Fuzzy Rough Theory And Modal Logic

Rough sets are acted as a new mathematical theory to dispose inaccuracy and indeterminacy and unperfected data ,there is a strong complementarily between rough sets and other theories which dispose indeterminacy and inaccuracy question such as probability ,fuzzy sets, information theory and testimony theory . This article has a research in intersect field about rough sets and fuzzy sets.There has a profound research about the structure nature and fuzziness of fuzzy rough sets with the structure nature and fuzziness of fuzzy sets. First ,the decomposition theorem and representation theorem of fuzzy rough sets are given ,second ,the relations among entropy .distance measure and similarity measure of fuzzy rough sets .third ,the relation between fuzzy rough sets and L-fuzzy sets are discussed.Rough sets describe the fuzzy measure of knowledge with membership function of rough sets. Let (U,R) be an approximation space and degree of rough belongingness of x in X is given byspeaking more generally, let R be an equivalence relation defined on X. Let us denote the equivalence class of x in X in the sense of R by Rs (x). Fuzziness in rough sets is discussed with the fuzzy entropy , and a new measure of fuzziness in rough sets is introduced with the farthest crisp set and the nearest crisp set, some characterizations of this measure are made.There is a close relation with other mathematical branch, through the continuous research about the rough sets. Some work about modal logic is done , a concept of rough sets is given through a new fuzzy relation ,the important characterizations about the modal logic are discussed. There are four chapters in this paper : In chapter 1, introduction.inIn chapter 2, the decomposition theorem and representation theorem of fuzzy rough sets are given ,second ,the relations among entropy ,distance measure and similarity measure of fuzzy rough sets .third ,the relation between fuzzy rough sets and L-fuzzy sets are discussed.In chapter 3, a new measure of fuzziness in rough sets is introduced.In chapter 4, a concept of rough sets is given through a new fuzzy relation ,the characterizations about the modal logic are discussed.