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Fuzzy Sets and Their Extensions

  • Urszula BentkowskaEmail author
Chapter
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 378)

Abstract

In this chapter basic notions regarding fuzzy calculus, its history and basic properties are recalled. Moreover, extensions of fuzzy sets are briefly described and the most important results concerning interval-valued fuzzy calculus are provided. Especially, the notions of diverse order and comparability relations for interval-valued settings are discussed.

References

  1. 1.
    Zadeh, L.A.: Fuzzy logic-a personal perspective. Fuzzy Sets Syst. 281, 4–20 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)zbMATHCrossRefGoogle Scholar
  3. 3.
    Klaua, D.: Über einen Ansatz zur mehrwertigen Mengenlehre. Monatsb. Deutsch. Akad. Wiss. Berlin 7, 859–876 (1965) A recent in-depth analysis of this paper has been provided by Gottwald, S.: An early approach toward graded identity and graded membership in set theory. Fuzzy Sets Syst. 161(18), 2369–2379 (2010)Google Scholar
  4. 4.
    Łukasiewicz, J: O logice trójwartościowej (in Polish). Ruch filozoficzny 5, 170–171 (1920) English translation: On three-valued logic. In: Borkowski L. (eds.) Selected works by Jan Łukasiewicz, pp. 87–88. North Holland, Amsterdam (1970)Google Scholar
  5. 5.
    Szpilrajn, E.: The characteristic function of a sequence of sets and some of its applications. Fund. Math. 31, 207–223 (1938)zbMATHCrossRefGoogle Scholar
  6. 6.
    Menger, K.: Ensembles flous et fonctions aléatoires. C. R. Acad. Sci. Paris 232, 2001–2003 (1951)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Rasiowa, H.: A generalization of a formalized theory of fields of sets on non-classical logics. Rozpr. Matemat. 42, 3–29 (1964)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Zadeh, L.A.: Similarity relations and fuzzy orderings. Inf. Sci. 3, 177–200 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Goguen, A.: L-fuzzy sets. J. Math. Anal. Appl. 18, 145–174 (1967)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Bandler, W., Kohout, L.J.: Semantics of implication operators and fuzzy relational products. Int. J. Man-Mach. Stud. 12, 89–116 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers, Dordrecht (2000)zbMATHCrossRefGoogle Scholar
  12. 12.
    Pradera, A., Beliakov, G., Bustince, H., De Baets, B.: A review of the relationships between implication, negation and aggregation functions from the point of view of material implication. Inf. Sci. 329, 357–380 (2016)zbMATHCrossRefGoogle Scholar
  13. 13.
    Drewniak, J., Król, A.: A survey of weak connectives and the preservation of their properties by aggregations. Fuzzy Sets Syst. 161, 202–215 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Bentkowska, U., Król, A.: Preservation of fuzzy relation properties based on fuzzy conjunctions and disjunctions during aggregation process. Fuzzy Sets Syst. 291, 98–113 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Baczyński, M., Jayaram, B.: Fuzzy Implications. Studies in Fuzziness and Soft Computing, vol. 231. Springer, Berlin (2008)Google Scholar
  16. 16.
    Bustince, H., Barrenechea, E., Pagola, M.: Image thresholding using restricted equivalence functions and maximizing the measures of similarity. Fuzzy Sets Syst. 158, 496–516 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Nguyen, H.T., Walker, E.: A First Course in Fuzzy Logic. CRC Press, Boca Raton (1996)Google Scholar
  18. 18.
    Bustince, H., Barrenechea, E., Pagola, M.: Restricted equivalence functions. Fuzzy Sets Syst. 157, 2333–2346 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Bentkowska, U., Król, A.: Fuzzy \(\alpha \)-\(C\)-equivalences. Fuzzy Sets Syst. (2018).  https://doi.org/10.1016/j.fss.2018.01.004Google Scholar
  20. 20.
    Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publisher, Dordrecht (1994)zbMATHCrossRefGoogle Scholar
  21. 21.
    Sambuc, R.: Fonctions \(\phi \)-floues: Application á l’aide au Diagnostic en Pathologie Thyroidienne. Ph.D. thesis, Universit\(\acute{e}\) de Marseille, France (1975) (in French)Google Scholar
  22. 22.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning-I. Inf. Sci. 8, 199–249 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Gorzałczany, M.B.: A method of inference in approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets Syst. 21, 1–17 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Bustince, H., Barrenechea, E., Pagola, M., Fernandez, J., Xu, Z., Bedregal, B., Montero, J., Hagras, H., Herrera, F., De Baets, B.: A historical account of types of fuzzy sets and their relationships. IEEE Trans. Fuzzy Syst. 24(1), 179–194 (2016)CrossRefGoogle Scholar
  25. 25.
