Skip to main content
SpringerLink
Log in

Fuzzy Set-Theoretic Operators and Quantifiers

  • Chapter
Fundamentals of Fuzzy Sets

Part of the book series: The Handbooks of Fuzzy Sets Series ((FSHS,volume 7))

Abstract

This chapter summarizes main ways to extend classical set-theoretic operations (complementation, intersection, union, set-difference) and related concepts (inclusion, quantifiers) for fuzzy sets. Since these extensions are mainly pointwisely defined, we review basic results on the underlying unary or binary operations on the unit interval such as negations, t-norms, t-conorms, implications, coimplications and equivalences. Some strongly related connectives (means, OWA, weighted, and prioritized operations) are also considered, emphasizing the essential differences between these and the formerly investigated operator classes. We also show other operations which have no counterpart in the classical theory but play some important role in fuzzy sets (like symmetric sums, weak t-norms and conorms, compensatory AND).

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 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Abel, N. H. (1826). Untersuchung der fuctionen zweier unabhängig veränderlichen grössen x und y wie f(x,y), welche die eigenschaft haben, dass f(z, f(x, y)) eine symmetrische function von x, y und z ist. J, Reine Angew. Math., 1:11–15. also as Oeuvres Complètes de N.H. Abel, Vol. 1, Christiana, 1881, 61-65.

    MATH  Google Scholar 

  • Aczél, J. (1948). Über eine klasse von funktionalgleichungen. Comment, Math. Helv., 54:247–256.

    Google Scholar 

  • Aczél, J. (1949). Sur les opérations définies pour des nombres réels. Bull. Soc. Math. de France, 76:59–64.

    Google Scholar 

  • Aczél, J. (1966). Lectures on Functional Equations and Applications. Academic Press, New York.

    MATH  Google Scholar 

  • Alsina, C., Trillas, E., and Valverde, L. (1983). On some logical connectives for fuzzy sets theory. J. Math. Anal Appl., 93:15–26.

    MathSciNet  MATH  Google Scholar 

  • Handler, W. and Kohout, L. (1980). Fuzzy power sets and fuzzy implication operators. Fuzzy Sets and Systems, 4:13–30.

    MathSciNet  Google Scholar 

  • Bellman, R. and Giertz, M. (1973). On the analytic formalism of the theory of fuzzy sets. Inform. Sci., 5:149–156.

    MathSciNet  MATH  Google Scholar 

  • Bellman, R. and Zadeh, L. (1977). Local and fuzzy logic. In Dunn, J. and Epstein, G., editors, Modern Uses of Multiple-valued Logic, pages 103–156. D. Reidel, Dordrecht.

    Google Scholar 

  • Birkhoff, G. (1967). Lattice Theory. American Mathematical Society, Providence, Rhode Island, third edition.

    MATH  Google Scholar 

  • Blom, W. B. A. (1979). Goals and Decisions in a Fuzzy Environment (in Dutch). Department of Electrical Engineering, Delft University of Technology.

    Google Scholar 

  • Czogala, E. and Drewniak, J. (1984). Associative monotonic operations in fuzzy set theory, Fuzzy Sets and Systems, 12:249–269.

    MathSciNet  MATH  Google Scholar 

  • De Baets, B. (1998). Idempotent uninorms. European Journal of Operational Research. (to appear).

    Google Scholar 

  • De Baets, B. and Fodor, J. (1997a). On the structure of uninorms and their residual iinplicators. In Gottwald, S. and Klement, E., editors, 18th Linz Seminar on Fuzzy Set Theory: Enriched Lattice Structures for Many-Valued and Fuzzy Logics, pages 81–87. (February 25-March 1, 1997), Johannes Kepler Universität, Linz, Austria.

    Google Scholar 

  • De Baets, B. and Fodor, J. (1997b). Residual operators of representable uni-norms. In Zimmermann, H.-J., editor, Proceedings of the 5th European Congress on Intelligent Techniques and Soft Computing, pages 52–56. (September 8–12, 1997, Aachen, Germany), Verlag Mainz, Wissenschaftsverlag, Aachen.

    Google Scholar 

  • De Baets, B. and Fodor, J. (1998). Residual operators of uninorms. Soft Computing, (to appear).

    Google Scholar 

  • Dombi, J. (1982a). Basic concepts for a theory of evaluation: the aggregative operator. European Journal of Operations Research, 10:282–293.

