Rewrite each of these statements so that negations appear only applied to predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives).

Question

1. ¬∃x∀yP(x, y)
2. ¬∀y∀x(P(x, y)∨Q(x, y))
3. ¬(∃x∀y¬P(x, y)∧∀x∀yQ(x, y))
4. ¬∀x(∃y∃zP(x, y, z)∨∃z∀yP(x, y, z))

Answer 1

Kai,

Your goal is to move the negation in towards the P(...)

Translate one quantifier to the other using 

¬∃x(stuff) <=> ∀x¬ (stuff)

1. a) ¬∃x∀yP(x, y) <=>

b) ∀x¬∀yP(x, y)

And again use 

¬∀x(stuff) <=> ∃x¬ (stuff)

c) ∀x∃y¬P(x, y)

2. ¬∀y∀x(P(x, y)∨Q(x, y))

∃y¬∀x(P(x, y)∨Q(x, y))

∃y∃x¬(P(x, y)∨Q(x, y))

Use DeMorgans Law

¬(stuff∨mush) <=> (¬stuff)∧ ( ¬mush)

∃y∃x(¬P(x, y)∧¬Q(x, y))

3. ¬(∃x∀y¬P(x, y)∧∀x∀yQ(x, y))

Now you do it: use DeMorgan

¬(stuff∧mush) <=> (¬stuff)∨( ¬mush)

Then for each term, move the negation to the right as we did in 1)

4. This is more complicated, but the same principles apply.

Does this help, Kai? Please ask again if you still need help on it.