A Mathematician's Guide to Fuzzy Logic with Applications in Fuzzy Additive Systems
A Mathematician's Guide to Fuzzy Logic with Applications in Fuzzy Additive Systems
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Date
2021
Authors
Thomas, Zachariah Alexander
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Journal ISSN
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Publisher
Middle Tennessee State University
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.
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Keywords
Fuzzy algebra,
Fuzzy logic,
Fuzzy measure,
Fuzzy metric,
Fuzzy set theory,
Fuzzy system,
Mathematics