I am new to the logic of compound statements, and I'm working on simplifying, but I'm having a tough time wrapping my head around how to reduce these statements.
I have the following, which thus far I think is accurate (it is supposed to simplify to $ \neg p $):
$ \neg(p \vee \neg q) \vee (\neg p \wedge \neg q) \equiv (\neg p \wedge q) \vee (\neg p \vee q)\ (De\ Morgan's\ Laws) \\ $
However, I'm not sure where to go from here. I've put together a few options which I think make decent sense, but I honestly have no clue which one is correct, or what particular law is being demonstrated.
$ \equiv \neg p \vee (\neg q \vee q) $
$ \equiv \neg p \vee (\neg q \wedge q) $
$ \equiv \neg p \wedge (\neg q \vee q) $
$ \equiv \neg p \wedge (\neg q \wedge q) $
Hopefully one of these is right, or maybe none of them are. I'm just wondering if someone can clarify if I'm heading in remotely the right direction or not.