On Some Comparison Properties of Rough Sets Based on Multigranulations and Types of Multigranular Approximations of Classifications

In this paper we consider the inclusion properties for upper and lower approximation of union and intersection of sets for both pessimistic and optimistic multigranulations. We find that two inclusions for pessimistic cases are actually equalities. For other six cases we provide examples to show that actually the proper inclusions hold true. On the approximation of classifications a theorem was proved in Tripathy et al to establish sufficient type properties. We establish here that actually the result is both necessary and sufficient one. Also, we consider types of elements in classifications with respect to both types of multigranulations and establish a general theorem on them.