A multisetMathworldPlanetmath 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, {1,1,3} as a set is actually equal to {1,3}. However, as a multiset, {1,1,3} is not simplifiable further.

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


A multiset is a pair (X,f), where X is a set, and f is a function mapping X to the cardinal numbersMathworldPlanetmath greater than zero. X is called the underlying set of the multiset, and for any xX, f(x) is the multiplicity of x.

Using this definition and expressing f as a set of ordered pairsMathworldPlanetmath, we see that the multiset {1,3} has X={1,3} and f={(1,1),(3,1)}. By contrast, the multiset {1,1,3} has X={1,3} and f={(1,2),(3,1)}.

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 infiniteMathworldPlanetmath multiplicity, which greatly complicates the definition of operationsMathworldPlanetmath such as unions, intersectionsMathworldPlanetmath, complements, etc.


  • 1 Kenneth P. Bogart, Introductory Combinatorics. Florence, Kentucky: Cengage Learning (2000): 93
  • 2 John L. Hickman, “A note on the conceptMathworldPlanetmath 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
Title multiset
Canonical name Multiset
Date of creation 2013-03-22 12:21:44
Last modified on 2013-03-22 12:21:44
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 13
Author PrimeFan (13766)
Entry type Definition
Classification msc 03E99
Synonym bag
Related topic AxiomsOfMetacategoriesAndSupercategories
Related topic ETAS
Defines multiplicity