<abstract xmlns="http://www.w3.org/1999/xhtml"><p> In this article, we continue the development of the theory of fuzzy sets [23], started with [14] with the future aim to provide the formalization of fuzzy numbers [8] in terms reflecting the current state of the Mizar Mathematical Library. Note that in order to have more usable approach in [14], we revised that article as well; some of the ideas were described in [12]. As we can actually understand fuzzy sets just as their membership functions (via the equality of membership function and their set-theoretic counterpart), all the calculations are much simpler. To test our newly proposed approach, we give the notions of (normal) triangular and trapezoidal fuzzy sets as the examples of concrete fuzzy objects. Also -cuts, the core of a fuzzy set, and normalized fuzzy sets were defined. Main technical obstacle was to prove continuity of the glued maps, and in fact we did this not through its topological counterpart, but extensively reusing properties of the real line (with loss of generality of the approach, though), because we aim at formalizing fuzzy numbers in our future submissions, as well as merging with rough set approach as introduced in [13] and [11]. Our base for formalization was [9] and [10].</p></abstract>

In this article, we continue the development of the theory of fuzzy sets [23], started with [14] with the future aim to provide the formalization of fuzzy numbers [8] in terms reflecting the current state of the Mizar Mathematical Library. Note that in order to have more usable approach in [14], we revised that article as well; some of the ideas were described in [12]. As we can actually understand fuzzy sets just as their membership functions (via the equality of membership function and their set-theoretic counterpart), all the calculations are much simpler. To test our newly proposed approach, we give the notions of (normal) triangular and trapezoidal fuzzy sets as the examples of concrete fuzzy objects. Also -cuts, the core of a fuzzy set, and normalized fuzzy sets were defined. Main technical obstacle was to prove continuity of the glued maps, and in fact we did this not through its topological counterpart, but extensively reusing properties of the real line (with loss of generality of the approach, though), because we aim at formalizing fuzzy numbers in our future submissions, as well as merging with rough set approach as introduced in [13] and [11]. Our base for formalization was [9] and [10].

The Formal Construction of Fuzzy Numbers

Institute of Informatics University of Białystok Akademicka 2, 15-267 Białystok Poland

Formalized Mathematics

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

{"article-title":"The Formal Construction of Fuzzy Numbers"}

In this article, we continue the development of the theory of fuzzy sets [23], started with [14] with the future aim to provide the formalization of fuzzy...

The Formal Construction of Fuzzy Numbers

Published Online: Dec 31, 2014

Page range: 321 - 327

Received: Dec 31, 2014

DOI: https://doi.org/10.2478/forma-2014-0032

Keywords
fuzzy sets, formal models of fuzzy sets, triangular fuzzy numbers

© by Adam Grabowski

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

The Formal Construction of Fuzzy Numbers

Published Online: Dec 31, 2014

Page range: 321 - 327

Received: Dec 31, 2014

DOI: https://doi.org/10.2478/forma-2014-0032

Keywordsfuzzy sets, formal models of fuzzy sets, triangular fuzzy numbers

© by Adam Grabowski

This work is licensed under the Creative Commons Attribution-ShareAlike 3.0 License.

Keywords
fuzzy sets, formal models of fuzzy sets, triangular fuzzy numbers