Is $\exists x \, \forall y : P(x,y)$ equivalent to $\exists x : P(x,y) \; \forall y$?

Question

Are the two statements \begin{align*} \exists x \, \forall y : P(x,y) \qquad \text{and} \qquad \exists x : P(x,y) \; \forall y \end{align*} equivalent?

Or is this more of a matter of what one understands by the second statement, which I think of as really $\exists x : (P(x,y) \, \forall y)$?

Answer 1

Let $P(x,y)$ be some property on two variables $x$ and $y$.

Note that when one writes $\exists x \, \forall y \, \colon P(x,y)$, it means that “There is an $x$, such that for all $y$, $P(x,y)$ holds.”. On the other hand, when one writes $\exists x \, \colon P(x,y) \, \forall y$, it means “There is an $x$ such that $P(x,y)$ holds for all $y$.”.

They are not strictly identical (they are different expressions), but they are equivalente, in the sense of being two ways of expressing the same idea.

Although, most people don’t write $\exists x \colon P(x,y) \, \forall y$, in mathematical notation.

Answer 2

Unfortunately, sometimes people write:

there exists $x$, $y=x^2$, for all $y$

but it is ambiguous. It might mean

there exists $x$, for all $y>0$, $y=x^2$

which is false,
or it might mean

for all $y>0$ there exists $x$, $y=x^2$

which is true.