2. Let X be a set, and let be a binary relation on X. (a) Define the binary relation 3 on X by xyyx. Show that: i.is transitive if is transitive; ii. is complete if is complete; iii. is a preference relation if is a preference relation. [The relation can be thought of as the "opposite" of . For example, if Ann and Bob are playing heads-up poker, and describes Ann's pref- erences over which starting hands they each have, then it's natural that Bob's preferences would be .] (b) Suppose i. If a ii. If a (c) Suppose is transitive. Suppose a, b, c E X. Show that: band bc, then a > c. b and bc, then a > c. is transitive. Explain why the relation ~ (given by x ~ y both xy and y x) is transitive. 3. Let X be a set, and let 1 and 2 be two binary relations on X. Define the lexicographic binary relation L as follows: for any x, y X, we have x Lif and only if at least one of the following two conditions holds:. x 1 Y x~1 y and x2 Y. [If 1 and 2 capture how a decision maker feels about different aspects of a decision, then L is one way of combining them: use aspect 1 to make decisions, but if aspect 1 does not make the decision easy, then use aspect 2 to break ties.] Explain why each of the following is true: (a) If x, y X have x Ly, then x1 y. (b) is complete if both 1 and 2 are complete. (c) L is transitive if both 1 and 2 are transitive. [Hint: part (a) of this question and parts (b,c) of the previous question may be helpful.] (d) is a preference relation if both 1 and 2 are preference relations. (e) If 2 is exactly 1, then L is exactly 1. (f) If 2 is exactly 21 (see question 2(a)), then L is exactly 1.