(p -> r) is equivalent to (¬p v r) i.e p implies r means that r is true or p is not true. (p v q) ^ (¬p v r) ^ (¬q v r) and (p ^ ¬p) v (r ^ q) ^ (¬q v r) both mean that p or q is true, and r is true or p is false, and r is true or q is false.
(p ^ ¬p) is false hence (r ^ q) ^ (¬q v r) is true, and r and q being true implies r is true.
________________________________ From: Ashraf Samhouri <firstname.lastname@example.org> To: email@example.com Sent: Monday, 19 March 2012, 8:51 Subject: Tautology Proof Question
While studying, I've passed through this proof (in the class notes):
Example: Proof that: ( (p v q) ^ (p -> r) ^ (q -> r) ) -> r is a tautology?
The solution is: ( (p v q) ^ (¬p v r) ^ (¬q v r) ) -> r ( (p ^ ¬p) v (r ^ q) ^ (¬q v r) ) -> r (F v (q ^ ¬q) v r) -> r r -> r ¬r v r .: T
Actually I can't understand the steps, am not sure if the notes are well-written here, but I'll appreciate any help in explaining the proof.