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Uncertainty Theory

An Introduction to its Axiomatic Foundations

  • Book
  • © 2004

Overview

Authors:
  1. Baoding Liu

    1. Uncertainty Theory Laboratory Dept. of Mathematical Science, Tsinghua University, Beijing, China

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  • Self contained, comprehensive and up-to-date presentation of uncertainty theory
  • Axiomatic approach to equip the reader to deal with uncertainty
  • Includes supplementary material: sn.pub/extras

Part of the book series: Studies in Fuzziness and Soft Computing (STUDFUZZ, volume 154)

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-xi
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  2. Measure and Integral

    • Baoding Liu
    Pages 1-20

  3. Probability Theory

    • Baoding Liu
    Pages 21-77

  4. Credibility Theory

    • Baoding Liu
    Pages 79-135

  5. Trust Theory

    • Baoding Liu
    Pages 137-190

  6. Fuzzy Random Theory

    • Baoding Liu
    Pages 191-213

  7. Random Fuzzy Theory

    • Baoding Liu
    Pages 215-244

  8. Bifuzzy Theory

    • Baoding Liu
    Pages 245-272

  9. Birandom Theory

    • Baoding Liu
    Pages 273-292

  10. Rough Random Theory

    • Baoding Liu
    Pages 293-310

  11. Rough Fuzzy Theory

    • Baoding Liu
    Pages 311-330

  12. Random Rough Theory

    • Baoding Liu
    Pages 331-348

  13. Fuzzy Rough Theory

    • Baoding Liu
    Pages 349-367

  14. Birough Theory

    • Baoding Liu
    Pages 369-386

  15. Some Remarks

    • Baoding Liu
    Pages 387-397

  16. Back Matter

    Pages 399-411
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Keywords

About this book

As a branch of mathematics that studies the behavior of random, fuzzy and rough events, uncertainty theory is the generic name of probability theory, credibility theory, and trust theory. The main purpose of this book is to provide axiomatic foundations of uncertainty theory. Itwasgenerallybelievedthatthestudyofprobabilitytheorywasstartedby Pascal and Fermat in 1654 when they succeeded in deriving the exact pro- bilities for certain gambling problem. Great progress was achieved when Von Mises initialized the concept of sample space, and ?lled the gape between probability theory and measure theory in 1931. A complete axiomatic fo- dation of probability theory was given by Kolmogoro? in 1933. Since then, probability theory has been developed steadily and has been widely applied in science and engineering. The axiomatic foundation of probability theory will be introduced in Chapter 2. Fuzzy set was initialized by Zadeh via membership function in 1965, and was well developed and applied in a wide variety of real problems. As a fuzzy set of real numbers, the term fuzzy variable was ?rst introduced by Kaufmann in 1975. In order to make a mathematical foundation, Nahmias gave three axioms to de?ne possibility measure in 1978, and Liu gave the fourth axiom to de?ne product possibility measure in 2002. There are three types of measure in the fuzzy world: possibility, necessity, and credibility.

Authors and Affiliations

  • Uncertainty Theory Laboratory Dept. of Mathematical Science, Tsinghua University, Beijing, China

    Baoding Liu

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