Properties of Rough Approximations

Rough set theory, introduced by Pawlak [10], is a mathematical formalism dealing with uncertainty and to some extent overlapping fuzzy set theory introduced by Zadeh [12]. In fuzzy set theory vagueness is expressed by a membership function. The rough set theory approach is based on indiscernibility relations and approximations. A major advantage of rough set theory is that it needs no preliminary or additional information about data, such as membership functions in fuzzy set theory. The basic idea of rough set theory is that knowledge about objects is represented by indiscernibility relations. Indiscernibility relations are usually assumed to be equivalences—reflexive, symmetric, and transitive binary relations—interpreted so that two objects are equivalent if we cannot distinguish them by their properties. This means that if we observe objects through knowledge given by an indiscernibility relation, our ability to distinguish objects is blurred—we cannot distinguish individual objects, only their equivalence classes. Let us consider the situation in Fig.1. Let X be a subset of a given universe of discourse U and let be an indiscernibility relation on U . Since the equivalence induces the partition U whose blocks are the equivalence classes of , the objects of the universe U are classified by in three classes for any subset X U :