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Continuous T-norms and T-conorms Satisfying the Principle of Inclusion and Exclusion

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Aggregation Functions in Theory and in Practise

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 228))

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Abstract

The classical principle of inclusion and exclusion is formulated for set-theoretic union and intersection. It is natural to ask if it can be extended to fuzzy sets. The answer depends on the choice of fuzzy logical operations (which belong to the larger class of aggregation operators). Further, the principle can be generalized to interval-valued fuzzy sets, resp. IF-sets (Atanassov’s intuitionistic fuzzy sets). The principle of inclusion and exclusion uses cardinality of sets (which has a natural extension to fuzzy sets, interval-valued fuzzy sets and IF sets) or, more generally, a measure, which can be defined in different ways. We also point up the question of the domain of the measure which has been neglected so far.

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References

  1. Atanassov, K.: Intuitionistic Fuzzy Sets: Theory and Applications. Springer, Heidelberg (1999)

    Book  MATH  Google Scholar 

  2. Butnariu, D., Klement, E.P.: Triangular Norm-Based Measures and Games with Fuzzy Coalitions. Kluwer Academic Publishers, Dordrecht (1993)

    Book  MATH  Google Scholar 

  3. Ciungu, L.C., Kelemenová, J., Riečan, B.: A New Point of View to the Inclusion - Exclusion Principle. In: 6th IEEE International Conference on Intelligent Systems IS 2012, Varna, Bulgaria, pp. 142–144 (2012)

    Google Scholar 

  4. Ciungu, L., Riečan, B.: General form of probabilities on IF-sets. In: Di Gesù, V., Pal, S.K., Petrosino, A. (eds.) WILF 2009. LNCS (LNAI), vol. 5571, pp. 101–107. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  5. Ciungu, L., Riečan, B.: Representation theorem for probabilities on IFS-events. Information Sciences 180, 793–798 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Deschrijver, G., Cornelis, C., Kerre, E.E.: On the representation of intuitionistic fuzzy t-norms and t-conorms. IEEE Trans. Fuzzy Syst. 12(1), 45–61 (2004)

    Article  MathSciNet  Google Scholar 

  7. Dubois, D., Gottwald, S., Hájek, P., Kacprzyk, J., Prade, H.: Terminological difficulties in fuzzy set theory—The case of “Intuitionistic Fuzzy Sets”. Fuzzy Sets and Systems 156(3), 485–491 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dubois, D., Prade, H.: Gradualness, uncertainty and bipolarity: Making sense of fuzzy sets. Fuzzy Sets and Systems 192, 3–24 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  9. Grzegorzewski, P., Mrówka, E.: Probability of intuitionistic fuzzy events. In: Grzegorzewski, P., et al. (eds.) Soft Methods in Probability, Statistics and Data Analysis, pp. 105–115. Springer, New York (2002)

    Chapter  Google Scholar 

  10. Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)

    Book  MATH  Google Scholar 

  11. Kuková, M.: The inclusion-exclusion principle for L-states and IF-events. Information Sciences 224, 165–169 (2013), http://dx.doi.org/10.1016/j.ins.2012.10.029

    Article  MathSciNet  Google Scholar 

  12. Kuková, M., Navara, M.: Principles of inclusion and exclusion for fuzzy sets. Fuzzy Sets Syst. (accepted)

    Google Scholar 

  13. Navara, M.: Triangular norms and measures of fuzzy sets. In: Klement, E.P., Mesiar, R. (eds.) Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms, pp. 345–390. Elsevier (2005)

    Google Scholar 

  14. Nguyen, H.T., Walker, E.: A First Course in Fuzzy Logic, 2nd edn. Chapman & Hall/CRC, Boca Raton (2000)

    MATH  Google Scholar 

  15. Riečan, B.: On some contributions to quantum structures by fuzzy sets. Kybernetika 43, 481–490 (2007)

    MATH  MathSciNet  Google Scholar 

  16. Riečan, B.: Probability theory on IF events. In: Aguzzoli, S., Ciabattoni, A., Gerla, B., Manara, C., Marra, V. (eds.) ManyVal 2006. LNCS (LNAI), vol. 4460, pp. 290–308. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  17. Riečan, B., Mundici, D.: Probability on MV-algebras. In: Pap, E. (ed.) Handbook of Measure Theory, ch. 21, pp. 869–910. Elsevier Science, Amsterdam (2002)

    Google Scholar 

  18. Riečan, B., Petrovičová, J.: On the Łukasiewicz Probability Theory on IF-sets. Tatra Mt. Math. Publ. 46, 125–146 (2010)

    MATH  MathSciNet  Google Scholar 

  19. Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland, New York (1983)

    MATH  Google Scholar 

  20. Zadeh, L.A.: Probability measures of fuzzy sets. J. Math. Anal. Appl. 23, 421–427 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  21. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning I. Inform. Sci. 8, 199–249 (1975)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, Banská Bystrica, Slovakia

    Mária Kuková

  2. Center for Machine Perception, Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical University, Prague, Czech Republic

    Mirko Navara

Authors
  1. Mária Kuková
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  2. Mirko Navara
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Corresponding author

Correspondence to Mária Kuková .

Editor information

Editors and Affiliations

  1. Department of Automatica y Computacion, Universidad Publica de Navarra, Pamplona, Spain

    Humberto Bustince

  2. Department of Automatica y Computacion, Universidad Publica de Navarra, Pamplona, Spain

    Javier Fernandez

  3. Department of Mathematics and, Slovak University of Technology, Bratislava, Slovakia

    Radko Mesiar

  4. Department of Computer Science, Universidad de Alcalá de Henares, Madrid, Spain

    Tomasa Calvo

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Kuková, M., Navara, M. (2013). Continuous T-norms and T-conorms Satisfying the Principle of Inclusion and Exclusion. In: Bustince, H., Fernandez, J., Mesiar, R., Calvo, T. (eds) Aggregation Functions in Theory and in Practise. Advances in Intelligent Systems and Computing, vol 228. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39165-1_20

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  • DOI: https://doi.org/10.1007/978-3-642-39165-1_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39164-4

  • Online ISBN: 978-3-642-39165-1

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