# Basic Formal Properties of Triangular Norms and Conorms

Formalized Mathematics (2017)

• Volume: 25, Issue: 2, page 93-100
• ISSN: 1426-2630

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## Abstract

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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].

## How to cite

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Adam Grabowski. "Basic Formal Properties of Triangular Norms and Conorms." Formalized Mathematics 25.2 (2017): 93-100. <http://eudml.org/doc/288374>.

abstract = {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].},
journal = {Formalized Mathematics},
keywords = {fuzzy set; triangular norm; triangular conorm; fuzzy logic},
language = {eng},
number = {2},
pages = {93-100},
title = {Basic Formal Properties of Triangular Norms and Conorms},
url = {http://eudml.org/doc/288374},
volume = {25},
year = {2017},
}

TY - JOUR
TI - Basic Formal Properties of Triangular Norms and Conorms
JO - Formalized Mathematics
PY - 2017
VL - 25
IS - 2
SP - 93
EP - 100
AB - 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].
LA - eng
KW - fuzzy set; triangular norm; triangular conorm; fuzzy logic
UR - http://eudml.org/doc/288374
ER -

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