Properties of two types of covering-based rough sets

Rough set theory is a tool to deal with the vagueness and granularity in information systems. The core concepts of classical rough sets are lower and upper approximations based on equivalence relations, or partitions. Recently, some sufficient and necessary conditions were given in Zhu (Proceedings of the first international workshop granular computing and brain informatics (GrBI'06), IEEE international conference on web intelligence (WI 06), pp. 494-497, 2006), Zhu and Wang (Proceedings of the IEEE international conference on data mining (ICDM'06) workshop foundation of data mining and novel techniques in high dimensional structural and unstructured data, pp. 407-411, 2006), (IEEE Trans Knowl Data Eng, 19, pp 1131-1144, 2007, Proceedings of the sixth international conference on machine learning and cybernetics, 33), under which a lower approximation operator and an upper approximation operator satisfy certain classical properties. In this paper, we give a counterexample to show that the condition given in Theorem 13 in (Proceedings of the first international workshop granular computing and brain informatics (Gr BI'06), IEEE international conference on web intelligence (WI 06), pp. 494-497, 2006) is not necessary. The correct formulation is stated. We also give other conditions for a covering, under which certain classical properties hold for the second and third types of covering-based lower and upper approximation operators.