Consider the axiomatic system where the undefined terms consist of elements of a set S, and a set P consisting of pairs of elements of S, (a, b), satisfying the following axioms: A1 If (a, b) is in P, then (b, a) is not in P. A2 If (a, b) and (b, c) are in P, then (a, c) is in P. Do the following questions:

(a) : Let S1 = {1, 2, 3, 4}, and let P1 = {(1, 2),(2, 3),(1, 3)}. Is this a model for the system? (Justify your answer—always justify your answers, unless specifically told otherwise.) Hints: Check if P1 satisfies the axioms.

(b) Let S2 = R, the set of all real numbers. Let P2 = {(x, y)|x < y}. Is this a model for the system?

(c) Use this to argue that the axiomatic system is not complete. In particular, can you add an axiom such that (S1, P1)