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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.
