Research On Fuzzy Quotient Space Theory And Approach

As a new mathematical tool to deal with human problems, fuzzy granular computing theory and approach is developing rapidly and has been widely applied in many fields. Its philosophy is to implement the problem solving at different levels of granularity. Based on quotient space theory, this thesis attmpts to extend fuzzy quotient space theory to fuzzyλquotient space theory, and focuses on rough equality problem of rough set theory.The definitions ofλquotient space and fuzzyλquotient space are proposed on the basis of fuzzy quotient space theory, and their properties are discussed. It is proved that a normalized isosceles distance given inλquotient space corresponds to a fuzzy equivalence relation given in universe and it can determine a fuzzy knowledge base. A method that shows how to gain a fuzzy equivalence relation from a series of given X cut relation is presented, also the advantage of obtaining fuzzy equivalence relation through the expression ofλsupremum is discussed. The experimental results indicate that fuzzyλquotient space theory is helpful for human being to solve fuzzy problems at different levels of granularity.Based on rough set theory, it is pointed out that different rough sets have the same upper and lower approximations, also the sufficient and necessary conditions of rough equality sets are proposed. The properties of rough equality are presented. By improving the expression of rough membership function and integrating with fuzzy membership function, two expressions of rough set which can distinguish rough sets are presented. Combined with quotient space theory, quotient space of rough equality classes is discussed. The propositions given in the thesis are meaningful to the research of rough equality problem.