Fuzzy Set Theory

  • James G. Shanahan
Part of the The Springer International Series in Engineering and Computer Science book series (SECS, volume 570)


This chapter presents the fundamental ideas behind fuzzy set theory. It begins with a review of traditional set theory (commonly referred to as crisp or classical set theory in the fuzzy set literature) and uses this as a backdrop, against which fuzzy sets and a variety of operations on fuzzy sets are introduced. Various justifications and interpretations of fuzzy sets as a form of knowledge granulation are subsequently presented. Different families of fuzzy set aggregation operators are then examined. The original notion of a fuzzy set can be generalised in a number of ways leading to more expressive forms of knowledge representation; the latter part of this chapter presents some of these generalisations, where the original idea of a fuzzy set is generalised in terms of its dimensionality, type of membership value and element characterisation. Finally, fuzzy set elicitation is briefly covered for completeness (Chapter 9 gives more detailed coverage of this topic).


Membership Function Fuzzy Relation Membership Grade Necessity Measure Knowledge Granulation 
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  1. Astanassov, K. T. (1986). “Intuitionistic fuzzy sets”, Fuzzy sets and systems, 20:87–96.MathSciNetCrossRefGoogle Scholar
  2. Baldwin, J. F. (1991). “A Theory of Mass Assignments for Artificial Intelligence”, In 1JCAI ′91 Workshops on Fuzzy Logic and Fuzzy Control, Sydney, Australia, Lecture Notes in Artificial Intelligence, A. L. Ralescu, ed., 22–34.Google Scholar
  3. Baldwin, J. F., and Lawry, J. (2000). “A fuzzy c-means algorithm for prototype induction.” In the proceedings of IPMU, Madrid, To appear.Google Scholar
  4. Baldwin, J. F., Martin, T. P., and Pilsworth, B. W. (1995). FRIL — Fuzzy and Evidential Reasoning in A.I. Research Studies Press (Wiley Inc.), ISBN 086380159 5.Google Scholar
  5. Baldwin, J. F., Martin, T. P., and Shanahan, J. G. (1997a). “Fuzzy logic methods in vision recognition.” In the proceedings of Fuzzy Logic: Applications and Future Directions Workshop, London, UK, 300-316.Google Scholar
  6. Baldwin, J. F., Martin, T. P., and Shanahan, J. G. (1997b). “Modelling with words using Cartesian granule features.” In the proceedings of FUZZ-IEEE, Barcelona, Spain, 1295-1300.Google Scholar
  7. Bonissone, P. P., and Decker, K. S. (1986). “Selecting uncertainty calculi and granularity: An experiment in trading-off precision and complexity”, In Uncertainty in Artificial Intelligence, L. N. Kanal and J. F. Lemer, eds., North-Holland, Amsterdam, 217–247.Google Scholar
  8. Borel, E. (1950). Probabilite et certitude. Press universite de France, Paris.Google Scholar
  9. Bruner, J. S., Goodnow, J. J., and Austin, G. A. (1956). A Study of Thinking. Wiley, New York.Google Scholar
  10. Dubois, D., and Prade, H. (1983). “Unfair coins and necessary measures: towards a possibilistic interpretation of histograms”, Fuzzy sets and systems, 10:15–20.MathSciNetMATHCrossRefGoogle Scholar
  11. Dubois, D., and Prade, H. (1988). An approach to computerised processing of uncertainty. Plenum Press, New York.Google Scholar
  12. Gaines, B. R. (1977). “Foundations of Fuzzy Reasoning”, In Fuzzy Automata and Decision Processes, M. Gupta, G. Saridis, and B. R. Gaines, eds., Elsevier, North-Holland, 19–75.Google Scholar
  13. Gaines, B. R. (1978). “Fuzzy and Probability Uncertainty Logics”, Journal of Information and Control, 38:154–169.MathSciNetMATHCrossRefGoogle Scholar
  14. Hamacher, H. (1978). Uber logiishe Aggregation nicht-binair expliziter Entscheidungskriterien. Main, Frankfurt.Google Scholar
  15. Hirota, K. (1981). “Concepts of probabilistic sets”, Fuzzy sets and systems, 5:31–46.MathSciNetMATHCrossRefGoogle Scholar
  16. Klir, G. J., and Yuan, B. (1995). Fuzzy Sets and Fuzzy Logic, Theory and Applications. Prentice Hall, New Jersey.MATHGoogle Scholar
