Multiset

In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set,^[1] allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that element in the multiset. As a consequence, an infinite number of multisets exist that contain only elements $a$ and $b$ , but vary in the multiplicities of their elements:

The set ${a, b}$ contains only elements $a$ and $b$ , each having multiplicity 1 when ${a, b}$ is seen as a multiset.
In the multiset ${a, a, b}$ , the element $a$ has multiplicity 2, and $b$ has multiplicity 1.
In the multiset ${a, a, a, b, b, b}$ , $a$ and $b$ both have multiplicity 3.

These objects are all different when viewed as multisets, although they are the same set, since they all consist of the same elements. As with sets, and in contrast to tuples, the order in which elements are listed does not matter in discriminating multisets, so ${a, a, b}$ and ${a, b, a}$ denote the same multiset. To distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset ${a, a, b}$ can be denoted by $[a, a, b]$ .^[2]

The cardinality or "size" of a multiset is the sum of the multiplicities of all its elements. 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 therefore the cardinality of this multiset is 6.

Nicolaas Govert de Bruijn coined the word multiset in the 1970s, according to Donald Knuth.^[3]^: 694 However, the concept of multisets predates the coinage of the word multiset by many centuries. Knuth himself attributes the first study of multisets to the Indian mathematician Bhāskarāchārya, who described permutations of multisets around 1150. Other names have been proposed or used for this concept, including list, bunch, bag, heap, sample, weighted set, collection, and suite.^[3]^: 694

^ Cantor, Georg; Jourdain, Philip E.B. (Translator) (1895). "beiträge zur begründung der transfiniten Mengenlehre" [contributions to the founding of the theory of transfinite numbers]. Mathematische Annalen (in German). xlvi, xlix. New York Dover Publications (1954 English translation): 481–512, 207–246. Archived from the original on 2011-06-10. By a set (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Gansen) M of definite and separate objects m (p.85)
^ Hein, James L. (2003). Discrete mathematics. Jones & Bartlett Publishers. pp. 29–30. ISBN 0-7637-2210-3.
^ ^a ^b Knuth, Donald E. (1998). Seminumerical Algorithms. The Art of Computer Programming. Vol. 2 (3rd ed.). Addison Wesley. ISBN 0-201-89684-2.

[Cantor-1] Cantor, Georg; Jourdain, Philip E.B. (Translator) (1895). "beiträge zur begründung der transfiniten Mengenlehre" [contributions to the founding of the theory of transfinite numbers]. Mathematische Annalen (in German). xlvi, xlix. New York Dover Publications (1954 English translation): 481–512, 207–246. Archived from the original on 2011-06-10. By a set (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Gansen) M of definite and separate objects m (p.85)

[2] Hein, James L. (2003). Discrete mathematics. Jones & Bartlett Publishers. pp. 29–30. ISBN 0-7637-2210-3.

[Knuth1998-3] Knuth, Donald E. (1998). Seminumerical Algorithms. The Art of Computer Programming. Vol. 2 (3rd ed.). Addison Wesley. ISBN 0-201-89684-2.

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