In fuzzy set theory the image of a fuzzy set induced by a fuzzy relation is usually obtained by a sup-t-norm composition. This corresponds to the upper image which gathers the elements in relation with at least one element of the fuzzy set. However the dual point of view, leading to the definition of the lower image as the fuzzy set of elements in relation with all the elements of the fuzzy set whose image is computed, is not often considered. Fuzzy sets and fuzzy relations, depending on the situations, can be interpreted either in a conjunctive manner or as subsets of mutually exclusive possible values for variables whose precise values are ill-known (disjunctive view). The applications of upper and lower images are investigated in both interpretations. The generalized modus ponens, used in fuzzy rule-based systems, corresponds to the disjunctive view. The interest of upper and lower images is also emphasized for diagnosis problems where the conjunctive interpretation is encountered. © 1992.
Dubois, D., & Prade, H. (1992). Upper and lower images of a fuzzy set induced by a fuzzy relation: Applications to fuzzy inference and diagnosis. Information Sciences, 64(3), 203–232. https://doi.org/10.1016/0020-0255(92)90101-D