The Study Of Probabilistic Fuzzy Sets And R-fuzzy Sets

Content:The theory of fuzzy sets was first introduced by control theory professor L.A.Zadeh.Since then,the theory of fuzzy sets and fuzzy systems was applied in many fields.In the field of fuzzy mathematics, many mathematical theory such as fuzzy topology,fuzzy analysis, fuzzy algebra and fuzzy logic,etc.,are obtained.In order to build the theory base of fuzzy sets,people presented many works such as the theory of nested sets,the theory of fuzzy sets and falling shadows of random sets and the theory of r-fuzzy sets.Their works have shown that the theory of fuzzy sets has intimate connection with probability theory.In 1975,Hirota introduced the concept of fuzzy probabilistic sets.Then,Gerstenkorn and Manko introduced the concept of bifuzzy probabilistic sets.Those works built a connection between fuzzy sets and probability theory,However,the relations between r-fuzzy sets and probabilistic fuzzy sets have not been described so far.The aim of this paper is to build a connection between probabilistic fuzzy sets and r-fuzzy sets.Firstly,The relations between fuzzy probabilistic sets and r-fuzzy sets are discussed.A new concept of r-fuzzy nested set is presented and it is pointed a fuzzy probabilistic set is an equivalent class of a r-fuzzy nested set.Secondly,we generalize the concept of r-fuzzy set on r-fuzzy set theory,and the concept of r-bifuzzy set and r-bifuzzy nested set is presented.Then the properties of it are studied.Finally,by the use of r-bifuzzy nested set,the relations between r-bifuzzy set and bifuzzy probabilistic set presented by Gerstenkorn are discussed.