Exercise 2.7 (Fuzzy Logic): There are many concepts that are used in fuzzy logic that sometimes become useful when studying fuzzy control. The following problems introduce some of the more popular fuzzy logic concepts that were not treated earlier in the chapter or were treated only briefly. (a) The complement (“not”) of a fuzzy set with a membership function has a membership function given by . Sketch the complement of the fuzzy set shown in Figure 2.6 on page 30. (b) There are other ways to define the “triangular norm” for representing the intersection operation (“and”) on fuzzy sets, different from the ones introduced in the chapter. Two more are given by defining “*” as a “bounded difference” (i.e.,x*y=max{0,x+y-1}) and “drastic intersection” (where x * y is x when y = 1, y when x = 1 , and zero otherwise). Consider the membership functions shown in Figure 2.9 on page 33. Sketch the membership function for the premise “error is zero and change-in-error is possmall” when the bounded difference is used to represent this conjunction (premise). Do the same for the case when we use the drastic intersection. Compare these to the case where the minimum operation and the product were used (i.e., plot these also and compare all four). (c) There are other ways to define the “triangular co-norm” for representing the union operation (“or”) on fuzzy sets, different from the ones introduced in the chapter. Two more are given by defining “⊕” as a “bounded sum” (i.e., ) and “drastic union” (where is  when y = 0, y when x = 0, and one otherwise). Consider the membership functions shown in Figure 2.9 on page 33. Sketch the membership function for “error is zero or change-in-error is possmall” when the bounded sum is used. Do the same for the case when we use the drastic union. Compare these to the case where the maximum operation and the algebraic sum were used (i.e., plot these also and compare all four).Exercise 2.7: (a)、  （b）、 使用“minimum”时  使用“product”时：  使用“ bounded difference”时：  使用”drastic intersection”时：  （c）、 使用“maximum”时：  使用”algebraic sum”时：  使用“bounded sum”时：  使用“drastic union”时