    MathSciNet  MATH  Google Scholar 

  • Dombi, J. (1982b). A general class of fuzzy operators, the De Morgan class of fuzzy operators and fuzziness measures induced by fuzzy operators. Fuzzy Sets and Systems, 8:149–163.

    MathSciNet  MATH  Google Scholar 

  • Dombi, J. and Zysno, P. (1982). Comments on the γ-model. In Trappl, R., editor, Cybernetics and Systems Research, pages 711–714. North-Holland, Amsterdam.

    Google Scholar 

  • Dubois, D. and Prade, H. (1980a). Fuzzy Sets and Systems: Theory and Applications. Academic Press, New York.

    MATH  Google Scholar 

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

    Google Scholar 

  • Dubois, D. and Prade, H. (1984a). Fuzzy logics and the generalized modus ponens revisited. Int. J. Cybernetics and Systems, 15:293–331.

    MathSciNet  MATH  Google Scholar 

  • Dubois, D. and Prade, H. (1984b). Fuzzy set-theoretic differences and inclusions and their use in the analysis of fuzzy equations. Control and Cybernetics, 13:129–145.

    MathSciNet  MATH  Google Scholar 

  • Dubois, D. and Prade, H. (1984c). A theorem on implication functions defined from triangular norms. Stochastica, VIII:267–279.

    MathSciNet  Google Scholar 

  • Dubois, D. and Prade, H. (1985). A review of fuzzy set aggregation connectives. Inform. Sci., 36:85–121.

    MathSciNet  MATH  Google Scholar 

  • Dubois, D. and Prade, H. (1991). Fuzzy sets in approximate reasoning, part 1: Inference with possibility distributions. Fuzzy Sets and Systems, 40:143–202.

    MathSciNet  MATH  Google Scholar 

  • Dyckhoff, H. and Pedrycz, W. (1984). Generalized means as model of compensative connectives. Fuzzy Sets and Systems, 14:143–154.

    MathSciNet  MATH  Google Scholar 

  • Esteva, F. and Domingo, X. (1980). Sobre funciones de negation en [0,1]. Stochastica, IV: 141–166.

    MathSciNet  Google Scholar 

  • Faucett, W. M. (1955). Compact semigroups irreducibly connected between two idempotents. Proc. Amer. Math. Soc., 6:741–747.

    MathSciNet  MATH  Google Scholar 

  • Filev, D. P. and Yager, R. R. (1994). Learning OWA operator weights from data. In Proceedings of the Third IEEE International Conference on Fuzzy Systems, pages 468–473, Orlando.

    Google Scholar 

  • Filev, D. P. and Yager, R. R. (1997). On the issue of obtaining OWA operator weights. Fuzzy Sets and Systems. (to appear).

    Google Scholar 

  • Fodor, J. (1991a). On fuzzy implication operators. Fuzzy Sets and Systems, 42:293–300.

    MathSciNet  MATH  Google Scholar 

  • Fodor, J. (1991b). Strict preference relations based on weak t-norrns. Fuzzy Sets and Systems, 43:327–336.

    MathSciNet  MATH  Google Scholar 

  • Fodor, J. (1993). A new look at fuzzy connectives. Fuzzy Sets and Systems, 57:141–148.

    MathSciNet  MATH  Google Scholar 

  • Fodor, J. (1995). Contrapositive symmetry of fuzzy implications. Fuzzy Sets and Systems, 69:141–156.

    MathSciNet  MATH  Google Scholar 

  • Fodor, J. (1996). An extension of Fung-Fu’s theorem. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems, 4:235–243.

    MathSciNet  MATH  Google Scholar 

  • Fodor, J. and Keresztfalvi, T. (1994). Characterization of the Hamacher family of t-norms. Fuzzy Sets and Systems, 65:51–58.

    MathSciNet  MATH  Google Scholar 

  • Fodor, J., Marichal, J.-L., and Roubens, M. (1995). Characterization of the ordered weighted averaging operators. IEEE Transactions on Fuzzy Systems, 3:236–240.

    Google Scholar 

  • Fodor, J. and Roubens, M. (1994). Fuzzy Preference Modelling and Multicriteria Decision Support Theory and Decision Library, Series D, Volume 14. Kluwer Academic Publishers, Dordrecht.

