(e, )-complex anti fuzzy subgroups and their applications

The complex anti -fuzzy set (CAFS) is an extension of the traditional anti -fuzzy set with a wider range for membership function beyond real numbers to complex numbers with unit disc aims to address the uncertainty of data. The complex anti -fuzzy set is more significant because it provides two dimensional information and versatile representation of vagueness and ambiguity of data. In terms of the characteristics of complex anti -fuzzy sets, we proposed the concept of (c, (5)-CAFSs that offer a more comprehensive representation of the uncertainty of data than CAFSs by considering both the magnitude and phase of the membership functions and explain the (c, (5) -complex anti fuzzy subgroups (CAFSG) in the context of CAFSs. Moreover, we showed that every CAFSG is a (c, (5)-CAFSG. Also, we used this approach to define (c, (5) -complex anti-fuzzy(CAF) cosets and (c, (5) -CAF normal subgroups of a certain group as well as to investigate some of their algebraic properties. We elaborated the (c, (5)-CAFSG of the classical quotient group and demonstrated that the set of all (c, (5) -CAF cosets of such a particular CAFs normal subgroup formed a group. Furthermore, the index of (c, (5)-CAFSG was demonstrated and (c, (5) -complex anti fuzzification of Lagrange theorem corresponding to the Lagrange theorem of classical group theory was briefly examined.