Skip to main content
SpringerLink
Log in

Membership Functions

  • Chapter
  • First Online:
Fuzzy Approaches for Soft Computing and Approximate Reasoning: Theories and Applications

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 394))

Abstract

The idea of a fuzzy set is formally modeled by a membership function that plays the same role as the characteristic function for an ordinary set, except that the membership function takes intermediary values between full membership and no membership. In this short note we first provide some references about the historical emergence of this notion, then discuss the nature of the scale for membership grades, and finally review their elicitation in relation with their intended meaning as a matter of similarity, uncertainty or preference.

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

Access this chapter

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

Institutional subscriptions

Notes

  1. 1.

    We omit references here, for the sake of conciseness; readers can find a lot of them by searching for the corresponding key-words.

References

  1. Aickin, M.: Connecting Dempster-Shafer belief functions with likelihood-based inference. Synthese 123(3), 347–364 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aladenise, N., Bouchon-Meunier, B.: Acquisition de connaissances imparfaites: mise en évidence d’une fonction d’appartenance. Rev. Int. Syst. 11(1), 109–127 (1997)

    Google Scholar 

  3. Atanassov, K.: Answer to D. Dubois, S. Gottwald, P., Hajek, J., Kacprzyk, H., Prade’s paper “Terminological difficulties in fuzzy set theory—the case of intuitionistic fuzzy sets”. Fuzzy Sets Syst. 156, 496–499 (2005)

    Google Scholar 

  4. Basu, K., Deb, R., Pattanaik, P.K.: Soft sets: an ordinal formulation of vagueness with some applications to the theory of choice. Fuzzy Sets Syst. 45, 45–58 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bellman, R.E., Zadeh, L.A.: Decision making in a fuzzy environment. Manag. Sci. 17, B141–B164 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bellman, R.E., Kalaba, R., Zadeh, L.A.: Abstraction and pattern classification. J. Math. Anal. Appl. 13, 1–7 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  7. Benferhat, S., Dubois, D., Prade, H., Williams, M.-A.: A framework for iterated Belief revision using possibilistic counterparts to Jeffrey’s rule. Fundam. Inform. 99(2): 147-168 (2010)

    Google Scholar 

  8. Bezdek, J., Keller, J., Krishnapuram, R., Pal, N.: Fuzzy Models for Pattern Recognition and Image Processing. Kluwer (1999)

    Google Scholar 

  9. Black, M.: Vagueness. Phil. Sci. 4, 427–455 (1937). Reprinted in Language and Philosophy: Studies in Method, Cornell University Press, Ithaca and London, pp. 23–58 (1949). Also in Int. J. Gener. Syst. 17, 107–128 (1990)

    Google Scholar 

  10. Cattaneo, G., Ciucci, D.: Basic intuitionistic principles in fuzzy set theories and its extensions (A terminological debate on Atanassov IFS). Fuzzy Sets Syst. 157(24), 3198–3219 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen, J.E., Otto, K.N.: Constructing membership functions using interpolation and measurement theory. Fuzzy Sets Syst. 73, 313–327 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Coletti, G., Scozzafava, R.: Coherent conditional probability as a measure of uncertainty of the relevant conditioning events. In: Proceedings of ECSQARU’03, Aalborg, LNAI, vol. 2711, pp. 407–418. Springer-Verlag (2003)

    Google Scholar 

  13. Coletti, G., Scozzafava, R.: Conditional probability, fuzzy sets, and possibility: a unifying view. Fuzzy Sets Syst. 144(1), 227–249 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Couso, I., Montes, S., Gil., P.: The necessity of the strong \(\alpha \)-cuts of a fuzzy set. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 9(2), 249–262 (2001)

    Google Scholar 

  15. Dombi, J.: Basic concepts for a theory of evaluation: the aggregative operator. Eur. J. Oper. Res. 10, 282–293 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dubois, D., Perny, P.: A review of fuzzy sets in decision sciences: achievements, limitations and perspectives. In: Greco S., et al. (eds.) Multiple Criteria Decision Analysis. State of the Art Surveys, pp. 637–691. Springer (2016)

    Google Scholar 

  17. Dubois, D., Prade, H., Rannou, E.: An improved method for finding typical values. In: Proceedings 7th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU’98), Paris, July 6–10, Editions EDK, Paris, pp. 1830–1837 (1998)

    Google Scholar 

  18. Dubois, D., Prade, H., Sandri, S.: On possibility/probability transformations. In: Lowen, R., Roubens, M. (eds.) Fuzzy Logic. State of the Art, pp. 103–112. Kluwer Academic Publishers, Dordrecht (1993)

