Rough set characterizes an uncertain set with two crisp boundaries, i.e., lower and upper-approximation sets, and it has successfully been applied into artificial intelligence fields. However, there is little discuss on how to establish a crisp set as approximation set of a target set instead of only giving out two boundary lines. Although many uncertain decision rules can be acquired from the information system based on lower-approximation set by keeping the positive region unchanged, the accuracy of these rules depends on the similarity between a target set and its lower-approximation set. In order to solve this problem, an approximation set model of rough set was proposed and applied into attribute reduction or uncertain image segmentation successfully in our previous research. In this paper, a kind of fuzzy similarity between a target set and its approximation set is presented based on Euclidean distance instead of the similarity based on cardinality of a finite set. Then many good properties of 0.5-approximation set of a rough set are presented, and the conclusion that 0.5-appoximation set is the best approximation set of a rough set in all definable sets is successfully proved with Euclidean similarity between two fuzzy sets. Finally, the change laws of fuzzy similarity between a target set and its 0.5-approximation set in different granularity spaces is analyzed in detail.
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