Logical Equivalences: $(P\wedge\neg Q)\vee P \equiv P$

Question

How do I approach/ complete logical equivalency problems such as these?

I can see that the left hand side Should end up becoming Idempotent law $P \vee P \equiv P$.

And have to use implication within $(P\wedge\neg Q)\vee P$, though I'm not sure how to forge it together.

Any help would be great.

Answer 1

Observe that, if the RHS is true, it implies that the LHS is true, once the disjunction is true. On the other hand, if the RHS is false, the LHS is also false, since the conjunction (and consequently the disjunction) is false. Therefore both formulas are equivalent.

Answer 2

If you want to use implication, you can take $P\wedge\neg Q \equiv\neg(\neg P \vee Q)$. Letting $U=(\neg P \vee Q)$, the LHS is $\neg U \vee P$, which is equivalent to $U \rightarrow P$. $U$ is equivalent to $P \rightarrow Q$, so the LHS is $(P \rightarrow Q)\rightarrow P$. We have that the RHS entails the LHS: if $P$ is true, then all implications with $P$ as the conclusion are true. We also have that the LHS entails the RHS: if $P$ is false, then $(P \rightarrow Q)$ is true, and $True\rightarrow False$ is false.