Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques

Abstract: The novelty of this paper is to construct the explicit combinatorial formula for the number of all distinct fuzzy matrices of finite order, which leads us to invent a new sequence. In order to achieve this new sequence, we analyze the behavioral study of equivalence classes on the set of all fuzzy matrices of a given order under a suitable natural equivalence relation. In addition this paper characterizes the properties of non-equivalent classes of fuzzy matrices of order n with elements having degrees of membership values anywhere in the closed unit interval [0,1]. Further, this paper also derives some important relevant results by enumerating the number of all distinct fuzzy matrices of a given order in general. And also, we achieve these results by incorporating the notion of k-level fuzzy matrices, chains, and flags (maximal chains). Keywords: Fuzzy matrices; k-level fuzzy matrices; Chains; Flags; Binomial numbers