# How to prove the following propositional logic equivalence?

The question asked me to prove the following equivalence. $$（r \rightarrow p)\rightarrow (p\land q)\equiv (\lnot r \rightarrow p)\land(\lnot r \rightarrow q)\land (p \rightarrow q)$$

And I did the followings:

For L.H.S., $（r\rightarrow p)\rightarrow (p\land q)\\\equiv \lnot (\lnot r\lor p)\lor (p\land q)\\ \equiv (r \land \lnot p) \lor (p \land q)$

For R.H.S., $(\lnot r \rightarrow p)\land(\lnot r \rightarrow q)\land (p \rightarrow q)\\ \equiv (r\lor p) \land (r \lor q) \land (\lnot p \lor q)\\ \equiv (r\lor (p\land q))\land (\lnot p\lor q)\\$

Now, I'm a bit stuck as to how to show that LHS = RHS.

• This is not a do my homework for me site. Please add your own workings into the post. It's okay if you get stuck midway through, but you do need to show us that you're willing to meat us half-way. – Namaste Apr 21 '18 at 17:38
• Lol actually I've been working on it and I think I did it in a wrong way so I decided not to post that up here – Cluyeia Apr 21 '18 at 17:40
• The left hand side is equivalent to $\lnot (\lnot r\lor p)\lor (p\land q)$. Why? Then this is equivalent to $(r \land \lnot p) \lor (p \land q).$ Why. Show me some of your work on the right-hand side? – Namaste Apr 21 '18 at 17:41
• Cluyeia It will help us understand the mistakes you may have made, so we can better help you. So please include your workings, right or wrong. And how do you know they are wrong? – Namaste Apr 21 '18 at 17:44
• The R.H.S is equivalent to $(r\lor p)\land (r\lor q)\land (\lnot p\lor q)$ and it’s equivalent to $(r\lor (p\land q))\land (\not p\lor q)$ right? – Cluyeia Apr 21 '18 at 17:46

I’ve done the proof and here is the answer.
L.H.S. = $（r\rightarrow p)\rightarrow (p\land q)\\ \equiv \lnot (\lnot r\lor p)\lor (p\land q)\\ \equiv (r \land \lnot p) \lor (p \land q)\\ \equiv (r\lor p)\land (\lnot p \lor p)\land (r\lor q)\land (\lnot p \lor q)\\ \equiv(r\lor p)\land T\land (r\lor q)\land(\lnot p\lor q)\\ \equiv(\lnot r\rightarrow p)\land (\lnot r\rightarrow q)\land (p\rightarrow q)\\ \equiv R.H.S.$

Thx for all your helps !

After writing out the left hand side using $\;\phi \rightarrow \psi \;\equiv\; \lnot \phi \lor \psi\;$ and DeMorgan, it is an $\;\lor\;$ of $\;\land\;$'s. Similarly, the right hand side becomes a $\;\land\;$ of $\;\lor\;$'s.
So you choose one of the two sides, and distribute $\;\land\;$ over $\;\lor\;$, or $\;\lor\;$ over $\;\land\;$, and work towards the other side.