Skip to main content
SpringerLink
Log in
Book cover

Interval-Valued Methods in Classifications and Decisions pp 3–23Cite as

Fuzzy Sets and Their Extensions

  • Chapter
  • First Online:

Part of the book series: Studies in Fuzziness and Soft Computing ((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.

In July of 1964, I was in New York, and was scheduled to have dinner with my friends. The dinner was canceled. I had a free evening. My thoughts turned to the issue of cointensive indefinability. At that point, a simple idea clicked in my mind the concept of a grade of membership. The concept of a grade of membership was a key to the development of the theory of fuzzy sets.

Lotfi A. Zadeh [1]

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Zadeh, L.A.: Fuzzy logic-a personal perspective. Fuzzy Sets Syst. 281, 4–20 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)

    Article  MATH  Google Scholar 

  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. 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. Szpilrajn, E.: The characteristic function of a sequence of sets and some of its applications. Fund. Math. 31, 207–223 (1938)

    Article  MATH  Google Scholar 

  6. Menger, K.: Ensembles flous et fonctions aléatoires. C. R. Acad. Sci. Paris 232, 2001–2003 (1951)

    MathSciNet  MATH  Google Scholar 

  7. Rasiowa, H.: A generalization of a formalized theory of fields of sets on non-classical logics. Rozpr. Matemat. 42, 3–29 (1964)

    MathSciNet  MATH  Google Scholar 

  8. Zadeh, L.A.: Similarity relations and fuzzy orderings. Inf. Sci. 3, 177–200 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  9. Goguen, A.: L-fuzzy sets. J. Math. Anal. Appl. 18, 145–174 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bandler, W., Kohout, L.J.: Semantics of implication operators and fuzzy relational products. Int. J. Man-Mach. Stud. 12, 89–116 (1980)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  15. Baczyński, M., Jayaram, B.: Fuzzy Implications. Studies in Fuzziness and Soft Computing, vol. 231. Springer, Berlin (2008)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Nguyen, H.T., Walker, E.: A First Course in Fuzzy Logic. CRC Press, Boca Raton (1996)

    Google Scholar 

  18. Bustince, H., Barrenechea, E., Pagola, M.: Restricted equivalence functions. Fuzzy Sets Syst. 157, 2333–2346 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bentkowska, U., Król, A.: Fuzzy \(\alpha \)-\(C\)-equivalences. Fuzzy Sets Syst. (2018). https://doi.org/10.1016/j.fss.2018.01.004

    Google Scholar 

  20. Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publisher, Dordrecht (1994)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Gorzałczany, M.B.: A method of inference in approximate reasoning based on interval-valued fuzzy sets. Fuzzy Sets Syst. 21, 1–17 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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. Liu, K.: Grey sets and stability of grey systems. J. Huazhong Univ. Sci. Technol. 10(3), 23–25 (1982)

    MathSciNet  Google Scholar 

  27. Atanassov, K.T.: Intuitionistic fuzzy sets. In: Proceedings of VII ITKRs Session, pp. 1684–1697. Sofia, Bulgaria (1983)

    Google Scholar 

  28. Atanassov, K.T.: Intuitionistic fuzzy sets. Fuzzy Sets Syst. 20, 87–96 (1986)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  30. Gau, W.L., Buehrer, D.J.: Vague sets. IEEE Trans. Syst. Man Cybern. 23(2), 610–614 (1993)

    Article  MATH  Google Scholar 

  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. Pawlak, Z.: Rough sets. Int. J. Comput. Inf. Sci. 11, 341–356 (1982)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  36. Birkhoff, G.: Lattice Theory. AMS Colloquium Publications XXV, Providence (1967)

    MATH  Google Scholar 

  37. Atanassov, K.T.: Intuitionistic Fuzzy Sets: Theory and Applications. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  38. Deschrijver, G., Kerre, E.E.: On the relationship between some extensions of fuzzy set thory. Fuzzy Sets Syst. 133(2), 227–235 (2003)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  41. Karmakar, S., Bhunia, A.K.: A comparative study of different order relations of intervals. Reliab. Comput. 16, 38–72 (2012)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  43. Dubois, D., Godo, L., Prade, H.: Weighted logics for artificial intelligence an introductory discussion. Int. J. Approx. Reason. 55, 1819–1829 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  46. Fishburn, P.C.: Intransitive indifference with unequal indifference intervals. J. Math. Psychol. 7, 144–149 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  47. Fishburn, P.C.: Utility Theory for Decision Making. Wiley, New York (1970)

    Book  MATH  Google Scholar 

  48. Fishburn, P.C.: Interval Orders and Interval Graphs. Wiley, New York (1985)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  51. Moore, R.E.: Interval Analysis, vol. 4. Prentice-Hall, Englewood Cliffs (1966)

    Google Scholar 

  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. 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. Kulish, U.W., Miranker, W.L.: Computer Arithmetic in Theory and Practice. Academic, New York (1981)

    Google Scholar 

  55. Moore, R.E.: Methods and Applications for Interval Analysis. SIAM, Philadelfia (1979)

    Book  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  57. Acióly, B.M.: Computational Foundation of Interval Mathematic. Ph.D. thesis (in Portugeese). CPGCC, UFRGS, Porto Allegre (1991)

    Google Scholar 

  58. Sengupta, A., Pal, T.K.: On comparing interval numbers. Eur. J. Oper. Res. 127(1), 28–43 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  59. Ishibuchi, H., Tanaka, H.: Multiobjective programming in optimization of the interval objective function. Eur. J. Oper. Res. 48(2), 219–225 (1990)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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. Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM, Philadelphia (2009)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  64. Sengupta, A., Pal, T.K.: Fuzzy Preference Ordering of Interval Numbers in Decision Problems. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  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. Xu, Z.S., Yager, R.R.: Some geometric aggregation operators based on intuitionistic fuzzy sets. Int. J. Gen. Syst. 35, 417–433 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  68. Bustince, H.: Construction of intuitionistic fuzzy sets with predetermined properties. Fuzzy Sets Syst. 109, 379–403 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  69. Bustince, H., Burillo, P.: Perturbation of intuitionistic fuzzy relations. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 9, 81–103 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  71. Bedregal, B.: On interval fuzzy negations. Fuzzy Sets Syst. 161(17), 2290–2313 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    MATH  Google Scholar 

  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. Bedregal, B., Dimuro, G., Santiago, R., Reiser, R.: On interval fuzzy S-implications. Inf. Sci. 180(8), 1373–1389 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. 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)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Natural Sciences, University of Rzeszów, Rzeszów, Poland

    Urszula Bentkowska

Authors
  1. Urszula Bentkowska
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Urszula Bentkowska .

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Bentkowska, U. (2020). Fuzzy Sets and Their Extensions. In: Interval-Valued Methods in Classifications and Decisions. Studies in Fuzziness and Soft Computing, vol 378. Springer, Cham. https://doi.org/10.1007/978-3-030-12927-9_1

Download citation

Publish with us

Policies and ethics

Search

Navigation