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Mathematics of Intuitionistic Fuzzy Sets

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Fuzzy Logic in Its 50th Year

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 341))

Abstract

Short firsthand remarks on the history and theory of Intuitionistic Fuzzy Sets (IFSs) are given. Influences of other areas of mathematics for development of the IFSs theory are discussed. On the basis of results in IFSs theory, some ideas for development of other mathematical areas are offered.

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References

  1. Atanassov, K.: Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia, June 1983 (Deposed in Central Sci.—Techn. Library of Bulg. Acad. of Sci. 1697/84) (in Bulgaria)

    Google Scholar 

  2. Atanassov, K., Stoeva S.: Intuitionistic fuzzy sets. In: Proceedings of polich symposium on interval & fuzzy mathematics, pp. 23–26, Poznan, (1983)

    Google Scholar 

  3. Zadeh, L.: Fuzzy sets. Inf. Cont. 8, 338–353 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  4. Kaufmann, A.: Introduction a la Theorie des Sour-ensembles Flous. Masson, Paris (1977)

    MATH  Google Scholar 

  5. Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York (1980)

    MATH  Google Scholar 

  6. Buhaescu, T.: Some observations on intuitionistic fuzzy relations. Itinerant Seminar on Functional Equations, pp. 111–118, Approximation and Convexity, Cluj-Napoca (1989)

    Google Scholar 

  7. Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  8. Atanassov, K.: 25 years of intuitionistic fuzzy sets or the most interesting results and the most important mistakes of mine. Advances in Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets and Related Topics, vol. I, pp. 1–35, Foundations, Academic Publishing House EXIT, Warszawa (2008)

    Google Scholar 

  9. Atanassov, K.: On some intuitionistic fuzzy negations. Notes Intuitionistic Fuzzy Sets 11(6), 13–20 (2005)

    Google Scholar 

  10. Atanassov, K.: On the temporal intuitionistic fuzzy sets. In: Proceedings of the Ninth International Conference IPMU 2002, vol. III, pp. 1833–1837, Annecy, France, 1–5 July (2002)

    Google Scholar 

  11. Klir, G., Yuan, B.: Fuzzy Sets and Fuzzy Logic. Prentice Hall, New Jersey (1995)

    MATH  Google Scholar 

  12. Atanassov, K.: Intuitionistic fuzzy sets. Springer, Heidelberg (1999)

    Book  MATH  Google Scholar 

  13. Antonov, I.: On a new geometrical interpretation of the intuitionistic fuzzy sets. Notes Intuitionistic Fuzzy Sets 1(1), 29–31 (1995). http://ifigenia.org/wiki/issue:nifs/1/1/29-31

  14. Danchev, S.: A new geometrical interpretation of some concepts in the intuitionistic fuzzy logics. Notes Intuitionistic Fuzzy Sets 1(2), 116–118 (1995)

    Google Scholar 

  15. Yang, Y., Chiclana, F.: Intuitionistic fuzzy sets: spherical representation and distances. Int. J. Intell. Syst. 24, 399–420 (2009)

    Google Scholar 

  16. Szmidt, E.: Applications of intuitionistic fuzzy sets in decision making. D.Sc. Dissertation, Technical University, Sofia (2000)

    Google Scholar 

  17. Szmidt, E., Kacprzyk, J.: Similarity of intuitionistic fuzzy sets and the Jaccard coefficient. In: Proceedings of Tenth International Conference IPMU’2004, vol. 2, 1405–1412, Perugia, 4–9 July 2004

    Google Scholar 

  18. Szmidt, E., Kacprzyk, J.: New measures of entropy for intuitionistic fuzzy sets. Notes Intuitionistic Fuzzy Sets 11(2), 12–20 (2005). http://ifigenia.org/wiki/issue:nifs/11/2/12-20

  19. Atanassova, V.: Representation of fuzzy and intuitionistic fuzzy data by Radar charts. Notes on Intuitionistic Fuzzy Sets 16(1), 21–26 (2010). http://ifigenia.org/wiki/issue:nifs/16/1/21-26

  20. Feys, R.: Modal Logics. Gauthier, Paris (1965)

    MATH  Google Scholar 

  21. Atanassov, K.: Temporal intuitionistic fuzzy sets. Comptes Rendus de l’Academie bulgare des Sciences, Tome 44(7), 5–7 (1991)

    Google Scholar 

  22. Atanassov, K.: On the temporal intuitionistic fuzzy sets. In: Proceedings of the ninth international conference IPMU 2002, vol. III, pp. 1833–1837, Annecy, France, 1–5 July 2002

    Google Scholar 

  23. Atanassov, K.: Temporal intuitionistic fuzzy sets (review and new results). In: Proceedings of the Thirty Second Spring of the Union of Bulgaria, pp. 79–88, Mathematicians, Sunny Beach, 5–8 Apr 2003

    Google Scholar 

  24. Atanassov, K.: A short remark on intuitionistic fuzzy operators X a,b,c,d,e,f and x a,b,c,d,e,f. Notes Intuitionistic Fuzzy Sets 19(1), 54–56 (2013) http://ifigenia.org/wiki/issue:nifs/19/1/54-56

  25. Dencheva, K.: Extension of intuitionistic fuzzy modal operators ⊞ and ⌧. In: Proceedings of the Second International IEEE Symposium: Intelligent Systems, vol. 3, pp. 21–22, Varna 22–24 June 2004

