1. Suppose that we have elicited the indifference relation ~ and the strict prefer- ence relation > from a decision-maker. Define the weak preference relation _ given relations ~ and >. That is, reconstruct the decision-maker's weak pref- erence statements from the decision-makers indifference and strict preference statements. 2. Assume that the set of alternatives is X = {a, b, c, d}, and consider the collection of budget sets B = {{a, b} , {b, c}, {c, d} , {a, c} , (a, d } , {b, d} , {a, c, d } }, along with the choice rule C(a, b}) = {a, b} C({b, c}) = {b, c} C({c, d}) = {c, d} C(a, c}) = {a} C(a, d}) = {a} C({b, d}) = {d} C({a, c, d}) = {d}. (a) Derive the revealed preference relation _* associated to this choice struc- ture. (b) Derive the choice structure (B, C*(., >*)) generated by the revealed pref- erence relation _*. (c) Does the revealed preference relation _* rationalise C(.) relative to B? (d) Derive the strict preference relation >* and the indifference relation ~* associated to _* (e) Is the indifference relation ~* from (d) transitive