Simplify the expression:
$\sim ((p\rightarrow (\sim q \vee r))\wedge (\sim p \wedge \sim q \wedge \sim r ))$
Correct answer: $p\vee q\vee r$
I don't know how to start on this problem. Are there some useful laws to use here or is there something else I should do when simplifying?
You can just simplify the expression. Knowing the equivalence $A\to B\equiv \lnot A\lor B$ and de Morgan's laws is a start.
\begin{align*} &\lnot ((p\rightarrow (\lnot q \lor r))\land (\lnot p \land \lnot q \land \lnot r ))\\ &= \lnot(p\rightarrow (\lnot q \lor r))\lor \lnot(\lnot p \land \lnot q \land \lnot r ) && (\text{de Morgan})\\ &= \lnot(\lnot p\lor (\lnot q \lor r))\lor (\lnot\lnot p \lor \lnot\lnot q \lor \lnot\lnot r ) &&(\text{definition, de Morgan})\\ &= (\lnot\lnot p\land\lnot(\lnot q \lor r))\lor (\lnot\lnot p \lor \lnot\lnot q \lor \lnot\lnot r ) &&(\text{de Morgan})\\ &= ((\lnot\lnot p\land\lnot(\lnot q \lor r))\lor \lnot\lnot p) \lor (\lnot\lnot q \lor \lnot\lnot r )\\ &= (\lnot\lnot p) \lor (\lnot\lnot q \lor \lnot\lnot r ) && (\text{absorption})\\ &=p\lor q\lor r && (\text{double negation}) \end{align*}