Roughness of fuzzy sets based on two new operators

In this paper, two new operators are introduced for the rough set theory. Using them, two inequalities well-known in the rough set theory can now be modified to become equalities. With this change, no information may be lost in the new expressions. Hence, many properties in the rough set theory can be improved and in particular, the union, the intersection, and the complement operations can be redefined based on the two equalities. Finally, the roughness properties of fuzzy sets are analyzed using the new operations.