Abstract
Complex fuzzy sets are the novel extension of Zadeh’s fuzzy sets. In this paper, we comprise the introduction to the concept of ξ -complex fuzzy sets and proofs of their various set theoretical properties. We define the notion of α , δ -cut sets of ξ -complex fuzzy sets and justify the representation of an ξ -complex fuzzy set as a union of nested intervals of these cut sets. We also apply this newly defined concept to a physical situation in which one may judge the performance of the participants in a given task. In addition, we innovate the phenomena of ξ -complex fuzzy subgroups and investigate some of their fundamental algebraic attributes. Moreover, we utilize this notion to define level subgroups of these groups and prove the necessary and sufficient condition under which an ξ -complex fuzzy set is ξ -complex fuzzy subgroup. Furthermore, we extend the idea of ξ -complex fuzzy normal subgroup to define the quotient group of a group G by this particular ξ -complex fuzzy normal subgroup and establish an isomorphism between this quotient group and a quotient group of G by a specific normal subgroup G A ξ .
Highlights
Considering the inaccuracy in decision-making, Zadeh [1] popularized the concept of fuzzy sets for the first time, in 1965
For more on fuzzy set theory, we suggest reading of [31,32,33,34,35]. e competency of the complex fuzzy set has played an effective role to solve many physical problems. It provides us the meaningful representations of measuring uncertainty and periodicity. Despite all of these advantages, we still face vast complications to counter various physical situations based on a complex-valued membership function. is motivates us to define the notion of ξ-complex fuzzy set (ξ-CFS) through which one can have multiple options to investigate a specific real-world situation in much efficient way by choosing appropriate value of the parameter ξ
We introduce the phenomenon of ξ-complex fuzzy subgroup (ξ-CFSG) over an ξ-complex fuzzy set and investigate some of their fundamental algebraic attributes
Summary
We initiate the idea of ξ-CFS as a powerful extension of classical fuzzy sets. (2) e intersection of ξ-CFS Aξ and Bξ is denoted by Aξ ∩ Bξ and is defined as follows: μAξ ∩ Bξ (m) rAξ ∩ Bξ (m)eiωAξ ∩ Bξ (m) min rAξ (m), rBξ (m)ei min(ωAξ (m),ωBξ (m)),. E (α, δ)-cut set of ξ-CFS Aξ is represented by Aξ(α,δ) and is defined as follows: Aξ(α,δ) m ∈ U: rAξ (m) ≥ α, ωAξ (m) ≥ δ, 0 ≤ α ≤ 1, 0 ≤ δ ≤ 2π. By applying Definition 10, in the above relations, we obtain rAξ (m) ≥ α, ωAξ (m) ≥ δ, rBξ (m) ≥ α, ωBξ (m) ≥ δ. We present a new approach to define ξ-CFS which is quite necessary to establish the proofs of decomposition theorems.
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