    MATH  Google Scholar 

  • Fodor, J., and Roubens, M. (1995). On meaningfulness of means, J. Comput. Appl. Math., 64:103–115.

    MathSciNet  MATH  Google Scholar 

  • Fodor, J., Yager, R., and Rybalov, A. (1997). Structure of uninorms. Int. J. of Uncertainty, Fuzziness and Knowledge-Based Systems, 5(4):411–427.

    MathSciNet  MATH  Google Scholar 

  • Frank, M. J. (1979). On the simultaneous associativity of f(x,y) and x + y − f(x,y), Aeq. Math., 19:194–226.

    MATH  Google Scholar 

  • Fung, L. W. and Fu, K. S. (1975). An axiomatic approach to rational decision-making in a fuzzy environment. In L.A. Zadeh, K.S. Fu, K. T. and Shimura, M., editors, Fuzzy Sets and their Applications to Cognitive and Decision Processes, pages 227–256. Academic Press, New York, San Francisco, London.

    Google Scholar 

  • Gaines, B. (1976). Foundations of fuzzy reasoning. Int. J. Man-Machine Stud., 8:623–668.

    MathSciNet  MATH  Google Scholar 

  • Giles, R. (1976). Lukasiewicz’s logic and fuzzy set theory. Int. J. Man-Machine Stud., 8:313–327.

    MathSciNet  MATH  Google Scholar 

  • Goguen, J. A. (1969). The logic of inexact concepts. Synthese, 19:325–373.

    MATH  Google Scholar 

  • Hamacher, H. (1978). Über logische Aggrationen nichtbinär explizierter Entscheidungskriterien; Ein axiomatischer Beitrag zur normativen Entscheidungstheorie. Rita G. Fischer Verlag, Frankfurt.

    Google Scholar 

  • Hintikka, R. (1978). Quantifiers in logic and quantifiers in natural languages. In Korner, S., editor, Philosophy of Logic. Basil Blackwell, Oxford.

    Google Scholar 

  • Höhle, U. (1988). Quotients with respect to similarity relations. Fuzzy Sets and Systems, 27:31–44.

    MathSciNet  MATH  Google Scholar 

  • Höhle, U. (1992). M-valued sets and sheaves over integral commutative CL-monoids. In Rodabaugh, S., Klement, E., and Höhle, U., editors, Applications of Category Theory to Fuzzy Subsets, pages 33–72. Kluwer Academic Publishers, Dordrecht.

    Google Scholar 

  • Höhle, U. and Stout, L. (1991). Foundations of fuzzy sets. Fuzzy Sets and Systems, 40:257–296.

    MathSciNet  MATH  Google Scholar 

  • Jenei, S. (1996). On discontinuous triangular norms. BUSEFAL, 66:38–42.

    Google Scholar 

  • Jenei, S. (1997a). Characterization and limit theorems on triangular norms. PhD thesis, Eötvös Loránd University, Budapest.

    Google Scholar 

  • Jenei, S. (1997b). New family of girard monoids on intervals via annihilation. BUSEFAL, 69:94–101.

    Google Scholar 

  • Jenei, S. (1998a). Fibred triangular norms. Fuzzy Sets and Systems. (to appear).

    Google Scholar 

  • Jenei, S. (1998b). New family of triangular norms via contrapositive sym-metrization of residuated implications. Fuzzy Sets and Systems. (to appear).

    Google Scholar 

  • Jenei, S. (1998c). On archimedean triangular norms. Fuzzy Sets and Systems. (to appear).

    Google Scholar 

  • Jenei, S. and Fodor, J. (1998). On continuous triangular norms. Fuzzy Sets and Systems. (to appear).

    Google Scholar 

  • Kaymak, U. and Van Nauta Lemke, H. R. (1993). A parametric generalized goal function for fuzzy decision making with unequally weighted objectives. In Proceedings of the Second IEEE International Conference on Fuzzy Systems, pages 1156–1160, San Francisco.

    Google Scholar 

  • Kitainik, L. (1986). Axiomatics and properties of fuzzy inclusions. Scientific Works of the Institute for System Studies, 10:97–107. (in Russian).

    Google Scholar 

  • Kitainik, L. (1987a). Comparative study of fuzzy inclusions. Scientific Works of the Institute for System Studies, 14:83–92. (in Russian).

