6.12. For the fuzzy sets defined in Example 64 (A), generate all of the nonzero values of (A+B) and Max(A,B) EXAMPLE 6.4 EXTENSION PRINCIPLE AND -CUTS A. Suppose X--3,-2,,0, 1,2, 3), thought of as a subset of the integers. Let fuzzy subsets A and B of X be defined by their membership vectors A- (0.0, 0.3. 0.8, 1.0, 0.80.3. 0.0) and B = (1.0. 0.9, 0.7, 0.5. 0.2. 0.0. 0.0). According to the extension principle, (A +B)OA(x) A B(y)). Since X is a finite set, the "sup" is just the maximum value, that is, (A+ B)(0)=Vs+y-o(A(x)^B(y)). Consider the following table of values with x and y coming from X and x+y=0: Clearly, the max of the mins of the columns 3 and 4 is 0.7 and so. (AB) (0)-0.7. Note that (A+B)(0) is NOT A(0)+B(0), since that would result in a value of 1.5, not a legal option for a fuzzy set. Exercise 6.12 will ask you to compute all of the values of A Bover all of the integers and not just X itself. Can we compute other extensions? What about Max(A,B)? Be careful, it's not the same as the union of A and B. For example, Max(A. B)(-1) supvyA(X) A By). The value of Max(A.B)) is computed to be B. Using this approach, you can do these tedious calculations on finite sets to get extensions of manv functions, such as A2, exp(A ln(A), and so on. Thankfully, Eqs. 6.19 and 6.20 make it possible to generate approximations to complex domains, like intervals of the real line. 4 0 2 0 Both A and B are triangular fuzzy numbers, and hence, their -cuts are closed intervals. To invoke the decomposition thoorem, we use the fact that (AB)- C"A) +"B) along with interval arithmetic to compute the right hand side of this expression. Now, for0