Fuzzy Probabilistic Sets and r-Fuzzy Sets

In this paper, a connection between the fuzzy probabilistic set and the r-fuzzy set is established. Firstly, a new concept of r-fuzzy nested set is presented and it is pointed that a fuzzy probabilistic set is an equivalent class of a r-fuzzy nested set. Secondly, the concepts of fuzzy probabilistic subgroup and r-fuzzy subgroup of a group are given respectively, the relations between fuzzy probabilistic subgroups and fuzzy subgroups presented by Rosenfeld and the properties of r-fuzzy subgroups are discussed. Finally, the membership function of fuzzy probabilistic sets and r-fuzzy subgroups are defined, the properties of membership function of r-fuzzy subgroups are studied.