Advertisement

Three-Valued Random Fuzzy Sets

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 367)

Abstract

The theory of fuzzy sets was founded by Zadeh as an approach to cope with the pressing need to deal with phenomena which cannot be modelled properly by conventional mathematics because they contain factors which are fuzzy in nature. However, in this frame of the theory of fuzzy sets, we cannot discern two conflicting fuzzy conceptions properly, and therefore, the excluded middle law is violated. In this theory, one does not differ fuzzy sets from their membership functions. That is, a fuzzy set in the sense of Zadeh is equivalent to its membership function. This may be the main reason why the excluded middle law was violated. In this paper, we use the concept of three-valued fuzzy sets to provide a mathematical frame of fuzzy sets theory, such that it can eliminate the suspicion for the objective reality of membership functions of fuzzy sets. In our frame, fuzzy sets and their membership functions are not equivalent conceptions.

Keywords

Three-valued random fuzzy set Random set Fuzzy set Random variable 

Notes

Acknowledgments

We thank the referee for several suggestions that improved the article.

References

  1. 1.
    Zadeh, L.A.: Fuzzy sets. Inf. Control 8, 338–353 (1965)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Landles, R.H., Wolfe, D.A.: Introduction to the Theory of Nonparamelvic Statistics. Wiley, New York (1979)Google Scholar
  3. 3.
    Hirota, K.: Concepts of probabilistic sets. Fuzzy Set Syst. 5, 31–46 (1981)CrossRefMathSciNetMATHGoogle Scholar
  4. 4.
    Luo, C.Z.: Fuzzy sets and set-embeddings. Fuzzy Math. 4, (Chinese) (1983)Google Scholar
  5. 5.
    Wang, P.Z., Liu, X.H.: Set valued statistics. J. Eng. Math. (first issue), (Chinese)Google Scholar
  6. 6.
    Wang, P.Z.: Fuzzy Sets and Random Falling Shadows. Beijing Normal University, Beijing (Chinese) (1985)Google Scholar
  7. 7.
    Suzuki, H.: Fuzzy sets and membership functions. Fuzzy Sets Syst. 58, 123–132 (1993)CrossRefMATHGoogle Scholar
  8. 8.
    Li, Q., Wang, P., Lee, E.S.: r-Fuzzy Set. Comput. Math. Appl. 31(2), 49–61 (1996)Google Scholar
  9. 9.
    Yuan, X.H., Li, H.X., Sun, K.B.: The cut sets, decomposition theorems and representation theorems on Intuitionistic fuzzy Sets and interval-valued fuzzy sets. Sci. China Inf. Sci. 53, 1–18 (2010). doi: 10.1007/s11432-010-4078-6

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Faculty of Electronic Information and Electrical EngineeringDalian University of TechnologyDalianChina
  2. 2.School of ScienceDalian Ocean UniversityDalianChina

Personalised recommendations