# Removing negations using logical equivalences

1. Sep 21, 2011

### idlackage

1. The problem statement, all variables and given/known data
Remove negations from this: ¬∀x∃y : (P (x, y) --> (∃z : ¬Q(x, y, z))) using logical equivalences.

2. Relevant equations
¬(∃x : P(x)) is equal to ∀x : ¬(P(x))
P --> Q is equal to ¬P v Q

3. The attempt at a solution
¬∀x∃y : (P (x, y) --> (∃z : ¬Q(x, y, z)))
≡ ¬∀x∃y : (¬P (x, y) v (∃z : ¬Q(x, y, z)))
≡ ∃x∀y : ¬ (¬P (x, y) v (∃z : ¬Q(x, y, z)))
≡ ∃x∀y : (P (x, y) v ¬(¬(∀z : Q(x, y, z))))
≡ ∃x∀y : (P (x, y) v (∀z : Q(x, y, z)))

I'm wondering if I've applied the logic right. I'm not sure if I can just shift the Not like that from line one to two (counting from the equivalency sign). On line two to line three, I'm assuming that (∃z : ¬Q(x, y, z)) would mean something like "There exists a person who doesn't like math", which would equal to ¬(∀z : Q(x, y, z)) or "Not everyone likes math". Then on line four the Not from the very outside cancels the Not in front of that ∀z. Am I doing this right? I feel like I'm missing something.

Thanks.