Granular Approximations of Partially-Known Concepts

The theory of rough sets concerns approximating a concept, represented by a set of objects as its instances, with a pair of lower and upper approximations. In some situations, we do not have complete knowledge of the set of instances of a concept, resulting in the need to study a partially-known concept. There are two ways to represent a partially-known concept. A triplet representation consists of a set of objects known to be instances, a set of objects known to be non-instances, and the rest objects of the concept. An interval set representation includes all possible sets that fall within a pair of a lower bound and an upper bound. In this paper, we systematically investigate rough set approximations of a partially-known concept under the two presentations. By approximating a partially-known concept, we can divide the universe into either seven or five regions. The analysis using seven regions or five regions can provide detailed analytics of a partially-known concept.