Multi-Fuzzy Complex Numbers and Multi-Fuzzy Complex Sets

Multi-Fuzzy Complex Numbers and Multi-Fuzzy Complex Sets

Asit Dey (Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, India) and Madhumangal Pal (Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore, India)
Copyright: © 2015 |Pages: 13
DOI: 10.4018/IJFSA.2015040102

Abstract

This paper presents a method to construct more general fuzzy complex numbers and sets from ordinary fuzzy complex numbers by introduced the ordered sequences of membership functions. It is concluded that multi-fuzzy complex set is an extension of Buckley fuzzy complex set. Also, two types of multi-fuzzy complex numbers based on the forms and are investigated.
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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 IJFSA.2015040102.m03 be a set of elements. Then a fuzzy bag IJFSA.2015040102.m04 drawn from IJFSA.2015040102.m05 can be characterized by a function IJFSA.2015040102.m06 such that IJFSA.2015040102.m07, where IJFSA.2015040102.m08 is the set of all crisp bags drawn from the unit intervals.

In this paper, IJFSA.2015040102.m09 stands for the set of complex numbers, IJFSA.2015040102.m10 stands for the set of natural numbers, IJFSA.2015040102.m11 and IJFSA.2015040102.m12 stands for the unit interval IJFSA.2015040102.m13 and the set of all functions from IJFSA.2015040102.m14 to IJFSA.2015040102.m15 respectively. IJFSA.2015040102.m16 stands for IJFSA.2015040102.m17 (IJFSA.2015040102.m18-times), IJFSA.2015040102.m19.

A fuzzy multiset IJFSA.2015040102.m20 can be characterized by a function IJFSA.2015040102.m21, where IJFSA.2015040102.m22, IJFSA.2015040102.m23 is the set of natural number including zero and IJFSA.2015040102.m24 consisting of all the mappings from IJFSA.2015040102.m25 to IJFSA.2015040102.m26. Now, one can demand that for each element IJFSA.2015040102.m27 there is only one membership degree and one multiplicity. In other word, a fuzzy multiset IJFSA.2015040102.m28 should be characterized by a function IJFSA.2015040102.m29.

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