    Google Scholar 

  26. Atanassov K., Çuvalc oğlu, G., Atanassova, V.: A new modal operator over intuitionistic fuzzy sets. Notes Intuitionistic Fuzzy Sets 20(5), 1–8 (2014). http://ifigenia.org/wiki/issue:nifs/20/5/1-8

  27. Çuvalcioğlu, G.: Some properties of E α,β operator. Adv. Stud. Contemp. Math. 14(2), 305–310 (2007)

    Google Scholar 

  28. Çuvalcioğlu, G.: On the diagram of one type modal operators on intuitionistic fuzzy sets: last expanding with Z α,β ωθ. Iran. J. Fuzzy Syst 10(1), 89–106 (2013)

    Google Scholar 

  29. Çuvalcioğlu, G.: The extension of modal operators’ diagram with last operators. Notes Intuitionistic Fuzzy Sets 19(3), 56–61 (2013). http://ifigenia.org/wiki/issue:nifs/19/3/56-61

  30. Atanassov, K.: On four intuitionistic fuzzy topological operators. Mathware Soft. Comput. 8, 65–70 (2001)

    MathSciNet  MATH  Google Scholar 

  31. Atanassov, K.: On two topological operators over intuitionistic fuzzy sets. Issues Intuitionistic Fuzzy Sets Generalized Nets 8, 1–7 (2010)

    Google Scholar 

  32. Atanassov, K., Vassilev, P., Tsvetkov, R.: Intuitionistic fuzzy sets, measures and integrals. In: Drinov, M. (ed.) Academic Publishing House, Sofia (2013)

    Google Scholar 

  33. Atanassov, K.: Cantor’s norms for intuitionistic fuzzy sets. Issues Intuitionistic Fuzzy Sets Generalized Nets 8, 36–39 (2010)

    Google Scholar 

  34. Asparoukhov, O., Atanassov K.: Intuitionistic fuzzy interpretation of confidential intervals of criteria for decision making. In: Lakov, D. (ed.) Proceedings of the First Workshop on Fuzzy Based Expert Systems, pp. 56-58, Sofia, 28–30 Sept

    Google Scholar 

  35. Koshelev, M., Kreinovich, V., Rachamreddy, B., Yasemis, H., Atanassov, K.: Fundamental justification of intuitionistic fuzzy logic and interval-valued fuzzy methods. Notes Intuitionistic Fuzzy Sets 4(2), 42–46 (1998). http://ifigenia.org/wiki/issue:nifs/4/2/42-46

  36. Kreinovich, V., Mukaidono, M., Atanassov, K.: From fuzzy values to intuitionistic fuzzy values, to intuitionistic fuzzy intervals etc.: can we get an arbitrary ordering? Notes Intuitionistic Fuzzy Sets 5(3), 11–18 (1999). http://ifigenia.org/wiki/issue:nifs/5/3/11-18

  37. Kreinovich, V., Nguyen, H., Wu, B, Atanassov, K.: Fuzzy justification of heuristic methods in inverse problems and in numerical computations, with applications to detection of business cycles from fuzzy and intuitionistic fuzzy data. Notes Intuitionistic Fuzzy Sets 4(2), 47–56 (1998). http://ifigenia.org/wiki/issue:nifs/4/2/47-56

  38. Atanassov, K.: Remark on an application of the intuitionistic fuzzy sets in number theory. Adv. Stud. Contemp. Math. 5(1), 49–55 (2002)

    Google Scholar 

  39. Atanassov, K.: Intuitionistic fuzzy logics as tools for evaluation of data mining processes. Knowl-Based Syst. 80, 122–130 (2015)

    Google Scholar 

  40. InterCriteria Research Portal.: http://www.intercriteria.net/publications. Accessed 1 Aug 2015

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Acknowledgements

This paper has been partially supported by the Bulgarian National Science Fund under the Grant Ref. No. DFNI-I-02-5 “InterCriteria Analysis: A New Approach to Decision Making”. The author is thankful to Evgeniy Marinov, Peter Vassilev and Vassia Atanassova for their valuable comments.

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Authors and Affiliations

  1. Department of Bioinformatics and Mathematical Modelling, Institute of Biophysics and Biomedical Engineering—Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Block 105, 1113, Sofia, Bulgaria

    Krassimir Atanassov

  2. Intelligent Systems Laboratory, Professor Asen Zlatarov University, 8000, Bourgas, Bulgaria

    Krassimir Atanassov

Authors
  1. Krassimir Atanassov
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Correspondence to Krassimir Atanassov .

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Editors and Affiliations

  1. Department of Industrial Engineering, Istanbul Technical University, Istanbul, Turkey

    Cengiz Kahraman

  2. School of Industrial Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

    Uzay Kaymak

  3. Department of Computer Engineering, Middle East Technical University (M, Ankara, Turkey

    Adnan Yazici

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Atanassov, K. (2016). Mathematics of Intuitionistic Fuzzy Sets. In: Kahraman, C., Kaymak, U., Yazici, A. (eds) Fuzzy Logic in Its 50th Year. Studies in Fuzziness and Soft Computing, vol 341. Springer, Cham. https://doi.org/10.1007/978-3-319-31093-0_3

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  • DOI: https://doi.org/10.1007/978-3-319-31093-0_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-31091-6

  • Online ISBN: 978-3-319-31093-0

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