Sets with partial memberships: a rough set view of fuzzy sets

Fuzzy set theory on a universe of granules (crisp sets) are considered. In this setting, the grade of a membership can be interpreted in the most elementary fashion. A grade of 1/2 membership means exactly one half of the granule (in a suitable measure) belongs to the fuzzy set. Based on such views, fuzzy set theoretical operations can be very naturally induced from the classical set theoretical operations; they are called R-operations. Some traditional operations, such as max and min, are R-operations of specific classes of fuzzy sets. The granular view of fuzzy sets provides a new and healthy insight into how fuzzy set theory might be structured