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Membership functions and operational law of uncertain sets

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Abstract

Uncertain set is a set-valued function on an uncertainty space, and attempts to model “unsharp concepts” that are essentially sets but their boundaries are not sharply described. This paper will propose a concept of membership function and define the independence of uncertain sets. This paper will also present an operational law of uncertain sets via membership functions or inverse membership functions. Finally, the linearity of expected value operator is verified.

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References

  • Gao X., Gao Y., Ralescu D. A. (2010) On Liu’s inference rule for uncertain systems. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 18(1): 1–11

    Article  MathSciNet  MATH  Google Scholar 

  • Gao, Y. (2012). Uncertain inference control for balancing inverted pendulum. Fuzzy Optimization and Decision Making (To be published).

  • Kolmogorov A. N. (1933) Grundbegriffe der Wahrscheinlichkeitsrechnung. Julius Springer, Berlin

    Google Scholar 

  • Liu B. (2007) Uncertainty theory, 2nd edn. Springer, Berlin

    MATH  Google Scholar 

  • Liu B. (2009) Some research problems in uncertainty theory. Journal of Uncertain Systems 3(1): 3–10

    Google Scholar 

  • Liu B. (2010a) Uncertain set theory and uncertain inference rule with application to uncertain control. Journal of Uncertain Systems 4(2): 83–98

    Google Scholar 

  • Liu B. (2010b) Uncertainty theory: A branch of mathematics for modeling human uncertainty. Springer, Berlin

    Google Scholar 

  • Liu B. (2011) Uncertain logic for modeling human language. Journal of Uncertain Systems 5(1): 3–20

    Google Scholar 

  • Liu B. (2012) Why is there a need for uncertainty theory?. Journal of Uncertain Systems 6(1): 3–10

    Google Scholar 

  • Matheron G. (1975) Random sets and integral geometry. Wiley, New York

    MATH  Google Scholar 

  • Peng Z. X., Iwamura K. (2010) A sufficient and necessary condition of uncertainty distribution. Journal of Interdisciplinary Mathematics 13(3): 277–285

    MathSciNet  MATH  Google Scholar 

  • Peng, Z. X., & Chen, X. W. (2012) Uncertain systems are universal approximators. http://orsc.edu.cn/online/100110.pdf.

  • Robbins H. E. (1944) On the measure of a random set. Annals of Mathematical Statistics 15(1): 70–74

    Article  MathSciNet  MATH  Google Scholar 

  • Tversky A., Kahneman D. (1986) Rational choice and the framing of decisions. Journal of Business 59: 251–278

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh L. A. (1978) Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1: 3–28

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Baoding Liu.

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Liu, B. Membership functions and operational law of uncertain sets. Fuzzy Optim Decis Making 11, 387–410 (2012). https://doi.org/10.1007/s10700-012-9128-7

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