Problem 3 doesn’t assume X is transitive. Can we assume that X is NOT necessarily transitive? Thanks.

Sorry, the question I asked just now should be rephrased as: can we assume that the preference operator on X is NOT necessarily transitive in problem 3 and 4?

On question 3, there is not actually a preference relation, there is only a (revealed preference) choice rule. That is to say, all you should be considering is a mapping from scriptB to subsets of X.

Similarly, in question 4, the question is about showing there can always be a rational preference relation that is consistent with any (revealed preference) choice rule.

I hope this clears things up some, but I’m not sure it will, so please let me know if not. I’m happy to try to be clearer.