I am given the problem of proving:
$p → (q\land r) \equiv (p→q) \land(p→r)$
Using known logical equivalences. I'm not well practiced in transforming logical statements that contain →'s in them into other forms, and i'm at a bit of a loss, so:
What logical equivalences are available the best to use when presented with →'s, as i'm certainly going to see them again soon and which would be best suited to solving my problem.
Thank you!
On
You can either work out the truth table for each of these propositions, fill in the assignments and you will see that the truth tables are identical, which means that these propositions are indeed equivalent.
Of you can use other equivalences and just unwind and rewind the proposition from one end to the other, for example: $$p\rightarrow (q\land r)\equiv \lnot p\lor(q\land r)\equiv \ldots\equiv(p\rightarrow q)\land(p\rightarrow r).$$
(And I leave you with the task of filling in those $\ldots$ there with the needed equivalences.)
you simply have $$p → (q\land r) \equiv p'\lor (q\land r)\\ \equiv (p'\lor q) \land(p'\land r)\\ \equiv (p→q) \land(p→r) $$