Fuzzy Rough Set Approximation Space Of Fuzzy Sets And Their Roughness

At present, the research of rough set theory focuses on its mathematical properties, extension of rough set model- connection and complementarity with other uncertainty methods, significant algorithm etc.In the study of relations between rough set theory and other methods to proceed ambiguity and uncertainty , we mainly discuss the relation and complementarity with fuzzy theory, Dempster-Shafer theory, probability theory and information theory.Both rough set and fuzzy set are mathematical tool to deal with ambiguity and uncertainty, they not exclude but complement each other. In 1990, Dubois pointed out that these two theories are different mathematical methods in handling two uncertainty(ambiguity and rough) ,and he put forward two kinds of mathematical model of fuzzy rough set and rough fuzzy set. In 1992, S.Nanda and S.Majumdar discussed the fuzzy rough set in lattice and proved its symmetric power law, commutative law, associative law, distribution law, De Morgan Law.In 1996,Banerjee and Paw.S.K gave the rough set of fuzzy set in approximation space,i.e.rough fuzzy set,and introduced the notion of roughness.Chengyi and Mo Zhiwen studied the fuzzy degree of fuzzy rough set and rough fuzzy set,and decomposition theorem and representation theorem of fuzzy rough set in 2001.afterwards,Zhang Wenxiu extended the rough fuzzy set to rough fuzzy set of triangle mould,and Liu Guilong extended it to rough fuzzy set on the fuzzy approximation space.Howerejn the rough fuzzy set introduced by Liu Guilong,he only gave the definition of lower approximation (RA) = {x U | [x] R A}, which is based on cut set, andupper approximation(RA) (x)= v R(x, y) A(y), which calculates subordinationdegree. This is his shortcoming. On the basis of this, we give the the lower and upper approximation of fuzzy set on the fuzzy approximation space again in this paper, and obtain another kind of extension of rough fuzzy set on the fuzzy approximation space. Following close behind, we introduce the concept ofroughness on the fuzzy approximation space and discuss its properties.Furthermore, for the rough fuzzy set model based on triangle mould gived by Zhang Wenxiu , It is very complex in proving relational properties using their definition of fuzzy T lower and upper approximation, hence,we give the definition of fuzzy T lower and upper approximation based on -cut set, and it simplifies the proof. Moreover, the notion of roughness is extended to fuzzy T approximation space and its properties are discussed.In the end , we compare the different definitions of the lower approximation operator on the approximation space, and point out the research directions.