and expressing as a set of ordered pairs, we see that the multiset has and . By contrast, the multiset has and .

Generally, a multiplicity of zero is not allowed, but a few mathematicians do allow for it, such as Bogart and Stanley. It is far more common to disallow infinite multiplicity, which greatly complicates the definition of operations such as unions, intersections, complements, etc.

Bibliography

1

Kenneth P. Bogart, Introductory Combinatorics. Florence, Kentucky: Cengage Learning (2000): 93

2

John L. Hickman, ``A note on the concept of multiset'' Bulletin of the Australian Mathematical Society 22 (1980): 211 - 217

3

Richard P. Stanley, Enumerative Combinatorics Vol 1. Cambridge: Cambridge University Press (1997): 15