Truthfulness of predicate logic statements

Question

Assume that ∀x∃yP(x, y) is true under an interpretation I. Which of the following formulas must also to be true under I? If the formula is true, explain. Otherwise, give a counterexample.

(i) ∀x∀yP(x, y)

(ii) ∃x∀yP(x, y)

(iii) ∃x∃yP(x, y)

For this question, I'm not sure if my answers are correct:

i) False: let interpretation I: D = natural numbers and let P(x,y) mean x divides y. Then this statement is obviously false, since every number does not divide every number (witness: x=20, y=3)

ii) False: same interpretation as above. Means that there is some number such that every number divides it which I don't believe is true (0 isn't divisible by 0).

iii) True: if every x has the property P(x,y), then there must be some subset of x (∃x) that has that property.

Are these answers on the right lines?

Answer 1

Yes they are. A more general approach is to abstract away from using numbers as examples and reason along the following lines:

For $\forall x \exists y P(x,y)$ says that every object in the model, there is at least one some object such $P(x,y)$ is true.

To show 1) is false, we take a two element model $\{x,y\}$ such that $P(x,x)$ and $ P(y,x)$ are true and $P(y,y)$ is false.

To show 2) is false, we take the same set of elements ${x,y}$ and let $P(x,x)$ and $P(y,y)$ be true but $P(x,y)$ and $P(y,x)$ be false.

To show 3) is true, assume the universe is non-empty, so there is an $x$ and by the only condition on the model it must be true that $P(x,x)$ so there is an element such that $P(x,y)$ is true, namely $x$ itself.

Answer 2

Perhaps you're expressing it a bit sloppy on (iii), but essentially you're correct except for one corner case: the (iii) is only true in an non-empty universe.

To be formal the implication $(\forall x\phi)\rightarrow(\exists x\phi)$ holds in a non-empty universe: the if $\phi$ is true for all $x$ and there exists one $x$ then one could just select that and for that $x$ $\phi$ would be true. It doesn't matter if $\phi$ contains an quantifier (eg being $\exists y P(x,y)$).