Knowledge reduction based on the equivalence relations defined on attribute set and its power set

One of the key problems of knowledge discovery is knowledge reduction. This paper proposes a new method for knowledge reduction in information systems. First, two families of closed sets CrCr and CRCR are defined, where r and R   are equivalence relations defined on the attribute set and its power set, respectively. The properties of CrCr and CRCR are also discussed. The necessary and sufficient condition for Cr=CRCr=CR is then given and employed to construct an approach to attribute reduction in information systems. It is also proved that under the condition Cr=CRCr=CR, the proposed approach to knowledge reduction is equivalent to the well-accepted one in reference [W.X. Zhang, Y. Leung, W.Z. Wu, Information Systems and Knowledge Discovery, Science Publishing Company, Beijing, 2003].