# Logically equivalent

• I
From the text it says (P -> Q) or (P -> R) is equivalent to P -> (Q or R)

I tried to see if this is true so I tried
$$(P \to Q) \lor (P \to R) \\ (P \lor \neg Q) \lor (P \lor \neg R) \\ P \lor \neg Q \lor \neg R \\ P \lor \neg(Q \land R) \\ P \to (Q \land R)$$
and
$$P \to (Q \lor R) \\ P \lor \neg(Q \lor R ) \\ P \lor (\neg Q \land \neg R) \\ (P \lor \neg Q) \land (P \lor \neg R) \\ (P \to Q) \land (P \to R)$$

From what I've done its seems like they're not equivalent ?!?!?

andrewkirk
Homework Helper
Gold Member
I tried
$$(P \to Q) \lor (P \to R) \\ (P \lor \neg Q) \lor (P \lor \neg R)$$
The second line does not follow from the first.

I think what you meant to write for the second line was
$$(\neg P\vee Q)\vee (\neg P\vee R)$$
which is not the same thing.

The second line does not follow from the first.

I think what you meant to write for the second line was
$$(\neg P\vee Q)\vee (\neg P\vee R)$$
which is not the same thing.

AHHHH thank you!