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Metrics of Symmetric Difference on Fuzzy Sets Based on R-implicators of the Usual Families of t-norms

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Advances in Fuzzy Logic and Technology 2017 (EUSFLAT 2017, IWIFSGN 2017)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 642))

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Abstract

Symmetric differences of two fuzzy sets by means of R-implications operators of a norm have been studied and their cardinalities provided classes of dissimilarity measures on fuzzy sets.

In this paper, we characterize t-norms which generate metrics from these measures. The obtained metrics are fuzzy versions of the well-known distance of cardinality of the symmetric difference of crisp sets. For an application point of view, we consider the seven usual parameterized families of t-norms and we use their R-implications to examine conditions on parameters under which these t-norms generate metrics on fuzzy sets from those measures.

L.A. Fono thanks French Government and Prof. Bernadette Bouchon Meunier. This paper was initiated when he was visiting scholar at LIP 6 of the University of Pierre Marie-Curie - France under the scholarship “Bourse de Stage du Gouvernement Francais 2007 du SCAC de l’Ambassade de France au Cameroun”.

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Author information

Authors and Affiliations

  1. Département de Mathématiques, Ecole Normale Supérieure, Université de Yaoundé I, B.P. 47, Yaoundé, Cameroon

    Siméon Fotso

  2. Laboratoire de Mathématiques et Applications Fondamentales, URFD MIBA-CRFD STG- Université de Yaoundé I, B.P. 812, Yaoundé, Cameroon

    Romuald Thierry Dzati Kamga

  3. Laboratoire de Mathématiques, Université de Douala, B.P. 24157, Douala, Cameroon

    Louis Aimé Fono

Authors
  1. Siméon Fotso
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  2. Romuald Thierry Dzati Kamga
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  3. Louis Aimé Fono
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Corresponding author

Correspondence to Louis Aimé Fono .

Editor information

Editors and Affiliations

  1. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Janusz Kacprzyk

  2. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Eulalia Szmidt

  3. Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

    Slawomir Zadrożny

  4. Department of Telecommunications and Information Processing, Bulgarian Academy of Sciences, Sofia, Bulgaria

    K. T. Atanassov

  5. WIT - Warsaw School of Information Technology, Warsaw, Poland

    Maciej Krawczak

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Fotso, S., Dzati Kamga, R.T., Fono, L.A. (2018). Metrics of Symmetric Difference on Fuzzy Sets Based on R-implicators of the Usual Families of t-norms. In: Kacprzyk, J., Szmidt, E., Zadrożny, S., Atanassov, K., Krawczak, M. (eds) Advances in Fuzzy Logic and Technology 2017. EUSFLAT IWIFSGN 2017 2017. Advances in Intelligent Systems and Computing, vol 642. Springer, Cham. https://doi.org/10.1007/978-3-319-66824-6_8

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  • DOI: https://doi.org/10.1007/978-3-319-66824-6_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66823-9

  • Online ISBN: 978-3-319-66824-6

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