    Hirota, K.: Concept of probabilistic sets. In: Proceedings of IEEE Conference on Decision and Control, pp. 1361–1366. New Orleans (1977)Google Scholar
  26. 26.
    Liu, K.: Grey sets and stability of grey systems. J. Huazhong Univ. Sci. Technol. 10(3), 23–25 (1982)MathSciNetGoogle Scholar
  27. 27.
    Atanassov, K.T.: Intuitionistic fuzzy sets. In: Proceedings of VII ITKRs Session, pp. 1684–1697. Sofia, Bulgaria (1983)Google Scholar
  28. 28.
    Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986)zbMATHCrossRefGoogle Scholar
  29. 29.
    Dubois, D., Gottwald, S., Hajek, P., Kacprzyk, J., Prade, H.: Terminological difficulties in fuzzy set theory - the case of intuitionistic fuzzy sets. Fuzzy Sets Syst. 156, 485–491 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Gau, W.L., Buehrer, D.J.: Vague sets. IEEE Trans. Syst. Man Cybern. 23(2), 610–614 (1993)zbMATHCrossRefGoogle Scholar
  31. 31.
    Yager, R.R.: Pythagorean fuzzy subsets. In: Proceedings of the Joint IFSA World Congress and NAFIPS Annual Meeting, pp. 57–61 (2013)Google Scholar
  32. 32.
    Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11, 341–356 (1982)zbMATHCrossRefGoogle Scholar
  33. 33.
    Sanz, J., Fernandez, A., Bustince, H., Herrera, F.: A genetic tuning to improve the performance of fuzzy rule-based classification systems with intervalvalued fuzzy sets: degree of ignorance and lateral position. Int. J. Approx. Reason. 52(6), 751–766 (2011)CrossRefGoogle Scholar
  34. 34.
    Bustince, H., Pagola, M., Barrenechea, E., Fernandez, J., Melo-Pinto, P., Couto, P., Tizhoosh, H.R., Montero, J.: Ignorance functions. An application to the calculation of the threshold in prostate ultrasound images. Fuzzy Sets Syst. 161(1), 20–36 (2010)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Barrenechea, E., Fernandez, J., Pagola, M., Chiclana, F., Bustince, H.: Construction of interval-valued fuzzy preference relations from ignorance functions and fuzzy preference relations: application to decision making. Knowl. Based Syst. 58, 33–44 (2014)CrossRefGoogle Scholar
  36. 36.
    Birkhoff, G.: Lattice Theory. AMS Colloquium Publications XXV, Providence (1967)zbMATHGoogle Scholar
  37. 37.
    Atanassov, K.T.: Intuitionistic Fuzzy Sets: Theory and Applications. Springer, Berlin (1999)zbMATHCrossRefGoogle Scholar
  38. 38.
    Deschrijver, G., Kerre, E.E.: On the relationship between some extensions of fuzzy set thory. Fuzzy Sets Syst. 133(2), 227–235 (2003)zbMATHCrossRefGoogle Scholar
  39. 39.
    Deschrijver, G., Kerre, E.E.: On the position of intuitionistic fuzzy set theory in the framework of theories modelling imprecision. Inf. Sci. 177, 1860–1866 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Lin, L., Yuan, X.-H., Xia, Z.-Q.: Multicriteria fuzzy decision-making methods based on intuitionistic fuzzy sets. J. Comput. Syst. Sci. 73, 84–88 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Karmakar, S., Bhunia, A.K.: A comparative study of different order relations of intervals. Reliab. Comput. 16, 38–72 (2012)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Dubois, D., Prade, H.: Gradualness, uncertainty and bipolarity: making sense of fuzzy sets. Fuzzy Sets Syst. 192, 3–24 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Dubois, D., Godo, L., Prade, H.: Weighted logics for artificial intelligence an introductory discussion. Int. J. Approx. Reason. 55, 1819–1829 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Pȩkala, B., Bentkowska, U., De Baets, B.: On comparability relations in the class of interval-valued fuzzy relations. Tatra Mt. Math. Publ. 66, 91–101 (2016)MathSciNetzbMATHGoogle Scholar
  45. 45.
