When using De Morgan's Law, does the negation go before or after the quantifier?

Question

For example, I want to use De Morgan's law on the following equation: $\forall x (P(x) \wedge R(x))$. Would it be number 1. $\neg \forall x(\neg P(x) \vee \neg R(x))$, or would it be number 2. $\forall x \neg(\neg P(x) \vee \neg R(x))$? I need to know which way is correct to complete one of my proofs, thanks.

Answer 1

We can distribute $\forall$ over $\land$ to get:

$∀x (P(x) ∧ R(x)) ≡ (∀x \ P(x) ∧ ∀x \ R(x))$.

We can get the same result in a more convoluted way using De Morgan.

We have to start with the equivalence:

$∀x \ (P(x) ∧ R(x)) ≡ ¬∃x¬ \ (P(x) ∧ R(x))$,

then use De Morgan to "move inside" the inner negation sign to get:

$¬∃x¬ \ (P(x) ∧ R(x)) ≡ ¬∃x \ (¬P(x) ∨ ¬R(x))$,

then distribute $\exists$ over $\lor$ to get:

$¬∃x \ (¬P(x) ∨ ¬R(x)) ≡ ¬(∃x¬ P(x) ∨ ∃x¬ R(x))$.

Finally, a further application of De Morgan will produce:

$¬∃x¬ P(x) ∧ ¬∃x¬ R(x)$.