The approximation set of a vague set in rough approximation space

Vague set is a further generalization of fuzzy set. In rough set theory, a target concept may be a defined set, fuzzy set or vague set. That the target concept is a defined set or fuzzy set was analyzed in detail in our other papers respectively. In general, we can only get two boundaries of an uncertain concept when we use rough set to deal with the uncertain problems and can not get a useable approximation defined set which is a union set with many granules in Pawlak's approximation space. In order to overcome above shortcoming, we mainly discuss the approximation set of a vague set in Pawlak's approximation space in the paper. Firstly, many preliminary concepts or definitions related to the vague set and the rough set are reviewed briefly. And then, many new definitions, such as 0.5-crisp set, step-vague set and average-step-vague set, are defined one by one. The Euclidean similarity degrees between a vague set and its 0.5-crisp set, step-vague set and average-step-vague set are analyzed in detail respectively. And then, the conclusion that the Euclidean similarity degree between a vague set and its 0.5-crisp set is better than the Euclidean similarity degree between the vague set and the other defined set in the approximation space ( U , R ) is drawn. Afterward, it is proved that average-step-vague set is an optimal step-vague set because the Euclidean similarity degree between a vague set and its average-step-vague set in the approximation space ( U , R ) can reach the maximum value. Finally, the change rules of the Euclidean similarity degree with the different knowledge granularities are discussed, and these rules are in accord with human cognitive mechanism in a multi-granularity knowledge space.