    Bustince, H., Fernandez, J., Kolesárová, A., Mesiar, R.: Generation of linear orders for intervals by means of aggregation functions. Fuzzy Sets Syst. 220, 69–77 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Fishburn, P.C.: Intransitive indifference with unequal indifference intervals. J. Math. Psychol. 7, 144–149 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Fishburn, P.C.: Utility Theory for Decision Making. Wiley, New York (1970)zbMATHCrossRefGoogle Scholar
  48. 48.
    Fishburn, P.C.: Interval Orders and Interval Graphs. Wiley, New York (1985)zbMATHCrossRefGoogle Scholar
  49. 49.
    Callejas-Bedregal, R., Callejas Bedregal, B.R.: Intervals as a domain constructor. TEMA - Tendências em Matemática Aplicada e Computacional 2(1), 43–52 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Dembczyński, K., Greco, S., Sowiński, R.: Rough set approach to multiple criteria classification with imprecise evaluations and assignments. Eur. J. Oper. Res. 198, 626–636 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Moore, R.E.: Interval Analysis, vol. 4. Prentice-Hall, Englewood Cliffs (1966)Google Scholar
  52. 52.
    Scot, D.S.: Outline of a mathematical theory of computation. In: 4th Annual Princeton Conference on Information Sciences and Systems, pp. 169–176 (1970)Google Scholar
  53. 53.
    Kulish, U.W., Miranker, W.L.: Computer Arithmetic in Theory and Practice. Technical report 33658, IBM Thomas L. Watson Research Center (1979)Google Scholar
  54. 54.
    Kulish, U.W., Miranker, W.L.: Computer Arithmetic in Theory and Practice. Academic, New York (1981)Google Scholar
  55. 55.
    Moore, R.E.: Methods and Applications for Interval Analysis. SIAM, Philadelfia (1979)zbMATHCrossRefGoogle Scholar
  56. 56.
    Dimuro, G.P., Costa, A.C.R., Claudio, D.M.: A coherent space of rational intervals for construction of IFR. J. Rielable Comput. 6, 139–178 (2000)zbMATHCrossRefGoogle Scholar
  57. 57.
    Acióly, B.M.: Computational Foundation of Interval Mathematic. Ph.D. thesis (in Portugeese). CPGCC, UFRGS, Porto Allegre (1991)Google Scholar
  58. 58.
    Sengupta, A., Pal, T.K.: On comparing interval numbers. Eur. J. Oper. Res. 127(1), 28–43 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Ishibuchi, H., Tanaka, H.: Multiobjective programming in optimization of the interval objective function. Eur. J. Oper. Res. 48(2), 219–225 (1990)zbMATHCrossRefGoogle Scholar
  60. 60.
    Chanas, S., Kuchta, D.: Multiobjective programming in optimization of interval objective functions - a generalized approach. Eur. J. Oper. Res. 94(3), 594–598 (1996)zbMATHCrossRefGoogle Scholar
  61. 61.
    Mahato, S.K., Bhunia, A.K.: Interval-arithmetic-oriented interval computing technique for global optimization. Appl. Math. Res. Express 1–19, (2006)Google Scholar
  62. 62.
    Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)zbMATHCrossRefGoogle Scholar
  63. 63.
    Karmakar, S., Bhunia, A.K.: An alternative optimization technique for interval objective constrained optimization problems via multiobjective programming. J. Egypt. Math. Soc. 22, 292–303 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    Sengupta, A., Pal, T.K.: Fuzzy Preference Ordering of Interval Numbers in Decision Problems. Springer, Berlin (2009)zbMATHCrossRefGoogle Scholar
  65. 65.
    Pȩkala, B.: Uncertainty Data in Interval-Valued Fuzzy Set Theory. Properties, Algorithms and Applications. Studies in Fuzziness and Soft Computing. Springer, Cham, Switzerland (2019)Google Scholar
  66. 66.
    Xu, Z.S., Yager, R.R.: Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gen. Syst. 35, 417–433 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Bentkowska, U., Bustince, H., Jurio, A., Pagola, M., Pȩkala, B.: Decision making with an interval-valued fuzzy preference relation and admissible orders. Appl. Soft Comput. 35, 792–801 (2015)CrossRefGoogle Scholar
  68. 68.
