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Proof. The theorem follows immediately from the equalities below. f (Aα ) = {f (u) : A(u) ≥ α} = {v ∈ V : A(u) ≥ α, f (u) = v} W −1 W ( Af )α = {v ∈ V : W Af −1 (v) ≥ α} = {v ∈ V : {A(u) : f (u) = v} ≥ α} W One should notice that for some α, it may not be true that A(P ) = α for any P . When U = V × V , f is a binary operation on V . If A and B are fuzzy subsets of V and the binary operation f is denoted ◦ and written in the usual way, then the theorem specifies exactly when Aα ◦ Bα = (A ◦ B)α , namely when certain sups are realized.

Let A be the fuzzy set and f : R → R defined by the equations 1 A(x) = χ{0} (x) + e− x χ(0,∞) (x) f (x) = xχ(0,1) (x) + χ[0,∞) (x) Recall that the fuzzy subset f (A) is defined by f (A)(y) = (∨Af −1 )(y) = sup{A(x) : x ∈ f −1 (y)} where f −1 (y) = {x : y = f (x)}. (a) Show that for each y ∈ [0, 1), sup{A(x) : x ∈ f −1 (y)} is attained. (b) Show that sup{A(x) : x ∈ f −1 (1)} is not attained. (c) Show that f (A)1 6= f (A1 ). 42 CHAPTER 2. SOME ALGEBRA OF FUZZY SETS 42. Let A and B be fuzzy sets, and f : R × R → R defined by the equations 1 A(x) = χ[3,4] (x) + e− x χ(0,3)∪(4,∞) (x) 1 B(x) = χ[−2,−1] (x) + e− x χ(−∞,−2)∪(−1,0) (x) f (x) = x + y Recall that f (A, B) is the fuzzy subset of R defined by f (A, B)(z) = sup{A(x) ∧ B(y) : x + y = z}.

They may or may not hold. But [0, 1] is a bounded distributive lattice which has an involution, namely x0 = 1−x, satisfying the De Morgan laws. Such a system (V, ∨, ∧,0 , 0, 1) is a De Morgan algebra. Every Boolean algebra is a De Morgan algebra, and in particular, the set of all subsets P(U ) of a set U is a De Morgan algebra. A De Morgan algebra that satisfies x ∧ x0 ≤ y ∨ y 0 for all x and y is a Kleene algebra. 6 Let (V, ∨, ∧,0 , 0, 1) be a De Morgan algebra and let U be any set. Let f and g be mappings from U into V .

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