Sufficient Conditions of Cut Sets on Intuitionistic Fuzzy Sets

  • Yiying Shi
  • Xuehai Yuan
  • Yuchun Zhang
  • Yuhong Zhang
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 367)

Abstract

In this paper, a monomorphism is established from the three-valued fuzzy sets to intuitionistic fuzzy sets, and with this monomorphism, the three-valued fuzzy sets can be embedded in intuitionistic fuzzy sets. Then the general definition of the cut sets on the intuitionistic fuzzy sets is proposed. Finally, the axiomatic descriptions for different cut sets are presented and three most intrinsic properties for each cut sets are listed.

Keywords

Intuitionistic fuzzy sets Cut sets Sufficient condition Monomorphism 

Notes

Acknowledgments

This work was supported by the National Science Foundation of China (61473327).

References

  1. 1.
    Zadeh, L.A.: Fuzzy Sets. Inform. Cont. 8, 338–353 (1965)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Atanassov, K.: Intuitionistic Fuzzy Sets. Fuzzy Sets Syst. 20, 87–96 (1986)CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Atanassov, K.: Intuitionistic Fuzzy Sets: Theory and Applications. Physica-Verlag, Heidelberg (1999)Google Scholar
  4. 4.
    Lai, Y.J., Hwang, C.L.: Fuzzy Mathematical Programming-Methods and Applications, pp. 1–156. Springer, Berlin (1992)Google Scholar
  5. 5.
    Xu, Z.S.: Uncertain Multiple Attribute Decision Making: Methods and Applications (in Chinese), pp. 3–236. Tsinghua University Press, Beijing (2004)Google Scholar
  6. 6.
    Mordeson, J.N., Malik, D.S.: Fuzzy Commutative Algebra, pp. 1–258. World Scientific Publishing, Singapore (1998)CrossRefMATHGoogle Scholar
  7. 7.
    Mordeson, J.N., Bhutani, K.R., Rosenfeld, A.: Fuzzy Group Theory, pp. 1–292. Springer, New York (2005)MATHGoogle Scholar
  8. 8.
    Dubois, D., Hüllermeier, E.: On the Representation of Fuzzy Rules in Terms of Crisp Rules. Inform. Sci. 151, 301–326 (2003)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Luo, C.Z., Wang, P.Z.: Representation of Compositional Relations in Fuzzy Reasoning. Fuzzy Sets Syst. 36, 77–81 (1990)CrossRefMATHGoogle Scholar
  10. 10.
    Wang, G.J.: L-fuzzy Topology Space Theory (in Chinese), pp. 1–406. Shanxi Normal University Press, Xi’an (1988)Google Scholar
  11. 11.
    Liu, Y.M., Luo, M.K.: Fuzzy Topology, pp. 1–306. World Scientific Publishing, Singapore (1990)Google Scholar
  12. 12.
    Zhang, G.Q.: Fuzzy Measure Theory (in Chinese), pp. 1–265. Guizhou Science and Technology Press, Guiyang (1994)Google Scholar
  13. 13.
    Wu, C.X., Ma, M.: The Basis of Fuzzy Analysis (in Chinese), pp. 1–147. National Defence Industry Press, Beijing (1991)Google Scholar
  14. 14.
    Bertoluzza, C., Solci, M., Capodieci, M.L.: Measure of a Fuzzy Set: the \(\alpha \)-cut Approach in the Finite Case. Fuzzy Sets Syst. 123, 93–102 (2001)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Garcia, J.N., Kutalik, Z., Cho, K.H.: Level Sets and Minimum Volume Sets of Probability Density Functions. Int. J. Appr. Reason 34, 25–47 (2003)CrossRefMATHGoogle Scholar
  16. 16.
    Pap, E., Surla, D.: Lebesgue Measure of \(\alpha \)-cuts Approach for Finding the Height of the Membership Function. Fuzzy Sets Syst. 111, 341–350 (2000)CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Wang, G.J.: Non-classical Logic and Approximate Reasoning (in Chinese), pp. 24–185. Science Press, Beijing (2000)Google Scholar
  18. 18.
    Luo, C.Z.: Introduction to Fuzzy Sets (1) (in Chinese), pp. 1–486. Beijing Normal University Press, Beijing (1989)Google Scholar
  19. 19.
    Li, M.: Cut Sets of Intuitionistic Fuzzy Sets (in Chinese). J Liaoning Norm Univ. (Nat. Sci. Ed.) 30, 152–154 (2007)Google Scholar
  20. 20.
    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. 54, 91–11 (2011)CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Yuan, X.H., Li, H.X., Lee, E.S.: Three New Cut Sets of Fuzzy Sets and New Theories of Fuzzy Sets. Comput. Math. Appl. 57, 691–701 (2009)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Yiying Shi
    • 1
    • 2
  • Xuehai Yuan
    • 1
    • 3
  • Yuchun Zhang
    • 2
  • Yuhong Zhang
    • 4
  1. 1.School of Control Science and EngineeringDalian University of TechnologyDalianPeople’s Republic of China
  2. 2.School of ScienceShenyang Ligong UniversityShenyangPeople’s Republic of China
  3. 3.School of ScienceDalian University of Technology at PanjinPanjinPeople’s Republic of China
  4. 4.Department of Basic Education, City InstituteDalian University of TechnologyDalianPeople’s Republic of China

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