Exercise 2.5 (Fuzzy Sets): There are many concepts that are used in fuzzy sets that sometimes become useful when studying fuzzy control. The following problems introduce some of the more popular fuzzy set concepts that were not treated earlier in the chapter. (a) The “support” of a fuzzy set with membership function  is the (crisp) set of all points x on the universe of discourse such that  and the “-cut” is the (crisp) set of all points on the universe of discourse such that . What is the support and 0.5-cut for the fuzzy set shown in Figure 2.6 on page 30? (b) The “height” of a fuzzy set with membership function  is the highest value that  reaches on the universe of discourse on which it is defined. A fuzzy set is said to be “normal” if its height is equal to one. What is the height of the fuzzy set shown in Figure 2.6 on page 30? Is it normal? Give an example of a fuzzy set that is not normal. (c) A fuzzy set with membership function  where the universe of discourse is the set of real numbers is said to be “convex” if and only if  (2.29) for all andand all . Note that just because a fuzzy set is said to be convex does not mean that its membership function is a convex function in the usual sense. Prove that the fuzzy set shown in Figure 2.6 on page 30 is convex. Prove that the Gaussian membership function is not convex. Give an example (besides the fuzzy set with a Gaussian membership function) of a fuzzy set that is not convex. (d) A linguistic “hedge” is a modifier to a linguistic value such as “very” or “more or less.” When we use linguistic hedges for linguistic values that already have membership functions, we can simply modify these membership functions so that they represent the modified linguistic values. Consider the membership function in Figure 2.6 on page 30. Suppose that we obtain the membership function for “error is very possmall” from the one for “possmall” by squaring the membership values (i.e.,) . Sketch the membership function for “error is very ) possmall.” For “error is more or less possmall” we could use  . Sketch the membership function for “error is more or less possmall.” Exercise 2.5.(Fuzzy Set) (a)、 , . (b)、1 , yes , (c)、证明：如图所示： 时    又   同理：当时    又   (d)、  ：  ：