T-Rough Fuzzy Subgroups of Groups

The rough set theory was introduced by Pawlak in 1982. It was proposed for presentation equivalence relations. But the concept of fuzzy set was introduced by Zadeh in 1965. In this paper,the concepts of the rough sets,T-rough sets,T-rough fuzzy sets, T-rough fuzzy subgroups, T-rough fuzzy ideals, and set-valued homomorphism of groups will be given. A necessery and sufficient condition for a fuzzy subgroup(ideal) and fuzzy prime ideal of a group under a set-valued homomorphism to be a T-rough fuzzy subgroup(ideal) and T-rough fuzzy prime ideal is stated. The purpose of this paper is to introduce and discuss the concept of T-rough fuzzy groups of groups that those have been proved in several papers. Also, we proved that intersection two fuzzy subgroups(ideals) of a set under a set-valued homomorphism is a T-rough fuzzy subgroup of other set.