The correct structures in fuzzy soft set theory

TL;DR

This paper revises fuzzy soft set theory by establishing correct definitions based on Molodtsov's original soft set framework, ensuring theoretical consistency and practical reliability.

Contribution

It provides a formal, accurate definition of fuzzy soft sets aligned with Molodtsov's correct soft set theory and introduces matrix representations for these sets.

Findings

Corrected the foundational definitions of fuzzy soft sets.

Presented a matrix form representation of fuzzy soft sets.

Derived significant theoretical results on fuzzy soft sets.

Abstract

In 1999, Molodtsov \cite{1} developed the idea of soft set theory, proving it to be a flexible mathematical tool for dealing with uncertainty. Several researchers have extended the framework by combining it with other theories of uncertainty, such as fuzzy set theory, intuitionistic fuzzy soft set theory, rough soft set theory, and so on. These enhancements aim to increase the applicability and expressiveness of soft set theory, making it a more robust tool for dealing with complex, real-world problems characterized by uncertainty and vagueness. The notion of fuzzy soft sets and their associated operations were introduced by Maji et al. \cite{7}. However, Molodtsov \cite{3} identified numerous incorrect results and notions of soft set theory that were introduced in the paper \cite{7}. Therefore, the derived concept of fuzzy soft sets is equally incorrect since the basic idea of soft sets in \cite{7} is flawed. Consequently, it is essential to address these incorrect notions and provide an exact and formal definition of the idea of fuzzy soft sets. This reevaluation is important to guarantee fuzzy soft set theory's theoretical stability and practical application across a range of domains. In this paper, we propose fuzzy soft set theory based on Molodtsov's correct notion of soft set theory and demonstrate a fuzzy soft set in matrix form. Additionally, we derive several significant findings on fuzzy soft sets.