Fuzzy approximation based on $ \tau- \mathfrak{K} $ fuzzy open (closed) sets

Abstract

Rough set theory can be generalized by induced topology through equivalence relations. Motivated by the work of generalization of the rough set via topology, the concept and properties of $ \tau-\mathfrak{K} $-fuzzy open (closed) sets are proposed. Considering the $ \tau-\mathfrak{K} $-fuzzy open (closed) sets, we have obtained the $ \tau-\mathfrak{K} $-fuzzy lower and upper approximations and also proved their properties. $ \tau-\mathfrak{K} $-fuzzy open sets can be represented as $ \tau-\mathfrak{K} $-open sets by $ \alpha $ level sets. The properties of $ \tau-\mathfrak{K} $-fuzzy approximations and fuzzy rough approximations on the basis of binary fuzzy relation are compared. Finally, an example and the decision method's algorithm to illustrate the $ \tau-\mathfrak{K} $-fuzzy approximation-based approach to decision making are presented.