Basic Operations of Fuzzy Multisets

Abstract

A multiset is a set-like entity which can contain repeated elements. Given a universal set, a multiset may be constructed by selecting elements of the universal set. Moreover an element of the universal set may be selected more than once. For example, let the universal set be X={a, b, c, d}, then A={a, a, b, b, b} is a multiset. the class of ordinary sets can be regarded as a subclass in the class of all multisets. Fuzzy multisets which are also called fuzzy bags have been studied by Yager who defined basic relations of inclusion and equality, and operations of union and intersection. However, his definitions are inappropriate, since the basic relations and operations are inconsistent with the corresponding relations and operations for ordinary fuzzy sets. In this paper, the basic relations of inclusion and equality, and the basic operatoins of union and intersection are newly defined using a grade sequence for each element of the universal set. Appropriateness of the new difinitions is shown by proving consistency with the ordinary fuzzy set relations and operations. Moreover and α-cut for fuzzy multisets is defined and the properties of the commutative law, the associative law, and the distributive law for fuzzy multisets are proved using the α-cut. Possible applications of fuzzy multisets are suggested.