Intelligent Classification Based on Tolerance Relations

Binary relations and its operations are important in set theory. But traditional research mainly focuses on pure mathematics operations on set theory. Pansystems strengthen the research work in set theory by combining mathematics with its philosophical background. In this paper, using pansystems PMT-combination method, we re-inspect the philosophical background of binary relations and give a new interpretation of equivalence relations and tolerance relations. Tolerance relations are more generic than equivalence relations. Equivalence relations can generate a division on a given set. Tolerance relations, however, cannot to be used to divide a set directly. Based on our research on tolerance relations and equivalence relations, we propose a new method of using tolerance relations to divide a given set. Furthermore we go back to the real world and give the philosophical explanation of this new kind of division.