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 0 exists such that -M S u(x) 0 exists such that u(x) - u(y) > m for any x > 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 x Z1 y but y X1 x; and we say a ~i y if x Z1 y and y Z1 2 2 ii. Explain why no number m > 0 exists such that u(x) - u(y) > m for any * > y