Multi-Fuzzy Complex Numbers and Multi-Fuzzy Complex Sets

1. Introduction

The elements of a crisp set are pairwise different but if we allow repeated occurrences of any element, then we get a mathematical structure. This mathematical structure is called multiset (Blizard, 1991). The numerous applications of multisets have found in mathematics and computer science. In addition multisets are used in concurrency theory (Nicola & Smolka 1996).

If we allow repeated occurrences of any element of a set integral number of times (includes a negative number of times), we get a structure that has been called hybrid set. Loeb (Loeb, 1992) introduced this mathematical structure and shown that one can use a hybrid set to describe the roots of a rational functions where elements that occur a positive number of times and describe the poles of a rational function where elements that occur a negative number of times.

Yager (Yager, 1985) developed the concept of fuzzy multisets from crisp multisets and it is a fuzzy subsets whose elements may occur more than once. The definition of fuzzy multisets given by Yager is as follows:

• Definition 1: Let be a set of elements. Then a fuzzy bag drawn from can be characterized by a function such that , where is the set of all crisp bags drawn from the unit intervals.

In this paper, stands for the set of complex numbers, stands for the set of natural numbers, and stands for the unit interval and the set of all functions from to respectively. stands for (-times), .

A fuzzy multiset can be characterized by a function , where , is the set of natural number including zero and consisting of all the mappings from to . Now, one can demand that for each element there is only one membership degree and one multiplicity. In other word, a fuzzy multiset should be characterized by a function .