A2 (25 points) There are two binary relations S and W defined over the three elements in the set X = {x1, x2, x3). They are given implicitly using two functions u : X - R and v : X - R as follows: For any pair r, y E X . rSy iff u(x) > u(y) and v(x) > v(y) . rWy iff u(x) > u(y) and v(x) 2 v(y) The function u satisfies u(x1) = 1, u(x2) = 2, u(x3) = 3. The function v satisfies v(1) = 2, v(x2) = 1, v(13) = 3. (a) Describe explicitly arSy and aWy. That is, use the functions u and v to determine which pairs are related according to S, and then repeat with W. [5] (b) Is S a strict preference relation, i.e. asymmetric and negatively transitive? [5] (c) Is W is a weak preference relation, i.e. complete and transitive? [5] Based on the relation S from above, define the choice function (A) = {x E A such that for all y ( A, -ySx.} This function assigns to every non-empty subset A of X an element of X. That is, we consider the choices of a decision maker who chooses the best elements according to S. Page 3 of 6 Copyright 2022 v01 @) University of Southampton ECON3039WI (d) Does this choice function satisfy Properties or and B, as defined in class? Discuss your finding in the light of your answer to question c). [10]