Need help with question 5, see question 3 for lexicographic preference relation as instructed in question 5.

3. Let X be a set. and let :1 and :3 be two binary relations on X. Define the lexicographic binary relation 2;} as follows: for any I. y E X , we have it :14 if and only if at least. one of the following two conditions holds:'. 0 1' >-1 y I :r ~1 y and :r :2 1;. [If :1 and :2 capture how a decision maker feels about different. aspects of a decision. then *1 is one wav of combining them: use aspect 1 to make decisions. but if aspect I does not make the decision e as}. then use aspect 2 to bre 1k ties. ] Explain why each of the. following is true: (a) If :rgy E X have .1\" *7 y. then .r >1 1;. (b) *1 is complete if both)\" ml and *2 are complete ((1.) >' is transitive if both>" MI and 2;; are transitive. [Hint: part. (a) of this Mfg question and parts (b.c) of the previous question may be helpful.] ((1) 14>" is a preference relation if both :1 and :2 are preference relations. (e) If *2 is exactlv>' then it}; is exactly :1 ml' (f) [1.21:2 is exactly :51 (see question 2(a)). then 2;; is exactly 35;. 5. Let X be a set. and let :1 and :3 be two preference relations on X. and let :L be the lexicographic preference relation as defined in question 3. Suppose 111 : X > R is a utility representation for :1. and 112 : X > R is a utility representation for :3. For every number 6 2 (1. dene the function ii'E : X > R by letting u'(.r) 2 111(1') + cux). ['Xe might say 2;; puts "much more weight" on :1 than on :3. This question gives a sense in which that interpretation is valid. but also shows that this interpretation can be a bit 111isleading.] (a) Consider the special case in which X is finite. Show that. for some 1' > U, the function u' is a utility representation for in. [Hint parts (ac) of the previous question might be helpful] (b) [Hard optional question. You don't need to know this. but you can think about it if you would nd it fun] Suppose X is eountably innite.2 Must there exist some E 2 0 such that n' is a utility representation for ESL