Consider the axiomatic system where the undefined terms consist of elements of a set S, and a
Question:
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)
Discrete Mathematics and Its Applications
ISBN: 978-0073383095
7th edition
Authors: Kenneth H. Rosen