    Google Scholar 

  19. Dubois, D., Prade, H.: New results about properties and semantics of fuzzy-set-theoretic operators. In: Wang, P.P., Chang, S.K. (eds.) Fuzzy Sets: Theory and Applications to Policy Analysis and Information Systems, pp. 59–75. Plenum Publishers (1980)

    Google Scholar 

  20. Dubois, D., Prade, H.: Possibility theory: qualitative et quantitative aspects. In: Gabbay, D., Smets, P. (eds.) Quantified Representation of Uncertainty et Imprecision. Handbook of Defeasible Reasoning et Uncertainty Management Systems, vol. 1, pp. 169–226. Kluwer Academic Publishers (1998)

    Google Scholar 

  21. Dubois, D., Prade, H.: Qualitative possibility theory and its applications to reasoning and decision under uncertainty. JORBEL (Belg. J. Oper. Res.) 37(1–2), 5–28 (1997)

    Google Scholar 

  22. Dubois, D., Prade, H.: Toll sets and toll logic. In: Lowen, R., Roubens, M. (eds.) Fuzzy Logic: State of the Art, pp. 169–177. Kluwer Academic Publishers (1993)

    Google Scholar 

  23. Dubois, D.: Possibility theory and statistical reasoning. Comput. Stat. Data Anal. 51, 47–69 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Dubois, D., Hüllermeier, E.: Comparing probability measures using possibility theory: a notion of relative peakedness. Int. J. Approx. Reason. 45, 364–385 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Dubois, D., Prade, H.: The three semantics of fuzzy sets. Fuzzy Sets Syst. 90, 141–150 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Dubois, D., Prade, H.: Possibility theory, probability theory and multiple valued logics: a clarification. Ann. Math. Artif. Intell. 32, 35–66 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Dubois, D., Prade, H.: On the use of aggregation operations in information fusion processes. Fuzzy Sets Syst. 142, 143–161 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. Dubois, D., Prade, H.: An introduction to bipolar representations of information and preference. Int. J. Intell. Syst. 23(8), 866–877 (2008)

    Article  MATH  Google Scholar 

  29. Dubois, D., Prade, H.: Gradual elements in a fuzzy set. Soft. Comput. 12, 165–175 (2008)

    Article  MATH  Google Scholar 

  30. 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 

  31. Dubois, D., Prade, H., Rossazza, J.P.: Vagueness, typicality and uncertainty in class hierarchies. Int. J. Intell. Syst. 6(2), 167–183 (1991)

    Article  Google Scholar 

  32. Dubois, D., Moral, S., Prade, H.: A semantics for possibility theory based on likelihoods. J. Math. Anal. Appl. 205, 359–380 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. Dubois, D., Foulloy, L., Mauris, G., Prade, H.: Possibility/probability transformations, triangular fuzzy sets, and probabilistic inequalities. Reliab. Comput. 10, 273–297 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  34. Dubois, D., Gottwald, S., Hájek, P., Kacprzyk, J., Prade, H.: Terminological difficulties in fuzzy set theory—the case of “Intuitionistic Fuzzy Set”. Fuzzy Sets Syst. 156(3), 485–491 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  35. Edwards, W.F.: Likelihood. Cambridge University Press, Cambridge, U.K. (1972)

    MATH  Google Scholar 

  36. Finch, P.D.: Characteristics of interest and fuzzy subsets. Inf. Sci. 24, 121–134 (1981)

    Article  MathSciNet  Google Scholar 

  37. Förstner, W.: Uncertain neighborhood relations of point sets and fuzzy Delaunay triangulation. In: Förstner, W., Buhmann, J.M., Faber, A., Faber, P. (eds.) Proceedings of the Mustererkennung 1999, 21. DAGM-Symposium, Bonn, 15–17 Sept., Informatik Aktuell, pp. 213–222. Springer (1999)

    Google Scholar 

  38. Freund, M.: On the notion of concept I & II. Artif. Intell. 172, 570–590, 2008 & 173, 167–179 (2009)

    Google Scholar 

  39. Ganter, B., Wille, R.: Formal Concept Analysis. Springer-Verlag (1999)

    Google Scholar 

  40. Giles, R.: The concept of grade of membership. Fuzzy Sets Syst. 25, 297–323 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  41. Goguen, J.A.: L-fuzzy sets. J. Math. Anal. Appl. 8, 145–174 (1967)

    Article  MATH  Google Scholar 

  42. Goodman, I.R.: Fuzzy sets as equivalence classes of random sets. In: Yager, R. (ed.) Fuzzy Sets and Possibility Theory: Recent Developments, pp. 327–342. Pergamon Press, Oxford (1981)

