Please do Exercise 17.2.8 part C and E and please show step by step and explain Example 17.2.3.(a) Consider the set of real numbers R. We know that every element of Rbelongs to one of two sets: the set of rational numbers, Q, or the set ofirrational numbers, I. The union of these two subsets, QUI = R, andQ and I are disjoint sets, so based on the definition {Q, I} is a partitionof R. Alternatively the word "partition" can be used as a verb, so wecould also say that Q and I partition R.(b) Let E and O be the even and odd integers, respectively. Then {E, O} isa partition of Z. Alternatively we could also say that E and O partitionZ.(c) Let S be the set of all single-element subsets of Z, so that for example{-552}, {7}, {1492} are all elements of S. Then S is also a partitionof Z. Here S has an infinite number of elements (all the single-elementsubsets of Z), but each element of S is a finite set.(d) Consider the set of complex numbers C. Every element of C has a realpart which we denote as Re[z] (as in Chapter 2). Let Ra be the set ofall complex numbers with real part a, i.e. Ra = {z € C | Re[z] = a}.Let P be the set consisting of all of the Ra's, i.e. P = {RaVa € R}.Then P is a partition of C. Here P has an infinite number of elements,where each element of P is an infinite set.Example 17.2.4.584CHAPTER 17 EQUIVALENCE RELATIONS AND EQUIVALENCE CLASSE(a) When making an inventory of the animals in a zoo, we may wish tocount the number of antelopes, the number of baboons, the number ofcheetahs, and so forth. In this case, all of the animals of the same speciesmight be grouped together in a single set. Each species give rise t adifferent set and these sets form a partition of the animals in that zoo.(b) If we are concerned only with people's given names (what Americanswould call "first name"), we can partition any set of people accordingto given name. Each set in the partition consists of all people who sharea particular given name.(c) In geometry, sometimes we are interested only in the shape of a triangleand not its location or orientation. In this case, we talk about congruenttriangles, where congruent means that corresponding sides of the twotriangles are equal, and corresponding angles are also equal. For anytriangle we may define the set of all triangles congruent to that triangle.There are an infinite number of such sets which form a partition of theset of all triangles. Exercise 17.2.8. Define a binary relation from the given partition, andshow that the relation has the above three properties.588 CHAPTER 17 EQUIVALENCE RELATIONS AND EQUIVALENCE CLASSES(a) The partition in Example 17.2.3(b)(b) The partition in Example 17.2.3(c)(c) The partition in Example 17.2.3(d)(d) The partition in Example 17.2.4(b)(e) The partition in Example 17.2.4(c)

In this question (17.2.8), we have to show three above properies to be satisfied but in this…

Exercise 17.2.8. Define a binary relation from the given partition, and show that the relation has the above three properties. 588 CHAPTER 17 EQUIVALENCE RELATIONS AND EQUIVALENC (a) The partition in Example 17.2.3(b) (b) The partition in Example 17.2.3(c) (c) The partition in Example 17.2.3(d) (d) The partition in Example 17.2.4(b) (e) The partition in Example 17.2.4(c)