Research on rough equality and rough inclusion of sets in multi-granulation spaces

Rough set theory proposed by professor Pawlak in 1982 is an important tool to solve uncertain problems. In order to make rough sets deal with uncertain problems better, by considering the stringent notions of mathematical equality and inclusion, the definitions of rough equality and rough inclusion were introduced by Pawlak. But the researches of rough equality and rough inclusion are on a certain granularity space. In this paper, some change rules will be given with respect to the relations of rough equality and rough inclusion between two uncertain target sets in multi-granulation spaces, and the similarity degree between two roughly equal sets is proposed to describe the similarity of two uncertain sets in multi-granulation spaces. In order to describe two uncertain sets which are rough equality by an approximation set at the same time, the definitions of optimistic lambda-approximation set and pessimistic lambda-approximation set will be defined from the points of optimism and pessimism, and some properties of them are discussed in detail.