To disprove a statement of the form $\exists x \forall y (x = ...y...)$, can I negate like this: $\forall x \exists y \lnot(x = ...y...)$?

Question

Can I use the negation, then give an example which proves $\forall x \exists y \lnot(x = ...y...)$ is true?

Can I simply give an example after negating the quantifiers to prove that the equation was impossible $(\lnot(x = ...y...) = T)$ as there won't exists any y to satisfy x in the equation?

Thanks!

Answer 1

Someone might say "To disprove a statement, you only need to give a counter example."

i.e. To disprove $\forall x P(x)$ we just show $\exists x\neg P(x)$ by an example.

This is true when we have $\forall$ in front of the statement, but when we want to disprove $$\exists x,P(x)$$ We will need to give a set of examples, that $\neg P(x)$ is true for every $x$.

Can I use the negation, then give an example which proves $∀x∃y,\neg(x=...y...)$ is true?

Can I simply give an example after negating the quantifiers to prove that the equation was impossible $\neg P(x,y) \equiv \top$ as there won't exists any y to satisfy x in the equation?

Write $P(x,y)\equiv(x=\dots y\dots)$

Here need to show $\forall x\exists y,\neg P(x,y)$

For every $x$, we need to give an example that exist $y$ such that $\neg P(x,y)$

It's really a set of examples, but how to prove this is also depend on what is $P(x,y)$.