A Mathematician's Guide to Fuzzy Logic with Applications in Fuzzy Additive Systems

Abstract

Traditional set theory, or crisp set theory, is built on the concept of crisp sets. These are sets for which the membership of an element within a set is defined to be either true or false; in or out; 1 or 0. This construction is extremely useful, as the vast majority of mathematics has shown, but struggles to model concepts of our world which possess vagueness or uncertainty. Therefore, as in the style of Lotfi Zadeh [51], we explore an expansion of set theory to allow an element to be partially within a set, thus constituting what is known as a fuzzy set. These fuzzy sets are namely used in modelling this vagueness. Definitions of core mathematical constructions can be expanded to be defined for these fuzzy sets. These expanded definitions prove to be equivalent to traditional, crisp definitions when crisp sets are used. Throughout this paper, we explore the results of fuzzy research in set theory, algebra, and analysis; as well as the selected topics of fuzzy systems, an application of fuzzy logic in computer science. It is our aim that the reader, with a moderate background in theoretical mathematics, will be able to read this paper as a guided entry into the world of fuzzy mathematics. There are many reviews of fuzzy logic that have a similar goal. These reviews frequently focus on a fuzzy set theory introduction along with a specified application (see [29] and [31]). In contrast, we aim in this paper to provide a comprehensive, concise exploration of recent fuzzy research in a variety of mathematical fields, while illustrating the parallels of our fuzzy constructions to corresponding traditional ones. We do this following the recent research of the international journal, Fuzzy Sets and Systems. Additionally, we conclude this paper by illustrating the logical structure of fuzzy inference systems, as an exploration of preliminary information needed by a researcher to interact with the recent work, “Fuzzy Additive Systems” [23], by Bart Kosko.