Fuzzy sets can be interpreted as limits of crisp sets, and this can help to fuzzify crisp notions

Abstract

Fuzzy sets have been originally introduced as generalizations of crisp sets, and this is how they are usually considered. From the mathematical viewpoint, the problem with this approach is that most notions allow many different generalizations, so every time we try to generalize some notions to fuzzy sets, we have numerous alternatives. In this paper, we show that fuzzy sets can be alternatively viewed as limits of crisp sets. As a result, for some notions, we can come up with a unique generalization - as the limit of the results of applying this notion to the corresponding crisp sets. © 2014 Springer International Publishing Switzerland.