Attribute Reduction Based on Equivalence Relation Defined on Attribute Set and Its Power Set

The knowledge discovery in information systems, essentially, is to classify the objects according to attributes and to study the relation among those classes. Attribute reduction, which is to find a minimum attribute set that can keep the classification ability, is one of the most important problems in knowledge discovery in information system. The general method to study attribute reduction in information system is rough set theory, whose theoretical basis is the equivalence relation created on universe. Novotny. M.(1998) [17] has proposed a new idea to study attribute reduction by creating equivalence relation on attribute set. In this paper, we develop this idea to study attribute reduction through creating equivalence relations on attribute set and its power set. This paper begins with the basis theory of information systems, including definitions of information systems, and equivalence relation R-B on universe. Furthermore, two equivalence relations r and R are defined on attribute set and its power set separately. In the next section, two closed operators - C(R) and C(r) are created. Using these two operators, we get two corresponding closed set families - C-r, C-R, which are defined as C-R = {B, C(R) (B) = B}, C-r = {B : C(r)(B) = B}. Further, we study properties of these two closed set families, and prove that CR is a subset of C-r. One of the most important result is the necessary and sufficient condition about C-r=CR. This equivalence condition is described by elements of attribute set's division. Finally, based on the equivalence proposition, we find an easy method to acquire attribute reduction when Cr=CR. This method is easy to understand and use.