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 $ (\epsilon, \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(\epsilon, \delta \right) $-complex anti fuzzy subgroups (CAFS) in the context of CAFSs. Moreover, we showed that everyCAFSGis a $ (\epsilon, \delta) $-CAFSG. Also, we used this approach to define $ (\epsilon, \delta) $-complex anti-fuzzy(CAF) cosets and $ (\epsilon, \delta) $-CAF normal subgroups of a certain group as well as to investigate some of their algebraic properties. We elaborated the $ (\epsilon, \delta) $-CAFSG of the classical quotient group and demonstrated that the set of all $ (\epsilon, \delta) $-CAF cosets of such a particular CAFs normal subgroup formed a group. Furthermore, the index of $ \left(\epsilon, \delta \right) $-CAFSG was demonstrated and $ (\epsilon, \delta) $-complex anti fuzzification of Lagrange theorem corresponding to the Lagrange theorem of classical group theory was briefly examined.
Citation: Arshad Ali, Muhammad Haris Mateen, Qin Xin, Turki Alsuraiheed, Ghaliah Alhamzi. $ (\epsilon, \delta) $-complex anti fuzzy subgroups and their applications[J]. AIMS Mathematics, 2024, 9(5): 11580-11595. doi: 10.3934/math.2024568
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 $ (\epsilon, \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(\epsilon, \delta \right) $-complex anti fuzzy subgroups (CAFS) in the context of CAFSs. Moreover, we showed that everyCAFSGis a $ (\epsilon, \delta) $-CAFSG. Also, we used this approach to define $ (\epsilon, \delta) $-complex anti-fuzzy(CAF) cosets and $ (\epsilon, \delta) $-CAF normal subgroups of a certain group as well as to investigate some of their algebraic properties. We elaborated the $ (\epsilon, \delta) $-CAFSG of the classical quotient group and demonstrated that the set of all $ (\epsilon, \delta) $-CAF cosets of such a particular CAFs normal subgroup formed a group. Furthermore, the index of $ \left(\epsilon, \delta \right) $-CAFSG was demonstrated and $ (\epsilon, \delta) $-complex anti fuzzification of Lagrange theorem corresponding to the Lagrange theorem of classical group theory was briefly examined.
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Arshad Ali, Muhammad Haris Mateen, Qin Xin, Turki Alsuraiheed, Ghaliah Alhamzi. $ (\epsilon, \delta) $-complex anti fuzzy subgroups and their applications[J]. AIMS Mathematics, 2024, 9(5): 11580-11595. doi: 10.3934/math.2024568
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