An uncertainty measure in partition-based fuzzy rough sets

This paper extends Pawlak's rough set onto the basis of a fuzzy partition of the universe of discourse. Some basic properties of partition-based fuzzy approximation operators are examined. To measure uncertainty in generalized fuzzy rough sets, a new notion of entropy of a fuzzy set is introduced. The notion is demonstrated to be adequate for measuring the fuzziness of a fuzzy event. The entropy of a fuzzy partition and conditional entropy are also proposed. These kinds of entropy satisfy some basic properties similar to those of Shannon's entropy. It is proved that the measure of fuzziness of a partition-based fuzzy rough set, FR(A), is equal to zero if and only if the set A is crisp and definable.