Need help with question 4d)See question 1e) for information needed to complete 4d), because it makes use of preference relation defined in 1e)Bear in mind Q4d has parts i and ii

1. Let X = Z+, the set of nonnegative integers. In each of the following parts, we will name a binary relation on X. For each one, tell me (i) whether or not it is complete, (ii) whether or not it is transitive, (iii) whether or not it is a preference relation, and if so, (iv) whether or not it admits a utility representation. Justify your answers. (a) The relation =. (b) The relation . (c) The relation YES where YESy for every a and y. (d) The relation NOPE where we don't have aNOPEy for any a and y. (e) The relation , where a ~ y if and only if either x = 0 or x 2 y > 1. 4. [In addition to giving you some practice thinking about utility representations, this question will give us a bit more intuition for the idea that utility is an "ordinal" concept. It will also show how some of our intuitions about utility might apply when X is finite but not when X is infinite.] Let X be a set, let _ be a preference relation on X, and let u be a utility representation for . (a) Suppose X is finite. Explain why some (large enough) number M > 0 exists such that -M J u(x) _ M for every r E X. That is, the utility is bounded. (b) Show that, even if X is infinite, some alternative utility representation u and some (large enough) number M > 0 exists such that -M S u(x) S M for every r E X. [Hint: The function 4 : R - R given by 4(t) = 1+14 is strictly increasing.] (c) Suppose X is finite. Show that some (small enough) number m > 0 exists such that u(x) - u(y) > m for any = > y. (d) Suppose X = Z+, and _ is the preference relation defined in question 1(e). i. Explain why any whole number k 2 3 has u(1) 1 y if I Z1 y but y X1 x; and we say a ~ y if z Z1 y and y X1 I 2 ii. Explain why no number m > 0 exists such that u(x) - u(y) > m for any x > y