  17. Klir, G. J., and Yuan, B. (1998). “Operations on Fuzzy Sets”, In Handbook of Fuzzy Computation, E. H. Ruspini, P. P. Bonissone, and W. Pedrycz, eds., Institute of Physics Publishing Ltd., Bristol, UK, B2.2:1–15.Google Scholar
  18. Klir, K. (1990). “A principle of uncertainty and information invariance”, International journal of general systems, 17(2, 3):249–275.MATHCrossRefGoogle Scholar
  19. Mizumoto, M., and Tanaka, K. (1976). “Some properties of fuzzy sets of type 2”, Information and control, 48(1):30–48.MathSciNetCrossRefGoogle Scholar
  20. Negoita, C. V., and Ralescu, D. (1975). “Representation theorems for fuzzy concepts”, Kybernetics, 4(3):169–174.MATHCrossRefGoogle Scholar
  21. Ralescu, A. L., ed. (1995a). “Applied Research in Fuzzy Technology”, Kluwer Academic Publishers, New York.MATHGoogle Scholar
  22. Ralescu, D. (1995b). “Cardinality, quantifiers, and the aggregation of fuzzy criteria”, Fuzzy sets and systems, 69:355–365.MathSciNetMATHCrossRefGoogle Scholar
  23. Ruspini, E. H., Bonissone, P. P., and Pedrycz, W., eds. (1998). “Handbook of Fuzzy Computation”, Institute of Physics Publishing Ltd., Bristol, UK.MATHGoogle Scholar
  24. Russell, B. (1923). “Vagueness”, Australasian journal of psychology and philosophy, 1:84–92.CrossRefGoogle Scholar
  25. Schweizer, B., and Sklar, A. (1961). “Associative functions and statistical triangle inequalities”, Publ. Math. Debrecen, 8:169–186.MathSciNetMATHGoogle Scholar
  26. Schweizer, B., and Sklar, A. (1963). “Associative functions and abstract semigroups”, Publ. Math. Debrecen, 10:69–81.MathSciNetGoogle Scholar
  27. Shanahan, J. G. (1998). “Cartesian Granule Features: Knowledge Discovery of Additive Models for Classification and Prediction”, PhD Thesis, Dept. of Engineering Mathematics, University of Bristol, Bristol, UK.Google Scholar
  28. Sudkamp, T. (1992). “On probability-possibility transformation”, Fuzzy Sets and Systems, 51:73–81.MathSciNetMATHCrossRefGoogle Scholar
  29. Sugeno, M., and Yasukawa, T. (1993). “A Fuzzy Logic Based Approach to Qualitative Modelling”, IEEE Trans on Fuzzy Systems, 1(1): 7–31.CrossRefGoogle Scholar
  30. Terano, T., Asai, K., and Sugeno, M. (1992). Applied fuzzy systems. Academic Press, New York.Google Scholar
  31. Yager, R. (1988). “On ordered weighted averaging aggregation operators in multicriteria decision making”, IEEE Transactions on Systems Man and Cybernetics, 18:183–190.MathSciNetMATHCrossRefGoogle Scholar
  32. Yager, R. R. (1994). “Generation of Fuzzy Rules by Mountain Clustering”, J. Intelligent and Fuzzy Systems, 2:209–219.Google Scholar
  33. Yager, R. R. (1998). “Characterisations of fuzzy set properties”, In Handbook of Fuzzy Computation, E. H. Ruspini, P. P. Bonissone, and W. Pedrycz, eds., Institute of Physics Publishing Ltd., Bristol, UK, B2.5:1–8.Google Scholar
  34. Yen, J., and Langari, R. (1998). Fuzzy logic: intelligence, control and information. Prentice Hall, London.Google Scholar
  35. Zadeh, L. A. (1965). “Fuzzy Sets”, Journal of Information and Control, 8:338–353.MathSciNetMATHCrossRefGoogle Scholar
  36. Zadeh, L. A. (1973). “Outline of a New Approach to the Analysis of Complex Systems and Decision Process”, IEEE Trans. on Systems, Man and Cybernetics, 3(1):28–44.MathSciNetMATHCrossRefGoogle Scholar
  37. Zadeh, L. A. (1978). “Fuzzy Sets as a Basis for a Theory of Possibility”, Fuzzy Sets and Systems, 1:3–28.MathSciNetMATHCrossRefGoogle Scholar
  38. Zadeh, L. A. (1983). “A Computational Approach to Fuzzy Quantifiers in Natural Languages”, Computational Mathematics Applications, 9:149–184.MathSciNetMATHCrossRefGoogle Scholar
  39. Zimmermann, H. J. (1996). Fuzzy set theory and its applications. Kluwer Academic Publishers, Boston, USA.MATHGoogle Scholar
  40. Zimmermann, H. J., and Zysno, P. (1980). “Latent connectives in human decision making”, Fuzzy Sets and Systems, 4(1):37–51.MATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • James G. Shanahan
    • 1
  1. 1.Xerox Research Centre Europe (XRCE)Grenoble LaboratoryMeylanFrance

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