Binary Relations as a Basis for Rule Induction in Presence of Quantitative Attributes

In original rough set theory, the notion of set approximation has been introduced by using indiscernibility relation defined on the set of objects. In some cases, it is necessary to generalize indiscernibility relation by using some other binary relations. In this paper, we consider similarity relations and tolerance relations among objects. These binary relations are defined from some similarity measures at the level of values of any quantitative attribute. The relations defined by single attribute are aggregated into a global relation at the level of the set of attributes. Then, we construct the lower approximation operation and the upper approximation operation generated by a binary relation and its inverse relation. In order to induce the minimal decision rules used to support the decision task, the nonsimilarity matrix of a decision table with respect to the lower approximation and boundary is defined to construct the nonsimilarity functions which are Boolean functions. The set of ''if … then …'' decision rules is decoded from prime implicants of the Boolean functions. An example is illustrated to demonstrate the application of this approach. Index Terms—binary relations, similarity, tolerance, rough sets, lower and upper approximations