By Didier Dubois and Henri Prade (Eds.)
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Extra info for Fuzzy Sets and Systems Theory and Applications
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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.
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