By Hung T. Nguyen, Elbert A. Walker

Utilizing fabric from a profitable path on fuzzy common sense, this e-book is an creation to the speculation of fuzzy units: mathematical gadgets modeling the vagueness of our average language once we describe phenomena that don't have sharply outlined barriers. The publication offers heritage details essential to practice fuzzy set thought in a number of parts, together with engineering, fuzzy common sense, and choice making. The workouts on the finish of every bankruptcy serve to deepen the reader's knowing of the suggestions, and to check their skill to make the required calculations.

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**Example text**

1 A =A B B 16. A + (−B) = A − B Proof. We prove some of these, leaving the others as exercises. The equations W 1 · A(x) = yz=x χ{1} (y) ∧ A(z) W = 1x=x χ{1} (1) ∧ A(x) = A(x) show that 1 · A = A. If (A(B + C))(x) > (AB + AC)(x), then there exist u, v, y with y(u + v) = x and such that A(y) ∧ B(u) ∧ C(v) > A(p) ∧ B(q) ∧ A(h) ∧ C(k) for all p, q, h, k with pq + hk = x. But this is not so for p = h = y, q = u, and v = k. Thus (A(B + C))(x) ≤ (AB + AC)(x) for all x, whence A(B + C) ≤ AB + AC. 1. FUZZY QUANTITIES 49 However, r(A + B) = rA + rB since ´ ³ W V χ{r} (A + B) (x) = uv=x (χ{r} (u) (A + B)(v)) V W = rv=x (χ{r} (r) (A + B)(v)) W V = s+t=v (A(s) B(t)) rv=x V W = s+t=v (χ{r} (r)A(s) χ{r} (r)B(t)) rv=x = (rA + rB)(x) There are a number of special properties that fuzzy quantities may have, and we need a few of them in preparation for dealing with fuzzy numbers and intervals.

At this point we need some notation. Suppose that f1 : X1 → Y1 and f2 : X2 → Y2 . Then f1 × f2 is standard notation for the mapping X1 × X2 → Y1 × Y2 : (x1 , x2 ) → (f1 (x1 ), f2 (x2 )) Now if A and B are fuzzy subsets of U and V, respectively, then A × B maps U × V into [0, 1] × [0, 1], and the image (A(u), B(v)) of an element of U × V is a pair of elements of [0, 1] and hence has a min. Thus the composition ∧(A × B) is a fuzzy subset of U × V. Sometimes in fuzzy set theory, the mapping ∧(A × B) is denoted simply A × B, but there are other binary operations besides ∧ that we will have occasion to follow A × B with.

F ∧ g)(x) = f (x) ∧ g(x), 3. f 0 (x) = (f (x))0 , 4. 0(x) = 0, 5. 1(x) = 1. Let V U be the set of all mappings from U into V . Then (V U , ∨, ∧,0 , 0, 1) is a De Morgan algebra. If V is a complete lattice, then so is V U . Proof. The proof is routine in all respects. For example, the fact that ∨ is an associative operation on V U comes directly from the fact 22 CHAPTER 2. SOME ALGEBRA OF FUZZY SETS that ∨ is associative on V . ) Using the definition of ∨ on V U and that ∨ is associative on V , we get (f ∨ (g ∨ h)) (x) = = = = = f (x) ∨ (g ∨ h) (x) f (x) ∨ (g(x) ∨ h(x)) (f (x) ∨ g(x)) ∨ h(x) (f ∨ g) (x) ∨ h(x) ((f ∨ g) ∨ h) (x) whence f ∨ (g ∨ h) = (f ∨ g) ∨ h, and so ∨ is associative on V U .