Fuzzy Equivalent Relation With Thresholds And Fuzzy Rough Set

Content: First, by the use of the relation between fuzzy points and fuzzysets, the definitions of (α,β)-fuzzy relation of equivalence and(β,α)-fuzzy relation of equivalence are introduced. Some properties of(β,α)-fuzzy relation of equivalence are obtained, whereα,βdenoteany one of∈, q,∈∨q,∈∧q. It has been found that acceptable non-trivialconcepts obtained in this manner are (∈,∈)-fuzzy relation of equivalence,(∈,∈∨q)-fuzzy relation of equivalence and ((?), (?)∨(?))-fuzzy relation ofequivalence.Second, the definition of (λ,μ]-fuzzy relation of equivalence isintroduced. The relation between (λ,μ]-fuzzy relation of equivalenceand classical relation of equivalence are discussed.Finally, a new kind of cut sets is introduced and some properties oftheses cut sets are presented. Based on representation theorem, theconcept of (λ,μ]-rough set is introduced and the relation between(λ,μ]-rough set and fuzzy set are discussed. By the constructive method,when R is a (λ,μ]-fuzzy relation of equivalence, fuzzy rough setmodels in the approximate space (X, R)are obtained and someproperties of them are presented.