    Google Scholar 

  43. Grabisch, M.: The Moebius transform on symmetric ordered structures and its application to capacities on finite sets. Discret. Math. 287, 17–34 (2004)

    Article  MATH  Google Scholar 

  44. Grattan-Guiness, I.: Fuzzy membership mapped onto interval and many-valued quantities. Z. Math. Logik. Grundladen Math. 22, 149–160 (1975)

    Article  MathSciNet  Google Scholar 

  45. Gutiérrez García, J., Rodabaugh, S.E.: Order-theoretic, topological, categorical redundancies of interval-valued sets, grey sets, vague sets, interval-valued “intuitionistic” sets, “intuitionistic” fuzzy sets and topologies. Fuzzy Sets Syst. 156(3), 445–484 (2005)

    Google Scholar 

  46. Hisdal, E.: Are grades of membership probabilities? Fuzzy Sets Syst. 25, 325–348 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  47. Jahn, K.U.: Intervall-wertige Mengen. Math. Nach. 68, 115–132 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  48. Kampé de Fériet, J.: Interpretation of membership functions of fuzzy sets in terms of plausibility and belief. In: Gupta, M., Sanchez, E. (eds.) Fuzzy Information and Decision Processes, pp. 93–98. North-Holland, Amsterdam (1982)

    Google Scholar 

  49. Kaplan, A.: Definition and specification of meanings. J. Phil. 43, 281–288 (1946)

    Article  Google Scholar 

  50. Kaplan, A., Schott, H.F.: A calculus for empirical classes. Methods III, 165–188 (1951)

    Google Scholar 

  51. Lawry, J.: Modelling and Reasoning with Vague Concepts. Springer (2006)

    Google Scholar 

  52. Lee, J.W.T.: Ordinal decomposability and fuzzy connectives. Fuzzy Sets Syst. 136, 237–249 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  53. Lee, J.W.T., Yeung, D.S., Tsang, E.C.C.: Ordinal fuzzy sets. IEEE Trans. Fuzzy Syst. 10(6), 767–778 (2002)

    Article  Google Scholar 

  54. Lesot, M.-J., Mouillet, L., Bouchon-Meunier, B.: Fuzzy prototypes based on typicality degrees. In: Reusch, B. (ed.) Proceedings of the International Conference on 8th Fuzzy Days, Dortmund, Germany, Sept. 29–Oct. 1, 2004. Advances in Soft Computing, vol. 33, pp. 125–138. Springer (2005)

    Google Scholar 

  55. Łukasiewicz, J.: Philosophical, remarks on many-valued systems of propositional logic (1930). In: Borkowski (ed.) Studies in Logic and the Foundations of Mathematics, pp. 153–179. North-Holland (1970) (Reprinted in Selected Works)

    Google Scholar 

  56. Maji, P.K., Biswas, R., Roy, A.R.: Soft set theory. Comput. Math. Appl. 45(4–5), 555–562 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  57. Marchant, T.: The measurement of membership by comparisons. Fuzzy Sets Syst. 148, 157–177 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  58. Marchant, T.: The measurement of membership by subjective ratio estimation. Fuzzy Sets Syst. 148, 179–199 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  59. Martin, T.P., Azvine, B.: The X-mu approach: fuzzy quantities, fuzzy arithmetic and fuzzy association rules. In: Proceedings of the IEEE Symposium on Foundations of Computational Intelligence (FOCI), pp. 24–29 (2013)

    Google Scholar 

  60. Mauris, G.: Possibility distributions: a unified representation of usual direct-probability-based parameter estimation methods. Int. J. Approx. Reason. 52(9), 1232–1242 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  61. Mauris, G., Foulloy, L.: A fuzzy symbolic approach to formalize sensory measurements: an application to a comfort sensor. IEEE Trans. Instrum. Measure. 51(4), 712–715 (2002)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  63. Molodtsov, D.: Soft set theory—first results. Comput. Math. Appl. 37(4/5), 19–31 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  64. Ruspini, E.H.: A new approach to clustering. Inform. Control 15, 22–32 (1969)

    Article  MATH  Google Scholar 

  65. Ruspini, E.H.: On the semantics of fuzzy logic. Int. J. Approx. Reason. 5(1), 45–88 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  66. Saaty, T.L.: Measuring the fuzziness of sets. J. Cybern. 4(4), 53–61 (1974)

    Article  MATH  Google Scholar 

  67. Sambuc, R.: Fonctions \(\phi \)-floues. Application à l’aide au diagnostic en pathologie thyroidienne. Ph.D. Thesis, Univ. Marseille, France (1975)

    Google Scholar 

  68. Sanchez, D., Delgado, M., Villa, M.A., Chamorro-Martinez, J.: On a non-nested level-based representation of fuzziness. Fuzzy Sets Syst. 192, 159–175 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  69. Savage, L.J.: The Foundations of Statistics. Wiley (1954)