    Google Scholar 

  • Kitainik, L. (1987b). Fuzzy inclusions and fuzzy dichotomous decision procedures. In Kacprzyk, J. and Oriovski, S., editors, Optimization Models using Fuzzy Sets and Possibility Theory, pages 154–170. D. Reidel,, Dordrecht.

    Google Scholar 

  • Kitainik, L. (1993). Fuzzy Decision Procedures with Binary Relations. Kluwer Academic Publishers, Dordrecht.

    MATH  Google Scholar 

  • Klawonn, F. and Kruse, R. (1993). Equality relations as a basis for fuzzy control Fuzzy Sets and Systems, 54:147–156.

    MathSciNet  MATH  Google Scholar 

  • Klement, E., Mesiar, R., and Pap, E. (1996). On the relationship of associative compensatory operators to triangular norms and conorms. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 4:129–144.

    MathSciNet  MATH  Google Scholar 

  • Klement, E., Mesiar, R., and Pap, E. (1997). A characterization of the ordering of continuous t-norms. Fuzzy Sets and Systems, 86:189–195.

    MathSciNet  MATH  Google Scholar 

  • Klement, E., Mesiar, R., and Pap, E. (1998). Triangular norms. (to appear).

    Google Scholar 

  • Klement, E. P. (1981a). An axiomatic theory of operations on fuzzy sets. Technical Report Institutsbericht 18, Institut für Mathematik, Johannes Kepler Universität Linz.

    Google Scholar 

  • Klement, E. P. (1981b). Operations on fuzzy sets arid fuzzy numbers related to triangular norms. In Proceedings of the Eleventh International Symposium on Multiple-valued Logic, pages 218–225.

    Google Scholar 

  • Klement, E. P. (1982). Operations on fuzzy sets-an axiomatix approach. Inf. Sci, 27:221–232.

    MathSciNet  MATH  Google Scholar 

  • Lemke, H. R. V. N., Dijkman, T. G., Haeringen, H. V., and Pleeging, M. (1983). A characteristic optimism factor in fuzzy decision making. In Proc. IFAC Symposium on Fuzzy Information, Knowledge Representation and Decision Analysis, pages 283–288, Marseille, France.

    Google Scholar 

  • Ling, C. H. (1965). Representation of associative functions. Publ. Math. Debrecen, 12:189–212.

    MathSciNet  Google Scholar 

  • Lowen, R. (1978). On fuzzy complements. Inform. Sci., 14:107–113.

    MathSciNet  MATH  Google Scholar 

  • Lukasiewicz, J. (1970). Selected Works. Reidel, Amsterdam.

    MATH  Google Scholar 

  • Menger, K. (1942). Statistical metric spaces. Proc. Nat Acad. Sci. USA, 28:535–537.

    MathSciNet  MATH  Google Scholar 

  • Miyakoshi, M. and Shimbo, M. (1985). Solutions of composite fuzzy relational operations with triangular norms. Fuzzy Sets and Systems, 16:53–63.

    MathSciNet  MATH  Google Scholar 

  • Mostert, P. S. and Shields, A. L. (1957). On the structure of semigroups on a compact manifold with boundary. Ann. of Math., 65:117–143.

    MathSciNet  MATH  Google Scholar 

  • O’Hagan, M. (1990). Using maximum entropy-ordered weighted averaging to construct a fuzzy neuron. In Proceedings 24th Annual IEEE Asilomar Conf. on Signals, Systems and Computers, pages 618–623, Pacific Grove, Ca.

    Google Scholar 

  • Ostasiewicz, W. (1988). Formalization of quantifier phrases. In Gupta, M. and Yamakawa, T., editors, Fuzzy Logic in Knowledge-Based Systems, Decision and Control, pages 271–286. North-Holland, Amsterdam.

    Google Scholar 

  • Ovchinnikov, S. (1981). Representations of synonymy and antonymy by automorphisms in fuzzy set theory. Stochastica, V:95–107.

    MathSciNet  Google Scholar 

  • Ovchinnikov, S. (1983). General negations in fuzzy set theory. J. Math. Anal. Appl., 92:234–239.

    MathSciNet  MATH  Google Scholar 

  • Ovchinnikov, S. and Roubens, M. (1991). On strict preference relations. Fuzzy Sets and Systems, 43:319–326.

