Suppose the following have been established or Axiom 1 p rig

Suppose the following have been established or Axiom 1: p rightarrow ~ y Axiom 2: ~ q rightarrow r Theorem 1: p rightarrow ~ z Theorem 2: x rightarrow either q or z Theorem 3: r rightarrow either x or y Write a two-column proof of the proposition p rightarrow q.

Solution

Solution :

1. p (given)
2. ~ y by axiom 1
3. ~z (Thm. 1)
4 . either q or ~x (contrapositve of Thm. 2)
5. if q we are done
6. if ~x, then ~r (Theorem. 3 contrapositive)
7. q (Axiom. 2 contrapositive)