Approximation Set of the Interval Set in Pawlak's Space

The interval set is a special set, which describes uncertainty of an uncertain concept or set Z with its two crisp boundaries named upper-bound set and lower-bound set. In this paper, the concept of similarity degree between two interval sets is defined at first, and then the similarity degrees between an interval set and its two approximations (i.e., upper approximation set [Formula: see text] (Z) and lower approximation set [Formula: see text] (Z)) are presented, respectively. The disadvantages of using upper-approximation set [Formula: see text] (Z) or lower-approximation set [Formula: see text] (Z) as approximation sets of the uncertain set (uncertain concept) Z are analyzed, and a new method for looking for a better approximation set of the interval set Z is proposed. The conclusion that the approximation set R (0.5)(Z) is an optimal approximation set of interval set Z is drawn and proved successfully. The change rules of R (0.5)(Z) with different binary relations are analyzed in detail. Finally, a kind of crisp approximation set of the interval set Z is constructed. We hope this research work will promote the development of both the interval set model and granular computing theory.