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

Mathematics

Tzung-Pei Hong¹  Rajesh Kumar Mohapatra² 

[1] Department of Computer Science and Information Engineering, National University of Kaohsiung, Kaohsiung 811726, Taiwan;Department of Mathematics, Kalasalingam Academy of Research and Education, Krishnankoil 626126, India;

关键词: finite fuzzy subsets;    α-cuts;    chains of crisp subsets;    binomial numbers;    integer sequences;   

DOI  :  10.3390/math10071161

来源: DOAJ

【 摘 要 】

This paper solves the issues of determining the number F_n 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 $F_n$, which is the sum of a finite series in the product of binomial numbers or the sum of $k$-level fuzzy subsets $F_{n,k}$ 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 $F_{n,k}$ and $F_n$.

【 授权许可】

Unknown