Fuzzy Rough Sets as a Pair of Fuzzy Numbers: A New Approach and New Findings

We study the properties of rough sets based on all the properties of sets including inclusion, strict inclusion and set equality ((Y. Y. Yao, 1996) and (Y. Y. Yao, 2003)). We add a counting function "count" to the rough set. We define rough inclusion, strict rough inclusion, rough equality, and rough inequality, based on the counting function associated with the rough set. Def: We say that a rough set A is roughly included in rough set B iff: I*(A) sube I*(B) iff Count(I*(A)) les Count(I*(B)), I*(A) sube I*(B) iff Count(I*(A)) les Count(I*(B)), Where I*(A) and I*(A) are the upper approximation (UA) and lower approximation (LA) of A. Def: We say that a rough set A is rougly strictly included in a rough set B iff: I*(A) sub I*(B) iff Count(I*(A)) < Count(I*(B)), I*(A) sub I*(B) iff Count(I*(A)) < Count(I*(B)) Def:We say that a rough set A is rougly equal to a rough set B iff: I*(A) = I*(B) iff Count (I*(A)) = Count (I*(B)), I*(A) = I*(B) iff Count (I*(A))=Count (I*(B)) Def:We say that A is roughly not equal to rough set B iff: I*(A) # I*(B)iffCount (I*(A)) # Count (I*(B)), I*(A) # I*(B) iff Count (I* (A)) # Count (I*(B)). Next we apply the above definitions to the framework of user modeling mainly comparing successful and unsuccessful users Web behavior on a fact based search "limbic functions of the brain". Once we collect such results, we query the database of users with queries that depend on one variable. An example of such a query is: "find the number of Web pages of users whose searching is near average". Next we extend the definition of the counting function to a fuzzy counting function more specifically fuzzy cardinality and thus extend the above definitions to fuzzy rough sets. The concept of fuzzy cardinality of a fuzzy set has been proposed in ([6]). The cardinality |Acirc| of a fuzzy set A, called FGCount(A), is defined by: forall n isin N, mu|Acirc|(n)=sup{alpha : |Aalpha| ges n}. Thus the above rough inclusion becomes as follows on the fuzzy rough sets A and B: A sube B iff muOmegaLA(A)(omega) les muOmegaLA(B)(omega)), forallomega sube N and mu OmegaLA(A)(omega) les muOmegaLA(B)(omega), forallomega sube N where mu of 2 fuzzy quantities Q and Q' is defined as: muQ#Q'(z)=sup(x,y)xney=z/ (min(muQ(x) ,muQ'(y)). We combine different fuzzy rough sets with different fuzzy cardinality operators the fuzzy number create new fuzzy rough sets. Fuzzy cardinality operators were used successfully in comparing the "fuzzy rough set" of the successful users to the "fuzzy rough set" of unsuccessful users to answer such a query. Thus fuzzy rough sets become a pair of fuzzy numbers. More specifically the intersection of fuzzy rough sets of successful and unsuccessful users becomes the intersection of two fuzzy numbers. The difference of the fuzzy rough set of successful and unsuccessful users becomes the difference between two fuzzy numbers. This granular approach yielded results that go beyond the obvious of comparing successful and unsuccessful users. For example for this query: "the difference between the LA of Successful Users and the LA of the Users who failed is a fuzzy number that is smaller than the fuzzy number that represents the difference between the UA of successful users from the UA of the users who failed"