Finding an equivalent statement in english

Question

Let $P(x,y)$ stand for the statement “$x$ is the parent of $y$”. Here, $x$ and $y$ each denote a human being.

Which of the following is equivalent to the statement $\neg\forall x\forall yP(x,y)$?

1. It is not the case that every human is the parent of every human.
2. Every human has a human parent.
3. There exists a human who is not the parent of any human.
4. There exists a human who is the parent of every human.

I got answer choice $3$ because I converted the statement to $\exists x \exists y\neg P(x,y)$ which in English is number $3$. Can my answer be confirmed?

Answer 1

The proposition $$\forall x \forall y P(x,y) $$ says that for any pair $ (x,y) $ of human

$ x $ is the parent of $ y $

Its negation $$\lnot \forall x \forall y P(x,y)$$ mean that it is not the case.

Answer 2

1. $\exists x \exists y \neg P(x,y) \iff \neg\forall x \forall y P(x,y)$

2. $\forall y \exists x P(x,y) \iff \neg \exists y \forall x \neg P(x,y)$

3. $\exists x \forall y \neg P(x,y)$

4. $\exists x \forall y P(x,y)$

Answer 3

Your negation $\exists x \, \exists y \, \lnot P(x,y)$ is correct, but your English translation of it is not correct. The correct translation of that negation would be "There exist two people so that the first is not the parent of the second".

But number 1 is a completely straightforward reading of $\lnot \forall x \, \forall y \, P(x,y)$.