The next definition given is fundamental to the concept of rough sets, since it addresses the central point of the approach, the vague classes. These are the ones with more than one value for the decision attribute.
The lower approximation of X is the collection of objects which can be classified with full certainty as members of the set X, using the attribute set B. Similarly, the upper approximation of X is the collection of objects that may possibly be classified as members of the set X. The boundary region comprises the objects that cannot be classified with certainty to be neither inside X, nor outside X, again using the attribute set B.
Properties of these approximations are given in [Paw91].