¬∃y(Q(y) ∧ ∀x¬R(x, y)) = ∀y¬(Q(y) ∧ ∀x¬R(x, y))? or ∀y¬(Q(y) v ∀x R(x, y))?

Question

Rewrite each of these statements so that negations ap- pear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives).

¬∃y(Q(y) ∧ ∀x¬R(x, y)) = ∀y¬(Q(y) ∧ ∀x¬R(x, y))?

or

∀y¬(Q(y) v ∀x R(x, y))?

My solution manual says the second one, however, I didn't see any parenthesis including the whole term, so shouldn't it be just the negative of the first term, and the second term unaffected?

Answer 1

The basic rule is the $\lnot \exists x T(x)=\forall x \lnot T(x)$ When you negate the parenthesized thing you don't change anything inside, so the first version is correct. You don't want to change the inner $\wedge$ to $\vee$ yet. Now you want to take the $\lnot$ inside the parentheses, which will involve negating each term and changing the connective.