By Lotfi A. Zadeh, King-Sun Fu, Kokichi Tanaka, Masamichi Shimura

**Read Online or Download Fuzzy Sets and their Applications to Cognitive and Decision Processes. Proceedings of the US–Japan Seminar on Fuzzy Sets and their Applications, Held at the University of California, Berkeley, California, July 1–4, 1974 PDF**

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**Extra info for Fuzzy Sets and their Applications to Cognitive and Decision Processes. Proceedings of the US–Japan Seminar on Fuzzy Sets and their Applications, Held at the University of California, Berkeley, California, July 1–4, 1974**

**Sample text**

The points in U at which y p (u) > 0 constitute the support of F. 5 are the crossover points of F. Example A9. 9b + d (A12) and as nonfuzzy and fuzzy subsets of U, respectively. 9 + 1/1 If U β [0,1], then F might be expressed as 28 (A14) FUZZY SETS AND THEIR APPLICATIONS 1 _1 / 0 1 + u2 /u (A15) which means that F îs a fuzzy subset of the unit interval [0,1] whose membership function is defined by U F (u) 1 (A16) 1 + u' In many cases, it is convenient to express the membership function of a fuzzy subset of the real line in terms of a standard function whose parameters may be adjusted to fît a specified membership function in an approximate fashion.

U. A. , If n«=6 then (1,3,6) Is complementary to (2,4,5)), and \f denotes the supremum over U. x U. · V r U. -,U. J i j. jn h i * If R Is a fuzzy subset of U. ,U. , then Its Ί 'k cylindrical extension In U. x U U. ,un) (A59) U. X U i n In terms of their cylindrical extensions, the composition of two binary relatlonsR and S (In U. x U- and IL x U-, respectively) Is expressed by R o S = Proj R / l S o n U j x U . (A60) where R and S are the cylindrical extensions of R and S In U. x LL x U~. Similarly, If R Is a binary relation In U.

Kalaba and L. A. Zadeh, "Abstraction and Pattern Classification," Jour. Math. » Vol. 13, pp. 1-7, 1966. 6. E. T. Jaynes, "Maximum Entropy for Hypothesis Formula tion," Annaj^^jjfJ^ Vol. 3*, pp. 911-930, 1963. 7. M. Tribus, Rational Descriptions, Decisions and Designs, Pergamon Press, New York and Oxford, 1969. 8. N. Rescher, Many-Valued Logic, McGraw-Hill, New York, 1969. 25 FUZZY SETS AND THEIR APPLICATIONS 9. I. A. Zadeh, M A Fuzzy-Set-Theoretic Interpretation of Linguistic Hedges," Jour, of Cybernetics, Vol.