Deriving fuzzy subsethood measures from violations of the implication between elements

  • Francisco Botana
Fuzzy Knowledge Representation and Inference
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1415)


The aim of this paper is to present a collection of new measures of subsethood between fuzzy sets. Starting from the relationship between crisp set containment and logical implication, some fuzzy approaches are reviewed. An excerpt of reasonable fuzzy implication operators is used to define fuzzy measures of inclusion using Kosko's fitviolation strategy. We test these measures on two axiomatics and derive, when possible, measures of fuzzy entropy. Once a subsethood measure between fuzzy sets is defined, other operations as set equality, similarity, disjointness, complement,... can be considered. The need for containment measures is present in wide areas as approximate reasoning and inference, image processing or learning.


Subsethood Measure Fuzzy Measure Logical Implication Fuzzy Reasoning Approximate Reasoning 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bandler, W., Kohout, L.: Fuzzy power sets and fuzzy implication operators. Fuzzy Sets and Systems 4 (1980) 13–30CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    De Luca, A., Termini, S.: A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory. Inform. Control 20 (1972) 301–312CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Dubois, D., Prade, H.: Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York (1980)zbMATHGoogle Scholar
  4. 4.
    Gaines, B.R.: Foundations of fuzzy reasoning. Int. J. Man-Mach. Stud. 8 (1976) 623–668CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Goguen, J. A.: The logic of inexact concepts. Synthese 19 (1969) 325–373CrossRefzbMATHGoogle Scholar
  6. 6.
    Halmos, P. R.: Naive Set Theory. Van Nostrand, Princeton (1960)zbMATHGoogle Scholar
  7. 7.
    Kleene, S. C.: On a notation for ordinal numbers. J. Symb. Logic 3 (1938) 150–155CrossRefGoogle Scholar
  8. 8.
    Kosko, B.: Neural networks and fuzzy systems. Prentice Hall, Englewood Cliffs (1992)zbMATHGoogle Scholar
  9. 9.
    Mamdani, E. H.: Application of fuzzy logic to approximate reasoning using linguistic systems. IEEE Trans. Comput. 26 (1977) 1182–1191CrossRefzbMATHGoogle Scholar
  10. 10.
    Mizurnoto, M., Zimmermann, H. J.: Comparison of fuzzy reasoning methods. Fuzzy Sets and Systems 8 (1982) 253–283CrossRefMathSciNetGoogle Scholar
  11. 11.
    Sánchez, E.: Inverses of fuzzy relations. Application to possibility distributions and medical diagnosis. Fuzzy Sets and Systems 2 (1979) 75–86CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Sinha, D., Dougherty, E. R.: Fuzzification of set inclusion: Theory and applications. Fuzzy Sets and Systems 55 (1993) 15–42CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Smets, Ph., Magrez, P.: Implication in fuzzy logic. Int. J. Approx. Reasoning 1 (1987) 327–347CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Willmott, R.: Two fuzzier implication operators in the theory of fuzzy power sets. Fuzzy Sets and Systems 4 (1980) 31–36CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Wu, W.: Fuzzy reasoning and fuzzy relational equations. Fuzzy Sets and Systems 20 (1986) 67–78CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Yager, R. R.: An approach to inference in approximate reasoning. Int. J. ManMach. Stud. 13 (1980) 323–338CrossRefMathSciNetGoogle Scholar
  17. 17.
    Young, V. R.: Fuzzy subsethood. Fuzzy Sets and Systems 77 (1996) 371–384CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Zadeh, L.A.: Fuzzy sets. Inform. Control 8 (1965) 338–353CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1998

Authors and Affiliations

  • Francisco Botana
    • 1
  1. 1.Departamento de Matemática AplicadaUniversidad de Vigo, Campus A XunqueiraPontevedraSpain

Personalised recommendations