Mathematical Theory of Fuzzy Sets presents the mathematical theory of non-normal fuzzy sets such that it can be rigorously used as a basic tool to study engineering and economic problems under a fuzzy environment. It may also be used as a textbook at the graduate level, or as a reference for researchers. The book explores the current state of affairs in set operations of fuzzy sets, arithmetic operations of fuzzy interval, and fuzzification of crisp functions, which are frequently adopted to model engineering and economic problems with fuzzy uncertainty. In particular, the concepts of gradual sets and gradual elements are presented in order to cope with the difficulty of considering elements of fuzzy sets like considering elements of crisp sets. Features Many extensions and equivalences for the essence of non-normal fuzzy sets Generalization of extension principle Presentation of the concepts of gradual sets and gradual elements