Suppose X = fx; y; zg and let U and V be any two linear functions defined on L(X)

that are NOT affine transformations of each other. Suppose U(x) > U(y) > U(z) and

V (x) > V (y) > V (z). Define the following two binary relations:

(i) p 1 q if and only if U(p) U(q) or V (p) V (q);

(ii) p 2 q if and only if U(p) U(q) and V (p) V (q).

For i = 1; 2, which vNM axioms does i satisfy? Which ones does it fail? Support your

answers with proofs or counter-examples as needed.