A New Trend of Bipolar-Valued Fuzzy Cartesian Products, Relations, and Functions

Abstract

A bipolar-valued fuzzy set (BVFS) is a generalization of the fuzzy set (FS). It has been applied to a wider range of problems that cannot be represented by FS. New forms of the bipolar-valued fuzzy Cartesian product (BVFCP), bipolar-valued fuzzy relations (BVFRs), bipolar-valued fuzzy equivalence relations (BVFERs), and Bipolar-valued fuzzy functions (BVFFs) are constructed to be a cornerstone of creating new approach of BVF group theory. Unlike other approaches, the definition of BVFCP “A×B” is exceptionally helpful at reclaiming again the subset A and B by using a fitting lattice. Also, the present approach reduced the calculations and numerical steps in contrast to fuzzy and classical BVF cases. Results relating to those on relations, equivalence relations, and functions in the fuzzy cases are proved for BVFRs, BVFERs, and BVFFs.