On the Number of Finite Fuzzy Subsets with Analysis of Integer Sequences

Abstract

This paper solves the issues of determining the number Fn of fuzzy subsets of a nonempty finite set X. To solve this, this paper incorporates the equivalence relation on the collection of all fuzzy subsets of X. We derive two closed explicit formulas for Fn, which is the sum of a finite series in the product of binomial numbers or the sum of k-level fuzzy subsets Fn,k by introducing a classification technique. Moreover, these explicit formulas enable us to find the number of the maximal chains of crisp subsets of X. Further, this paper presents some elementary properties of Fn,k and Fn.