$(ϵ,\delta )$-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 $(ϵ,\delta )$-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 $\left(ϵ,\delta \right)$-complex anti fuzzy subgroups (CAFS) in the context of CAFSs. Moreover, we showed that everyCAFSGis a $(ϵ,\delta )$-CAFSG. Also, we used this approach to define $(ϵ,\delta )$-complex anti-fuzzy(CAF) cosets and $(ϵ,\delta )$-CAF normal subgroups of a certain group as well as to investigate some of their algebraic properties. We elaborated the $(ϵ,\delta )$-CAFSG of the classical quotient group and demonstrated that the set of all $(ϵ,\delta )$-CAF cosets of such a particular CAFs normal subgroup formed a group. Furthermore, the index of $\left(ϵ,\delta \right)$-CAFSG was demonstrated and $(ϵ,\delta )$-complex anti fuzzification of Lagrange theorem corresponding to the Lagrange theorem of classical group theory was briefly examined.