Verification of Some of the Inclusion Based Properties of Optimistic and Pessimistic Multigranular Rough Sets Using A Database

摘要

Uncertainty and impreciseness are well handled by the rough set theory which was introduced by Pawlak [1, 2]. The rough set theory was extended to multi-granular based rough sets by Qian, Y.H and Liang, J.Y [3, 4]. In order to more widely apply the rough set theory in practical applications Qian et al. extended Pawlak's single granulation rough set model to a multigranulation rough set model where the set approximations were defined by multiple equivalence relation on the universe. There are two types of multigranulations have been found in literature. After the introduction of the second one these are now called as optimistic multigranulation which was introduced first and pessimistic multigranulation. The properties of both optimistic and pessimistic multigranular rough sets have been studied and established, some of the inclusion property of pessimistic multigranular rough sets can be replaced with equalities not by sufficient conditions but with the help of other properties in [10]. Also some of inclusion properties of optimistic multigranular rough sets cannot be replaced by equalities was provided with examples in [10]. There were eight relations between the upper and lower multigranular approximations of union and intersections of rough sets found in [10] and in that paper the proof was given to show that out of eight relations two can be replaced with equalities and for other six cases were supported with examples to show that proper inclusions hold true. In this paper database based validation for the replacement of some of the inclusion property with equalities is provided. Also in this paper for the other appropriate inclusion properties of optimistic and pessimistic multigranular rough sets that cannot be replaced by equalities is validated with a faculty database table with assumed data.