I'm given three sentences:
(a) If Frodo destroys the ring, then the world will be saved. (b) Gollum stole the ring from Frodo or Frodo destroyed the ring. (c) The world will be saved or Gollum stole the ring from Frodo.
I have to prove that sentences (a) and (b) jointly entail (c). I'm not quite sure how to do this.
I started by assigning the following variables:
$p$ = Frodo destroys the ring
$q$ = The world will be saved
$r$ = Gollum stole the ring.
Sentence (a) translates to: ($p \rightarrow q$)
Sentence (b) translates to: ($r \lor p$)
Sentence (c) translates to: ($q \lor r$)
I wrote up a truth table for all three variables, and then for each statement. I tried making the truth table for the statement $( p \rightarrow q) \land (r \lor p)$ equivalent to the one for $(q \lor r)$, but was not able to. Any ideas what I am doing wrong?