REPOZYTORIUM UNIWERSYTETU
W BIAŁYMSTOKU

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 Tytuł: Some Properties of Membership Functions Composed of Triangle Functions and Piecewise Linear Functions Autorzy: Mitsuishi, Takashi Słowa kluczowe: membership functionpiecewise linear function Data wydania: 2021 Data dodania: 4-sty-2022 Wydawca: DeGruyter Open Źródło: Formalized Mathematics, Volume 29, Issue 2, Pages 103-115 Abstrakt: IF-THEN rules in fuzzy inference is composed of multiple fuzzy sets (membership functions). IF-THEN rules can therefore be considered as a pair of membership functions [7]. The evaluation function of fuzzy control is composite function with fuzzy approximate reasoning and is functional on the set of membership functions. We obtained continuity of the evaluation function and compactness of the set of membership functions [12]. Therefore, we proved the existence of pair of membership functions, which maximizes (minimizes) evaluation function and is considered IF-THEN rules, in the set of membership functions by using extreme value theorem. The set of membership functions (fuzzy sets) is defined in this article to verifier our proofs before by Mizar [9], [10], [4]. Membership functions composed of triangle function, piecewise linear function and Gaussian function used in practice are formalized using existing functions. On the other hand, not only curve membership functions mentioned above but also membership functions composed of straight lines (piecewise linear function) like triangular and trapezoidal functions are formalized. Moreover, different from the definition in [3] formalizations of triangular and trapezoidal function composed of two straight lines, minimum function and maximum functions are proposed. We prove, using the Mizar [2], [1] formalism, some properties of membership functions such as continuity and periodicity [13], [8]. Afiliacja: University of Marketing and Distribution Sciences, Kobe, Japan Sponsorzy: This work has been partially supported in 2019-2020 by the domestic research grant of University of Marketing and Distribution Sciences in Kobe (Japan). URI: http://hdl.handle.net/11320/12393 DOI: 10.2478/forma-2021-0011 ISSN: 1426-2630 e-ISSN: 1898-9934 Typ Dokumentu: Article metadata.dc.rights.uri: https://creativecommons.org/licenses/by-sa/3.0/ Właściciel praw: © 2021 University of BiałymstokuCC-BY-SA License ver. 3.0 or later Występuje w kolekcji(ach): Formalized Mathematics, 2021, Volume 29, Issue 2

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