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Decomposition of Intersections with Fuzzy Sets of Operands

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Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

We investigate the operation of intersection of fuzzy sets with a fuzzy set of operands. This is a natural generalization of the corresponding operation which involves a crisp set of operands. The decomposition approach was used to study the intersections of fuzzy sets with a fuzzy set of operands. The result of this operation is a type-2 fuzzy set (T2FS). We prove several results which enable us to simplify constructing the type-2 membership function. It is shown that the resulting T2FS can be decomposed according to secondary membership grades into a finite collection of type-1 fuzzy sets. Each of these sets is the intersection of the original sets with a crisp set of operands. This crisp set is the corresponding \(\alpha \)-cut of the fuzzy set of operands. Illustrative examples are given.

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References

  1. Aisbett, J., Rickard, J.T., Morgenthaler, D.G.: Type-2 fuzzy sets as functions on spaces. IEEE Trans. Fuzzy Syst. 18(4), 841–844 (2010)

    Article  Google Scholar 

  2. Chen, Q., Kawase, S.: On fuzzy-valued fuzzy reasoning. Fuzzy Sets Syst. 113, 237–251 (2000)

    Article  MathSciNet  Google Scholar 

  3. Coupland, S., John, R.: Geometric type-1 and type-2 fuzzy logic systems. IEEE Trans. Fuzzy Syst. 15(1), 3–15 (2007)

    Article  Google Scholar 

  4. Hamrawi, H., Coupland, S.: Non-specificity measures for type-2 fuzzy sets. IEEE Trans. Fuzzy Syst. 732–737 (2009)

    Google Scholar 

  5. Hamrawi, H., Coupland, S.: Type-2 fuzzy arithmetic using alpha-planes. In: Proceedings of the IFSA-EUSFLAT, Portugal, pp. 606–611 (2009)

    Google Scholar 

  6. Hamrawi, H., Coupland, S., John, R.: Extending operations on type-2 fuzzy sets. In: Proceedings of the UKCI, Nottingham, UK (2009)

    Google Scholar 

  7. Hamrawi, H., Coupland, S., John, R.: A novel alpha-cut representation for type-2 fuzzy sets. IEEE Trans. Fuzzy Syst. 1–8 (2010)

    Google Scholar 

  8. Hamrawi, H., Coupland, S., John, R.: Type-2 fuzzy alpha-cuts. IEEE Trans. Fuzzy Syst. 25(3), 682–692 (2017)

    Article  Google Scholar 

  9. Harding, J., Walker, C., Walker, E.: The variety generated by the truth value algebra of T2FSs. Fuzzy Sets Syst. 161, 735–749 (2010)

    Article  Google Scholar 

  10. Karnik, N.N., Mendel, J.M.: Introduction to type-2 fuzzy logic systems. IEEE Trans. Fuzzy Syst. 2, 915–920 (1998)

    Google Scholar 

  11. Liu, F.: An efficient centroid type-reduction strategy for general type-2 fuzzy logic system. Inf. Sci. 178(9), 2224–2236 (2008)

    Article  MathSciNet  Google Scholar 

  12. Mashchenko, S.O.: Generalization of Germeyer’s criterion in the problem of decision making under the uncertainty conditions with the fuzzy set of the states of nature. J. Autom. Inf. Sci. 44(10), 26–34 (2012)

    Article  Google Scholar 

  13. Mashchenko, S.O.: A mathematical programming problem with the fuzzy set of indices of constraints. Cybern. Syst. Anal. 49(1), 62–70 (2013)

    Article  MathSciNet  Google Scholar 

  14. Mashchenko, S.: Intersections and unions of fuzzy sets of operands. Fuzzy Sets Syst. 352, 12–25 (2018)

    Article  MathSciNet  Google Scholar 

  15. Mashchenko, S.O., Morenets, V.I.: Shapley value of a cooperative game with fuzzy set of feasible coalitions. Cybern. Syst. Anal. 53(3), 432–440 (2017)

