Multiset

In mathematics, a multiset (aka bag or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The positive integer number of instances, given for each element is called the multiplicity of this element in the multiset. As a consequence, an infinite number of multisets exist, which contain only elements a and b, but vary by the multiplicity of their elements: In mathematics, a multiset (aka bag or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The positive integer number of instances, given for each element is called the multiplicity of this element in the multiset. As a consequence, an infinite number of multisets exist, which contain only elements a and b, but vary by the multiplicity of their elements: 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, order 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 as . The cardinality of a multiset is constructed by summing up 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.:694However, the use of the concept for multisets predates the coinage of 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. Knuth also lists other names that were proposed or used for this concept, including list, bunch, bag, heap, sample, weighted set, collection, and suite.:694 Wayne Blizard traced multisets back to the very origin of numbers, arguing that “in ancient times, the number n was often represented by a collection of n strokes, tally marks, or units.” These and similar collections of objects are multisets, because strokes, tally marks, or units are considered indistinguishable. This shows that people implicitly used multisets even before mathematics emerged. Practical needs for this structure have caused multisets to be rediscovered several times, appearing in literature under different names.:323 For instance, they were important in early AI languages, such as QA4, where they were referred to as bags, a term attributed to Peter Deutsch. A multiset has been also called an aggregate, heap, bunch, sample, weighted set, occurrence set, and fireset (finitely repeated element set).:320 Although multisets were used implicitly from ancient times, their explicit exploration happened much later. The first known study of multisets is attributed to the Indian mathematician Bhāskarāchārya circa 1150, who described permutations of multisets.:694 The work of Marius Nizolius (1498–1576) contains another early reference to the concept of multisets. Athanasius Kircher found the number of multiset permutations when one element can be repeated. Jean Prestet published a general rule for multiset permutations in 1675. John Wallis explained this rule in more detail in 1685. Multisets appeared explicitly in the work of Richard Dedekind.:114 Other mathematicians formalized multisets and began to study them as precise mathematical structures in the 20th century. For example, Whitney (1933) described generalized sets ('sets' whose characteristic functions may take any integer value - positive, negative or zero).:326:405 Monro (1987) investigated the category Mul of multisets and their morphisms, defining a multiset as a set with an equivalence relation between elements 'of the same sort', and a morphism between multisets as a function which respects sorts. He also introduced a multinumber: a function f(x) from a multiset to the natural numbers, giving the multiplicity of element x in the multiset. Monro argued that the concepts of multiset and multinumber are often mixed indiscriminately, though both are useful.:327-328