Fuzzy Sets and Systems Theory and Applications by Didier Dubois and Henri Prade (Eds.)

By Didier Dubois and Henri Prade (Eds.)

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On the analytic formalism of the theory of fuzzy sets. Inf. Sci. 5, 149–157. Brown, J. G. (1971). A note on fuzzy sets. Inf. Control 18, 32–39. Capocelli, R. , and De Luca, A. (1973). Fuzzy sets and decision theory. Inf. Control 23, 446–473. Chapin, E. W. (1974). Set-valued set-theory. Part I. Notre Dame J. Formal Logic 4, 619–634. Chapin, E. W. (1975). Set-valued set-theory. Part II. Notre Dame J. Formal Logic 5, 255–267. , and Termini, S. (1972a). Algebraic properties of fuzzy sets. J. Math. Anal.

So it is possible to show that ˜ ([1, + ∞)), ∀M ∈ ᏼ ∀Λ > 0, ∀P > 0 MΛ ᭪ MP = MΛ  P and ∀M ∈ ᏼ˜ (]0, 1]), ∀Λ < 0, ∀P < 0, MΛ ᭪ MP = MΛ  P (because x yt u and x y + u are increasing operations for x у 1, t у 1, y у 0, and u у 0 and decreasing for 0 < x р 1, 0 р t < 1, y р 0, and u р 0; hence, Theorem 1 can be applied). Owing to (M ᭪ N) – 1 = (M – 1 ) ᭪ (N – 1) and ∀M > 0, (MΛ) – 1 = M(–Λ), the formula MΛ ᭪ MP = MΛ  P holds as ˜ (]0, 1]) ʜ ᏼ ˜ ([1, + ∞)) and both Λ and P are positive or soon as M ∈ ᏼ negative.

J. Math. Anal. Appl. 18, 145–174. Goguen, J. A. (1974). Concept representation in natural and artificial languages: Axioms, extensions and applications for fuzzy sets. Int. J. Man-Mach. Stud. 6, 513–561. ) Gottinger, H. W. (1973). Towards a fuzzy reasoning in the behavioral sciences. Cybernetica 16, No. 2, 113–135. Grattan-Guiness, I. (1975). Fuzzy membership mapped onto interval and many-valued quantities. Z. Math. Logik Grundlag. Math. 22, 149–160. Hirota, K. (1977). Concepts of probabilistic sets.

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