Multiset

In mathematics, a multiset (or bag) is a generalization of the concept of a set that, unlike a set, allows multiple instances of the multiset's elements. For example, {a, a, b} and {a, b} are different multisets although they are the same set. However, order does not matter, so {a, a, b} and {a, b, a} are the same multiset.

The multiplicity of an element is the number of instances of the element in a specific multiset. For example, an infinite number of multisets exist which contain only elements a and b, varying only by multiplicity:

Nicolaas Govert de Bruijn coined the word multiset in the 1970s, according to Donald Knuth. However, the use of multisets predates the word multiset by many centuries. Knuth attributes the first study of multisets to the Indian mathematician Bhāskarāchārya, who described permutations of multisets around 1150. Knuth also lists other names that were proposed or used for multisets, including list, bunch, bag, heap, sample, weighted set, collection, and suite.

The number of times an element belongs to the multiset is the multiplicity of that member. The total number of elements in a multiset, including repeated memberships, is the cardinality of 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, and the cardinality of the multiset is 6. To distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset {2, 2, 3} can be represented as [2, 2, 3]. In multisets, as in sets and in contrast to tuples, the order of elements is irrelevant: The multisets {a, a, b} and {a, b, a} are equal.

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