having difficulty proving logical equivalence of this.

Question

Here is the question I'm stuck on

$\lnot q \land (p \lor q) \equiv p \land \lnot q$

I can not even figure out which law to apply first.

Answer 1

\begin{align} \lnot q \land (p \lor q) &\equiv (\lnot q \land p) \lor (\lnot q \land q) &\text{distributivity}\\ &\equiv (\lnot q \land p) \lor \bot &\text{contradiction}\\ &\equiv \lnot q \land p &\text{neutral element for }\lor\\ &\equiv p \land \lnot q &\text{commutativity} \end{align} where $\bot$ stands for absurdity or contradiction, i.e. a formula that is always false.