    MathSciNet  MATH  Google Scholar 

  • Ovchinnikov, S. and Roubens, M. (1992). On fuzzy strict preference, indifference and incomparability relations. Fuzzy Sets and Systems, 47:313–318.

    MathSciNet  MATH  Google Scholar 

  • Paalman-De Miranda, A. B. (1964). Topological semigroups. Technical report, Mathematisch Centrum, Amsterdam.

    Google Scholar 

  • Perny, P. (1992). Modélisation, agrégation et exploitation des préférences floues dans une problématique de rangement PhD thesis, Université Paris-Dau-phine.

    Google Scholar 

  • Prade, H. (1982). Modèles mathématiques de l’imprécis et de lincertain en vue d’applications au raisonnement naturel PhD thesis, Université P. Sabatier, Toulouse.

    Google Scholar 

  • Schweizer, B. and Sklar, A. (1961). Associative functions and statistical triangle inequalities. Publ. Math. Debrecen, 8:169–186.

    MathSciNet  MATH  Google Scholar 

  • Schweizer, B. and Sklar, A. (1963). Associative functions and abstract semigroups. Publ. Math. Debrecen, 10:69–81.

    MathSciNet  Google Scholar 

  • Schweizer, B. and Sklar, A. (1983). Probabilistic Metric Spaces,. North-Holland, Amsterdam.

    Google Scholar 

  • Sikorski, R. (1964). Boolean Algebras. Springer, Berlin.

    MATH  Google Scholar 

  • Silvert, W. (1979). Symmetric summation: a class of operations on fuzzy sets. IEEE Transactions on Systems, Man and Cybernetics, 9:659–667.

    MathSciNet  Google Scholar 

  • Sinha, D. and Dougherty, E. (1993). Fuzzification of set inclusion: theory and applications. Fuzzy Sets and Systems, 55:15–42.

    MathSciNet  MATH  Google Scholar 

  • Smets, P. and Magrez, P. (1987). Implication in fuzzy logic. Int. J. of Approximate Reasoning, 1:327–347.

    MathSciNet  MATH  Google Scholar 

  • Sugeno, M. (1974). Theory of fuzzy integrals and its applications. PhD thesis, Tokyo Institute of Technology, Tokyo.

    Google Scholar 

  • Sugeno, M. (1977). Fuzzy measures and fuzzy initegrals: a survey. In M.M. Gupta, G. S. and Gaines, B., editors, Fuzzy Automata and Decision Processes, pages 89–102. North-Holland, Amsterdam.

    Google Scholar 

  • Trillas, E. (1979). Sobre funciones de negación en la teoría de conjuntos difusos. Stochastica, III:47–60.

    MathSciNet  Google Scholar 

  • Trillas, E. and Riera, T. (1981). Towards a representation of “synonyms” and “antonyms” by fuzzy sets. BUSEFAL, 5:42–68.

    Google Scholar 

  • Trillas, E. and Valverde, L. (1981). On some functionally expressable implications for fuzzy set theory. In Proc. of the 3rd Intern. Seminar on Fuzzy Set Theory, pages 173–190. Johannes Kepler Universität

    Google Scholar 

  • Trillas, E. and Valverde, L. (1985). On implication and indistinguishability in the setting of fuzzy logic. In Kacprzyk, J. and Yager, R., editors, Management Decision Support Systems using Fuzzy Sets and Possibility Theory, pages 198–212. Verlag TÜV Rheinland, Köln.

    Google Scholar 

  • Turksen, L. (1986). Interval valued fuzzy sets based on normal forms. Fuzzy Sets and Systems, 20:191–210.

    MathSciNet  MATH  Google Scholar 

  • Turksen, I. (1992). Interval-valued fuzzy sets and ‘compensatory AND’. Fuzzy Sets and Systems, 51:295–307.

    MathSciNet  Google Scholar 

  • Vilenkin, N. and Shreider, Y. (1977). Majority spaces and quantifier “majority”, Semiotika i Informatika, 8:45–82. (in Russian).

    MATH  Google Scholar 

  • Wangming, W. (1992). Representation of Archimedean implications, (unpublished).

    Google Scholar 

  • Weber, S. (1983). A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms. Fuzzy Sets and Systems, 11:115–134.

    MathSciNet  MATH  Google Scholar 

  • Yager, R. (1979). On the measure of fuzziness and negation, part I: membership in the unit interval. Internat. J. General Systems, 5:221–229.

