Problem 3: (a) Let X be a finite set and be a binary relation on X. Show that if U : X R represents then> must be asymmetric and negatively transitive. (b) Let X be a set and-a binary relation on X. Defineas x x' if x, (ie, we define "weak preference" as: z is weakly preferred to if it is not the case that ' is strictly preferred to z). We sayis complete if, for any z,z' e X we get z z' OR z. Prove that if s is asylnmetric thenis complete. Prove also that if e is negatively transitive thenis transitive. (c) Given a weak preference on a set X define ~ as follows: x ~ x, if and only if x x, AND x, (ie, we define "indifference" as sorne alternative r is indifferent to some alternative if a is weakly preferred to ' and vice versa). Fronone can then obtain a preference-defined by x-x, if and only if x z' but not(x ~ x'). Question: if we start with a preference relation , we derive fron as above, and fromwe derive "; is ,"? If "yes", prove the statement; if "no", give a counter-example. (Not really hard to do, but you've got to wrap your mind around it first)