Rough Representations of Ill-Known Sets and Their Manipulations in Low Dimensional Space

Abstract

Ill-known sets are subsets whose members are not known exactly. They can be represented by a family of subsets that can be true. When each subset is assigned a possible degree, the ill-known set is called a graded ill-known set. In this chapter, we focus on manipulations of graded ill-known sets, a possibility distribution on the power set. Two fuzzy sets on the universe called lower and upper approximations are uniquely defined from a graded ill-known set. On the contrary, a graded ill-known set is not uniquely determined by given lower and upper approximations but the maximal one is. Under a certain condition, we explicitly represent the maximal graded ill-known set having given lower and upper approximations. To utilize graded ill-known sets in decision and information sciences, possibility and necessity measures of graded ill-known sets are described. Simple computation formulae of possibility and necessity measures of graded ill-known sets are shown when lower and upper approximations are given.