    Bustince, H.: Construction of intuitionistic fuzzy sets with predetermined properties. Fuzzy Sets Syst. 109, 379–403 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Bustince, H., Burillo, P.: Perturbation of intuitionistic fuzzy relations. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 9, 81–103 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Bentkowska, U.: New types of aggregation functions for interval-valued fuzzy setting and preservation of pos-B and nec-B-transitivity in decision making problems. Inf. Sci. 424, 385–399 (2018)MathSciNetCrossRefGoogle Scholar
  71. 71.
    Bedregal, B.: On interval fuzzy negations. Fuzzy Sets Syst. 161(17), 2290–2313 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Deschrijver, G., Cornelis, C., Kerre, E.: On the representation of intuitonistic fuzzy t-norms and t-conorms. IEEE Trans. Fuzzy Syst. 12, 45–61 (2004)CrossRefGoogle Scholar
  73. 73.
    Asiaín, M.J., Bustince, H., Mesiar, R., Kolesárová, A., Takáč, Z.: Negations with respect to admissible orders in the interval-valued fuzzy set theory. IEEE Trans. Fuzzy Syst. 26(2), 556–568 (2018)CrossRefGoogle Scholar
  74. 74.
    Zapata, H., Bustince, H., Montes, S., Bedregal, B., Dimuro, G.P., Takáč, Z., Baczyński, M., Fernandez, J.: Interval-valued implications and interval-valued strong equality index with admissible orders. Int. J. Approx. Reason. 88, 91–109 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    Bustince, H., Montero, J., Pagola, M., Barrenechea, E., Gomez, D.: A survey of interval-valued fuzzy sets. In: Pedrycz, W., Skowron, A., Kreinovich, V. (eds.) Handbook of Granular Computing, pp. 489–515. Wiley, New York (2008)CrossRefGoogle Scholar
  76. 76.
    Bustince, H., Barrenechea, E., Pagola, M.: Generation of interval-valued fuzzy and atanassovs intuitionistic fuzzy connectives from fuzzy connectives and from \(K_{\alpha }\) operators: laws for conjunctions and disjunctions, amplitude. Int. J. Intell. Syst. 23, 680–714 (2008)zbMATHGoogle Scholar
  77. 77.
    Bedregal, B., Dimuro, G., Santiago, R., Reiser, R.: An approach to interval-valued R-implications and automorphisms. In: Carvalho, J.P., Dubois, D., Kaymak, U., Sousa, J.M.C. (eds.) Proceedings of the Joint 2009 International Fuzzy Systems Association World Congress and 2009 European Society of Fuzzy Logic and Technology Conference, Lisbon, Portugal, pp. 1–6. ISBN: 978-989-95079-6-8 (20–24 July, 2009)Google Scholar
  78. 78.
    Bedregal, B., Dimuro, G., Santiago, R., Reiser, R.: On interval fuzzy S-implications. Inf. Sci. 180(8), 1373–1389 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    Cornelis, C., Deschrijver, G., Kerre, E.E.: Implication in intuitionistic fuzzy and interval-valued fuzzy set theory: construction, classification, application. Int. J. Approx. Reason. 35(1), 55–95 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  80. 80.
    Reiser, R.H.S., Dimuro, G.P., Bedregal, B.C., Santiago, R.H.N.: Interval valued QL-implications. In: Leivant D., De Queiroz R. (eds.) Logic, Language, Information and Computation. WoLLIC 2007. Lecture Notes in Computer Science, vol. 4576, pp. 307–321. Springer, Berlin (2007)CrossRefGoogle Scholar
  81. 81.
    Jurio, A., Pagola, M., Paternain, D., Lopez-Molina, C., Melo-Pinto, P.: Interval-valued restricted equivalence functions applied on clustering technique. In: Carvalho, J.P., Dubois, D., Kaymak, U., Sousa, J.M.C. (eds.) Proceedings of the Joint 2009 International Fuzzy Systems Association World Congress and 2009 European Society of Fuzzy Logic and Technology Conference, Lisbon, Portugal, pp. 831–836. ISBN: 978-989-95079-6-8 (20–24 July, 2009)Google Scholar
  82. 82.
    Bustince, H., Galar, M., Bedregal, B., Kolesárová, A., Mesiar, R.: A new approach to interval-valued Choquet integrals and the problem of ordering in interval-valued fuzzy sets applications. IEEE Trans. Fuzzy Syst. 21(6), 1150–1162 (2013)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Faculty of Mathematics and Natural SciencesUniversity of RzeszówRzeszówPoland

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