# Logical equivalences/proof

So I am working on logical equivalences for the first time and it was all making sense, until I was given the exercise:

Verify the following equivalence by writing an equivalence proof. That is, start on one side and use known equivalences to get to the other side. $(p \to q) \land (p \lor q) ≡ q$.

I am aware of the various laws, however in what order do I apply these laws? Is there an order? If there is not an order then what are the steps to take in order to prove the equivalence?

• It's not entirely clear to me what you mean, but have you tried replacing $(p\to q)$ by $(\neg p \vee q)$ and then apply a distributivity law ? – Mathieu Huot Feb 6 '18 at 17:44
• I am mainly wanting to know if there is a formula to proof an equivalence or do I just have to compare already known logical equivalences? – Danny Sanderson Feb 6 '18 at 17:53
• What have you tried so far? What are the various laws you're aware of? It's not the purpose of this forum to just dump your problems for others to solve - especially when you don't share information about what you actually know about the problem so far. – skyking Feb 6 '18 at 17:54
• Associative, distribute, identity, negation, double negation, idempotent, De Morgans. Sorry that my post isn't up to your standards, skyking. I do not want this exercise solved for me, I simply want to know how I can solve it myself. – Danny Sanderson Feb 6 '18 at 17:58

\begin{align} (p \to q) \land (p \lor q) &\equiv (\lnot p \lor q) \land (p \lor q) &&\text{classical definition of }\to \\ &\equiv (\lnot p \land p) \lor q &&\text{distributivity of } \lor \text{ over } \land \\ &\equiv \bot \lor q \equiv q \end{align} where $\bot$ is the absurdity, i.e. a proposition that is always false.