    Google Scholar 

  70. Shafer, G.: A Mathematical Theory of Evidence. Princeton University Press (1976)

    Google Scholar 

  71. Smets, P.: Possibilistic inference from statistical data. In: Ballester, A. (ed.) Proceedings of the 2nd World Conference on Mathematics at the Service of Man, Las Palmas (Spain) pp. 611–613 (1982)

    Google Scholar 

  72. Smets, Ph: Constructing the pignistic probability function in a context of uncertainty. In: Henrion, M., et al. (eds.) Uncertainty in Artificial Intelligence, vol. 5, pp. 29–39. North-Holland, Amsterdam (1990)

    Chapter  Google Scholar 

  73. Smith, E.E., Osherson, D.N.: Conceptual combination with prototype concepts. Cognit. Sci. 8, 357–361 (1984)

    Google Scholar 

  74. Sudkamp, T.: Similarity and the measurement of possibility. In: Actes Rencontres Francophones sur la Logique Floue et ses Applications (Montpellier, France), Toulouse: Cepadues Editions, pp. 13–26 (2002)

    Google Scholar 

  75. Trillas, E., Alsina, C.: A reflection on what is a membership function. Mathw. Soft Comput. VI(2–3), 201–215 (1999)

    Google Scholar 

  76. Türksen, I.B., Bilgic, T.: Measurement of membership functions: theoretical and empirical work. In: Dubois, D., Prade, H. (eds.) Fundamentals of Fuzzy Sets. The Handbooks of Fuzzy Sets, pp. 195–230. Kluwer Publ. Comp. (2000)

    Google Scholar 

  77. Vasudev Murthy, S., Kandel, A.: Fuzzy sets and typicality theory. Inform. Sci. 51(1), 61–93 (1990)

    Google Scholar 

  78. Wang, G.-J., He, Y.-H.: Intuitionistic fuzzy sets and L-fuzzy sets. Fuzzy Sets Syst. 110, 271–274 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  79. Weyl, H.: Mathematics and logic- a brief survey serving as preface to a review of "the Philosophy of Bertrand Russell". Amer. Math. Month. 53, 2–13 (1946)

    MathSciNet  MATH  Google Scholar 

  80. Yager, R.R., Rybalov, A.: Uninorm aggregation operators. Fuzzy Sets Syst. 80(1), 111–120 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  81. Zadeh, L.A.: Fuzzy sets and systems. In: Fox, J. (ed.) Systems Theory; Proceedings of the Simposium on System Theory, New York, April 20–22, pp. 29–37 (1965). Polytechnic Press, Brooklyn, N.Y. (1966) (reprinted in Int. J. Gener. Syst. 17, 129–138 (1990))

    Google Scholar 

  82. Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning. Inf. Sci. Part I 8(3), 199–249; Part II 8(4), 301–357; Part III 9(1), 43–80 (1975)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  84. Zadeh, L.A.: Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1, 3–28 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  85. Zadeh, L.A.: PRUF—A meaning representation language for natural languages. Int. J. Man-Mach. Stud. 10, 395–460 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  86. Zadeh, L.A.: A note on prototype theory and fuzzy sets. Cognition 12(3), 291–297 (1982)

    Article  Google Scholar 

Download references

Dedication

The authors are very glad to dedicate this chapter to Bernadette for her continuous and endless efforts to develop and promote the fuzzy set methodology in an open-minded way through her publications, and also for the launching of important international conferences she organized such as IPMU, which have been essential forums for exchanges between various approaches to uncertainty and vagueness, including fuzzy set theory, over more than four decades. Thank you so much, Bernadette.

Author information

Authors and Affiliations

  1. IRIT, Université Paul Sabatier, 118 Route de Narbonne, 31062, Toulouse Cedex 9, France

    Didier Dubois & Henri Prade

Authors
  1. Didier Dubois
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Henri Prade
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Didier Dubois .

Editor information

Editors and Affiliations

  1. Sorbonne Université, CNRS, LIP6, Paris, France

    Marie-Jeanne Lesot

  2. Sorbonne Université, CNRS, LIP6, Paris, France

    Christophe Marsala

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Dubois, D., Prade, H. (2021). Membership Functions. In: Lesot, MJ., Marsala, C. (eds) Fuzzy Approaches for Soft Computing and Approximate Reasoning: Theories and Applications. Studies in Fuzziness and Soft Computing, vol 394. Springer, Cham. https://doi.org/10.1007/978-3-030-54341-9_2

Download citation

Publish with us

Policies and ethics

Search

Navigation