Development of the fuzzy sets theory: weak operations and extension principles

Анық емес жиындар теориясының дамуы: әлсiз операциялар және жалпылау принциптерi
Развитие теории нечетких множеств: слабые операции и принципы обобщения
Katsyv S. Kukharchuk V. Kondratenko N. Kucheruk V. Kulakov P. Karabekova D.
2023 E.A. Buketov Karaganda University Publish house

Bulletin of the Karaganda University. Mathematics Series
2023 #111 Issue 3 59 - 71 pp.

The paper considers the problems that arise when using the theory of fuzzy sets to solve applied problems. Unlike stochastic methods, which are based on statistical data, fuzzy set theory methods make sense to apply when statistical data are not available. In these cases, algorithms should be based on membership functions formed by experts who are specialists in this field of knowledge. Ideally, complete information about membership functions is required, but this is an impractical procedure. More often than not, even the most experienced expert can determine only their carriers or separate sets of the α-cuts for unknown fuzzy parameters of the system. Building complete membership functions of unknown fuzzy parameters on this basis is risky and unreliable. Therefore, the paper proposes an extension of the fuzzy sets theory axiomatics in order to introduce non-traditional (less demanding on the completeness of data on membership functions) extension principles and operations on fuzzy sets. The so-called α-weak operations on fuzzy sets are proposed, which are based on the use of separate sets of the α-cuts. It is also shown that all classical theorems of Cantor sets theory apply in the extended axiomatic theory. New extension principles of generalization have been introduced, which allow solving problems in conditions of significant uncertainty of information.

Cantor set , core of fuzzy set , function of belonging , fuzzy set , set of α-cut , α-weak operation

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Vinnytsia National Technical University, Vinnytsia, Ukraine
Uman National University of Horticulture, Uman, Ukraine
Karaganda University of the Name of Academician E.A. Buketov, Karaganda, Kazakhstan

Vinnytsia National Technical University
Uman National University of Horticulture
Karaganda University of the Name of Academician E.A. Buketov

10 лет помогаем публиковать статьи Международный издатель

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