I have two homework questions that I've been struggling with. For the first I need to prove that
$((p \lor q) \land (\lnot p \lor r)) \to (q \lor r)$
is a tautology.
I've tried two approaches. First I tried substituting other logically equivalent statements for the propositions on the LHS. Once that failed, I tried assuming that the LHS is true and I tried to show the RHS must also be true. I wasn't able to do that either. There is nothing saying I can't use a truth table, but I'd prefer not to. Any help would be appreciated.
The second question is to decide whether
$\forall x \exists y(P(x) \to P(y)) \to \exists y \forall x(P(x) \to P(y))$
is logically valid or not. Does logically valid mean tautology? If so, I don't even know where to start.
EDIT: This question originally asked to prove the second expression as true, which was incorrect.