Am I on the right track simplifying this logical expression?

Question

I am just learning logic and sets, and I am not completely familiar with them yet, that's why I am asking you here, if at least I am on the right track. Is this first of all correctly simplified?

$P \land (((P \land \neg Q) \rightarrow \neg P) \land (Q \lor P))$ [equivalent]

$P \land ((\neg(P \land \neg Q) \lor \neg P) \land (Q \lor P))$ [morgan]

$ P \land ((\neg P \lor Q \lor \neg P) \land (P \lor Q))$

$ P \land (P \lor Q) \land (\neg P \lor Q) $[absorption]

$ P \land (\neg P \lor Q)$

Answer 1

Your simplification is okay, but not your last step:

$(P \vee Q) \wedge (\neg P \vee Q) \equiv (\neg P \vee Q)$.

Instead, you can try some distribuitivity:

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

Then, since we have contradiction $\vee Q$:

$P \wedge Q$