# The Formal Construction of Fuzzy Numbers

Formalized Mathematics (2014)

• Volume: 22, Issue: 4, page 321-327
• ISSN: 1426-2630

top

## Abstract

top
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].

## How to cite

top

Adam Grabowski. "The Formal Construction of Fuzzy Numbers." Formalized Mathematics 22.4 (2014): 321-327. <http://eudml.org/doc/270967>.

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].},
journal = {Formalized Mathematics},
keywords = {fuzzy sets; formal models of fuzzy sets; triangular fuzzy numbers},
language = {eng},
number = {4},
pages = {321-327},
title = {The Formal Construction of Fuzzy Numbers},
url = {http://eudml.org/doc/270967},
volume = {22},
year = {2014},
}

TY - JOUR
TI - The Formal Construction of Fuzzy Numbers
JO - Formalized Mathematics
PY - 2014
VL - 22
IS - 4
SP - 321
EP - 327
AB - 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].
LA - eng
KW - fuzzy sets; formal models of fuzzy sets; triangular fuzzy numbers
UR - http://eudml.org/doc/270967
ER -

## References

top
1. [1] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
2. [2] Józef Białas. Properties of the intervals of real numbers. Formalized Mathematics, 3(2): 263-269, 1992.
3. [3] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
4. [4] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
5. [5] Czesław Bylinski. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521-527, 1990.
6. [6] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
7. [7] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
8. [8] Didier Dubois and Henri Prade. Operations on fuzzy numbers. International Journal of System Sciences, 9(6):613-626, 1978. Zbl0383.94045
9. [9] Didier Dubois and Henri Prade. Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York, 1980. Zbl0444.94049
10. [10] Didier Dubois and Henri Prade. Rough fuzzy sets and fuzzy rough sets. International Journal of General Systems, 17(2-3):191-209, 1990. Zbl0715.04006
11. [11] Adam Grabowski. Efficient rough set theory merging. Fundamenta Informaticae, 135(4): 371-385, 2014. doi:10.3233/FI-2014-1129.[WoS][Crossref] Zbl1329.68247
12. [12] Adam Grabowski. On the computer certification of fuzzy numbers. In M. Ganzha, L. Maciaszek, and M. Paprzycki, editors, 2013 Federated Conference on Computer Science and Information Systems (FedCSIS), Federated Conference on Computer Science and Information Systems, pages 51-54, 2013.
13. [13] Adam Grabowski. Basic properties of rough sets and rough membership function. Formalized Mathematics, 12(1):21-28, 2004.
14. [14] Takashi Mitsuishi, Noboru Endou, and Yasunari Shidama. The concept of fuzzy set and membership function and basic properties of fuzzy set operation. Formalized Mathematics, 9(2):351-356, 2001.
15. [15] Takashi Mitsuishi, Katsumi Wasaki, and Yasunari Shidama. Basic properties of fuzzy set operation and membership function. Formalized Mathematics, 9(2):357-362, 2001.
16. [16] Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787-791, 1990.
17. [17] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777-780, 1990.
18. [18] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.
19. [19] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.
20. [20] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
21. [21] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.
22. [22] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.
23. [23] Lotfi Zadeh. Fuzzy sets. Information and Control, 8(3):338-353, 1965.

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.