    Article  Google Scholar 

  16. Mendel, J.M.: Type-2 fuzzy sets: some questions and answers. IEEE Connect. Newsl. IEEE Neural Netw. Soc. 1, 10–13 (2003)

    Google Scholar 

  17. Mendel, J.M., John, R.: Type-2 fuzzy sets made simple. IEEE Trans. Fuzzy Syst. 10(2), 117–127 (2002)

    Article  Google Scholar 

  18. Mendel, J., Liu, F., Zhai, D.: \( \alpha \)-plane representation for type-2 fuzzy sets: theory and applications. IEEE Trans. Fuzzy Syst. 17(5), 1189–1207 (2009)

    Google Scholar 

  19. Tahayori, H., Tettamanzi, A., Antoni, G.: Approximated type-2 fuzzy set operations. In: Proceedings of IEEE World Congress on Computational Intelligence, Vancouver, Canada, pp. 1910–1917 (2006)

    Google Scholar 

  20. Wagner, C., Hagras, H.: zSlices towards bridging the gap between interval and general type-2 fuzzy Logic. IEEE Trans. Fuzzy Syst. 489–497 (2008)

    Google Scholar 

  21. Wagner, C., Hagras, H.: Toward general type-2 fuzzy logic systems based on \( z \)-slices. IEEE Trans. Fuzzy Syst. 18(4), 637–660 (2010)

    Google Scholar 

  22. Wu, D., Mendel, J.M.: Aggregation using the linguistic weighted average and interval type-2 fuzzy sets. IEEE Trans. Fuzzy Syst. 15(6), 1145 (2007)

    Article  Google Scholar 

  23. Wu, D., Mendel, J.: Corrections to aggregation using the linguistic weighted average and interval type-2 fuzzy sets. IEEE Trans. Fuzzy Syst. 16(6), 1664–1666 (2008)

    Article  Google Scholar 

  24. Yager, R.R.: Level sets and the extension principle for interval valued fuzzy sets and its application to uncertain measures. Inf. Sci. 178(18), 3565–3576 (2008)

    Article  Google Scholar 

  25. Zadeh, L.A.: Fuzzy sets. Inf. Control 46(3), 338–353 (1965)

    Article  Google Scholar 

  26. Zadeh, L.A.: Quantitative fuzzy semantics. Inf. Sci. 3, 159–176 (1971)

    Article  MathSciNet  Google Scholar 

  27. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning. Inf. Sci. 8(3), 199–249 (1975)

    Article  MathSciNet  Google Scholar 

  28. Zeng, W., Li, H.: Representation theorem of interval-valued fuzzy set. Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 14(3), 259–269 (2006)

    Article  MathSciNet  Google Scholar 

  29. Zeng, W., Li, H., Zhao, Y., Yu, X.: Extension principle of lattice-valued fuzzy set. In: Proceedings of the 4th International Conference on Fuzzy Systems and Knowledge Discovery, FSKD 2007, vol. 1, pp. 140–144 (2007)

    Google Scholar 

  30. Zeng, W., Shi, Y.: Note on interval-valued fuzzy set. Lect. Notes Artif. Intell. (Sub-Ser. Lect. Notes Comput. Sci.) 3613, 20–25 (2005)

    Google Scholar 

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Authors and Affiliations

  1. Taras Shevchenko National University of Kyiv, Volodymyrska Street, 60, Kyiv, 01601, Ukraine

    S. O. Mashchenko & D. O. Kapustian

Authors
  1. S. O. Mashchenko
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  2. D. O. Kapustian
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Correspondence to S. O. Mashchenko .

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Editors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    Victor A. Sadovnichiy

  2. National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine

    Michael Z. Zgurovsky

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Mashchenko, S.O., Kapustian, D.O. (2021). Decomposition of Intersections with Fuzzy Sets of Operands. In: Sadovnichiy, V.A., Zgurovsky, M.Z. (eds) Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-50302-4_20

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