<abstract xmlns="http://www.w3.org/1999/xhtml">
<p><bold>Summary</bold>. 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.</p>
<p>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].</p>
</abstract>

Summary. 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].

Some Properties of Membership Functions Composed of Triangle Functions and Piecewise Linear Functions

University of Marketing and Distribution Sciences

Formalized Mathematics

Summary. 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...

Some Properties of Membership Functions Composed of Triangle Functions and Piecewise Linear Functions

Article Category: Artikel

Published Online: Dec 30, 2021

Page range: 103 - 115

Accepted: Jun 30, 2021

DOI: https://doi.org/10.2478/forma-2021-0011

Keywords
membership function, piecewise linear function

© 2021 University of Białystok, published by Sciendo

CC-BY-SA License ver. 3.0 or later

Some Properties of Membership Functions Composed of Triangle Functions and Piecewise Linear Functions

Article Category: Artikel

Published Online: Dec 30, 2021

Page range: 103 - 115

Accepted: Jun 30, 2021

DOI: https://doi.org/10.2478/forma-2021-0011

Keywordsmembership function, piecewise linear function

© 2021 University of Białystok, published by Sciendo

CC-BY-SA License ver. 3.0 or later

Keywords
membership function, piecewise linear function