Where do I start with $\sim((P\wedge Q)\vee \sim(P\vee Q))$?

Question

can anyone tell me in a table form how to start with this $\sim((P\wedge Q)\vee \sim(P\vee Q))$ I am confused on how to do this part $\sim(P\wedge Q)$, which one we do first, inside brackets or outside?

Answer 1

$$\lnot ((P\wedge Q)\vee \lnot(P\vee Q))$$

Which "negation" $\lnot$ you apply first doesn't matter with respect to the end result. But outer, then inner is less work.

Using DeMorgan's with respect to the "outermost" negation sign, and then the inner one:

$$\begin{align} \lnot ((P\wedge Q)\vee \lnot (P\vee Q))&\equiv \lnot (P\land Q) \land \lnot \lnot (P\lor Q)\\ \\ &\equiv (\lnot P \lor \lnot Q) \land (P\lor Q)\end{align}$$

Using DeMorgan's first with respect to the innermost application, and then the outer one:

$$\begin{align} \lnot ((P\wedge Q)\vee \lnot (P\vee Q))& \equiv \lnot ((P \land Q) \lor (\lnot P \land \lnot Q))\\ \\ &\equiv \lnot (P \land Q) \land \lnot (\lnot P \land \lnot Q)\\ \\ & \equiv (\lnot P \lor \lnot Q)\land (\lnot \lnot P \lor \lnot \lnot Q) \\ \\ &\equiv (\lnot P \lor \lnot Q) \land (P \lor Q)\end{align}$$