# Basic Formal Properties of Triangular Norms and Conorms

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## Summary

In the article we present in the Mizar system [1], [8] the catalogue of triangular norms and conorms, used especially in the theory of fuzzy sets [13]. The name triangular emphasizes the fact that in the framework of probabilistic metric spaces they generalize triangle inequality [2].

After defining corresponding Mizar mode using four attributes, we introduced the following t-norms:

• minimum t-norm minnorm (Def. 6),

• product t-norm prodnorm (Def. 8),

• Łukasiewicz t-norm Lukasiewicz_norm (Def. 10),

• drastic t-norm drastic_norm (Def. 11),

• nilpotent minimum nilmin_norm (Def. 12),

• Hamacher product Hamacher_norm (Def. 13),

and corresponding t-conorms:
• maximum t-conorm maxnorm (Def. 7),

• probabilistic sum probsum_conorm (Def. 9),

• bounded sum BoundedSum_conorm (Def. 19),

• drastic t-conorm drastic_conorm (Def. 14),

• nilpotent maximum nilmax_conorm (Def. 18),

• Hamacher t-conorm Hamacher_conorm (Def. 17).

Their basic properties and duality are shown; we also proved the predicate of the ordering of norms [10], [9]. It was proven formally that drastic-norm is the pointwise smallest t-norm and minnorm is the pointwise largest t-norm (maxnorm is the pointwise smallest t-conorm and drastic-conorm is the pointwise largest t-conorm).

This work is a continuation of the development of fuzzy sets in Mizar [6] started in [11] and [3]; it could be used to give a variety of more general operations on fuzzy sets. Our formalization is much closer to the set theory used within the Mizar Mathematical Library than the development of rough sets [4], the approach which was chosen allows however for merging both theories [5], [7].

[1] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-817.

[2] Didier Dubois and Henri Prade. Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York, 1980.

[3] Adam Grabowski. The formal construction of fuzzy numbers. Formalized Mathematics, 22(4):321–327, 2014. doi: 10.2478/forma-2014-0032.

[4] Adam Grabowski. On the computer-assisted reasoning about rough sets. In B. Dunin-Kȩplicz, A. Jankowski, A. Skowron, and M. Szczuka, editors, International Workshop on Monitoring, Security, and Rescue Techniques in Multiagent Systems Location, volume 28 of Advances in Soft Computing, pages 215–226, Berlin, Heidelberg, 2005. Springer-Verlag. doi: 10.1007/3-540-32370-815.

[5] Adam Grabowski. Efficient rough set theory merging. Fundamenta Informaticae, 135(4): 371–385, 2014. doi: 10.3233/FI-2014-1129.

[6] 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.

[7] Adam Grabowski and Takashi Mitsuishi. Initial comparison of formal approaches to fuzzy and rough sets. In Leszek Rutkowski, Marcin Korytkowski, Rafal Scherer, Ryszard Tadeusiewicz, Lotfi A. Zadeh, and Jacek M. Zurada, editors, Artificial Intelligence and Soft Computing - 14th International Conference, ICAISC 2015, Zakopane, Poland, June 14-18, 2015, Proceedings, Part I, volume 9119 of Lecture Notes in Computer Science, pages 160–171. Springer, 2015. doi: 10.1007/978-3-319-19324-315.

[8] Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi: 10.1007/s10817-015-9345-1.

[9] Petr Hájek. Metamathematics of Fuzzy Logic. Dordrecht: Kluwer, 1998.

[10] Erich Peter Klement, Radko Mesiar, and Endre Pap. Triangular Norms. Dordrecht: Kluwer, 2000.

[11] 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.

[12] Takashi Mitsuishi, Katsumi Wasaki, and Yasunari Shidama. Basic properties of fuzzy set operation and membership function. Formalized Mathematics, 9(2):357–362, 2001.

[13] Lotfi Zadeh. Fuzzy sets. Information and Control, 8(3):338–353, 1965.

# Formalized Mathematics

## (a computer assisted approach)

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