    MathSciNet  MATH  Google Scholar 

  • Yager, R. (1980a). An approach to inference in approximate reasoning. Int. J. Man-Machine Stud., 13:323–338.

    MathSciNet  Google Scholar 

  • Yager, R. (1980b). On the measure of fuzziness and negation IL lattices. Information and Control, 44:236–260.

    MathSciNet  MATH  Google Scholar 

  • Yager, R. (1998). On ordered weighted averaging aggregation operators in multi-criteria decision making. IEEE Transactions on Systems, Man and Cybernetics, 18:183–190.

    MathSciNet  Google Scholar 

  • Yager, R. (1991a). Connectives and quantifiers in fuzzy sets. Fuzzy Sets and Systems, 40:39–75.

    MathSciNet  MATH  Google Scholar 

  • Yager, R. (1993). MAM and MOM operators for aggregation. Information Sciences, 69:259–273.

    MATH  Google Scholar 

  • Yager, R. (1995). A unified approach to aggregation based upon MOM and MAM operators. International Journal of Intelligent Systems, 10:809–855.

    Google Scholar 

  • Yager, R. and Rybalov, A. (1996a). Uninorm aggregation operators. Fuzzy Sets and Systems, 80:111–120.

    MathSciNet  MATH  Google Scholar 

  • Yager, R. R. (1977). Multiple objective decision making using fuzzy sets. International J. of Man-Machine Studies, 9:375–382.

    MATH  Google Scholar 

  • Yager, R. R. (1987a). A note on weighted queries in information retrieval systems. J. of the American Society of Information Sciences, 38:23–24.

    Google Scholar 

  • Yager, R. R. (1987b). Using approximate reasoning to represent default knowledge. Artificial Intelligence, 31:99–112.

    MathSciNet  MATH  Google Scholar 

  • Yager, R. R. (1991b), Nonmonotonic set theoretic operations, Fuzzy Sets and Systems, 42:173–190.

    MathSciNet  MATH  Google Scholar 

  • Yager, R. R. (1994). Misrepresentations and challenges: A response to Elkan. IEEE Expert, August:41–42.

    Google Scholar 

  • Yager, R. R. (1997a). On a class of weak triangular norm operators. Information Sciences, 96:47–78.

    MathSciNet  MATH  Google Scholar 

  • Yager, R. R. (1997b). Structures for prioritized fusion of fuzzy information. Technical Report Mil-1719, Machine Intelligence Institute, lona College, New Rochelle, NY.

    Google Scholar 

  • Yager, R. R. and Filev, D. P. (1994a). Essentials of Fuzzy Modeling and Control John Wiley, New York.

    Google Scholar 

  • Yager, R. R. and Filev, D. P. (1994b). Parameterized “andlike“ and “orlike“ OWA operators. International Journal of General Systems, 22:297–316.

    Google Scholar 

  • Yager, R. R. and Rybalov, A. (1996b). Uni-norms: a unification of t-norms and t-conorms. In Proceedings of Nafips Conference, pages 50–54, Berkeley.

    Google Scholar 

  • Zadeh, L. (1965). Fuzzy sets. Information and Control, 8:338–353.

    MathSciNet  MATH  Google Scholar 

  • Zadeh, L. (1983). A computational approach to fuzzy quantifiers in natural languages. Computing and Mathematics with Applications, 9:149–184.

    MathSciNet  MATH  Google Scholar 

  • Zimmermann, H.-J. and Zysno, P. (1980). Latent connectives in human decision making. Fuzzy Sets and Systems, 4:37–51.

    MATH  Google Scholar 

Download references

Authors
  1. János Fodor
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ronald R. Yager
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut de Recherche en Informatique de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse Cedex 4, France

    Didier Dubois  & Henri Prade  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Science+Business Media New York

About this chapter

Cite this chapter

Fodor, J., Yager, R.R. (2000). Fuzzy Set-Theoretic Operators and Quantifiers. In: Dubois, D., Prade, H. (eds) Fundamentals of Fuzzy Sets. The Handbooks of Fuzzy Sets Series, vol 7. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-4429-6_3

Download citation

  • DOI: https://doi.org/10.1007/978-1-4615-4429-6_3

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-6994-3

  • Online ISBN: 978-1-4615-4429-6

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation