Rules and Algorithms for Objective Construction of Fuzzy Sets

Abstract

This paper aims to present objective methods for constructing new fuzzy sets from known fuzzy or classical sets, defined over the elements of a finite universe's superstructure. The paper proposes rules for assigning membership functions to these new fuzzy sets, leading to two important findings. Firstly, the property concerning the cardinality of a power set in classical theory has been extended to the fuzzy setting, whereby the scalar cardinality of a fuzzy set $\tilde B$ defined on the power set of a finite universe of a fuzzy set $\tilde A$ satisfies $\text{card}(\tilde B)=2^{\text{card}(\tilde A)}$. Secondly, the novel algorithms allow for an arbitrary membership value to be objectively achieved and represented by a specific binary sequence.