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Roughness and Fuzziness

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Computational Intelligence and Mathematics for Tackling Complex Problems

Part of the book series: Studies in Computational Intelligence ((SCI,volume 819))

Abstract

Between roughness and fuzziness, the rough membership function may establish a connection. Rough membership functions can be viewed as a special type of fuzzy membership functions. This paper addresses possible coincidences between rough membership and fuzzy membership functions regarding not only classical cases but their different extensions as well. Roughness is treated in a general set approximation framework.

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Notes

  1. 1.

    We follow the notation where the set, later fuzzy set and its generalizations are distinguished from the symbols of their membership functions. The togetherness of the symbol and membership function is denoted by such a simple equation as \(S=\chi _S\).

  2. 2.

    \(S^c\) is the complement of S with respect to U. If \(f:U\rightarrow V\), the complement of f(S) with respect to V is denoted by \(f^c(S)\) instead of \((f(S))^c\).

  3. 3.

    The notion of the classical rough membership function was explicitly introduced by Pawlak and Skowron in [1, 30]. Nevertheless, it had been used and studied earlier by many authors. For more historical remarks, see [31]. Moreover, such a coefficient had already been considered by Łukasiewicz in 1913 [32, 33].

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Acknowledgements

The authors would like to thank the anonymous referees for their useful comments and suggestions.

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Authors and Affiliations

  1. Department of Health Informatics, Faculty of Health, University of Debrecen, Sóstói út 2-4, Nyíregyháza, 4406, Hungary

    Zoltán Ernő Csajbók & József Ködmön

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  1. Zoltán Ernő Csajbók
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  2. József Ködmön
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Correspondence to Zoltán Ernő Csajbók .

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Editors and Affiliations

  1. Faculty of Engineering Sciences, Széchenyi István University, Gyõr, Hungary

    László T Kóczy

  2. Science Faculty, Department of Mathematics, University of Cádiz, Cádiz, Spain

    Jesús Medina-Moreno

  3. Faculty of Economic and Business Sciences, Department of Mathematics, University of Cádiz, Cádiz, Spain

    Eloísa Ramírez-Poussa

  4. Institute of Mathematics and Computer Science, University of Latvia, Riga, Latvia

    Alexander Šostak

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Csajbók, Z.E., Ködmön, J. (2020). Roughness and Fuzziness. In: Kóczy, L., Medina-Moreno, J., Ramírez-Poussa, E., Šostak, A. (eds) Computational Intelligence and Mathematics for Tackling Complex Problems. Studies in Computational Intelligence, vol 819. Springer, Cham. https://doi.org/10.1007/978-3-030-16024-1_4

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