Optimal Approximations with Rough Sets

When arbitrary sets are approximated by more structured sets, it may not be possible to obtain an exact approximation that is equivalent to a given set. A proposal is presented for a ‘metric’ approach to Rough Sets. This includes a definition of the ‘optimal’ or best approximation with respect to a measure of similarity, and an algorithm to find it using the Jaccard Index. A definition of consistency also allows the algorithm to work for a larger class of similarity measures. Several consequences of these definitions are also presented.