Regarding covering as a collection of binary relations

Covering-based rough set is an important generalization of Pawlak´s rough set theory. By the fact that there have defined eight pairs lower and upper approximation operations in the case of covering-based rough set. Checking which pair of upper and lower approximations is the best definition among them is very necessary. We took advantage of the 2nd GrC Model and 6th pairs of upper and lower approximations of covering-based rough set, which have the best upper and lower bounds, two cases are analyzed. An equivalence relation´s geometric equivalence is a partition. A natural generalization of equivalence relation is a binary relation and geometric generalization of partition is covering. So, we considered two cases: regarding the covering as a neighborhood system and a binary relation, respectively. Reduction is one of the most important issues in covering rough set theory. Reducible element solves the problem of redundant covering-element in covering rough sets effectively. This is the reduct of covering. Taking an example to compare two cases of definitions. Results show that we can get the same lower and upper approximation bounds if there exists a reducible element, otherwise the first case defined by the 2nd GrC Model has the better lower bound, however all the pairs have the same upper approximation.