$\forall$ At the beginning or at the end?

Question

I have a set of real numbers $x_1, x_2, \ldots, x_n$ and two functions $f:\mathbb{R} \rightarrow \mathbb{R}$ and $g:\mathbb{R} \rightarrow \mathbb{R}$.

What are the differences between the following statements?

1. $\forall v \in \{1, \ldots, n\} ~f(x_v) = 0 \vee g(x_v) = 0$
2. $f(x_v) = 0 \vee g(x_v) = 0 ~\forall v \in \{1, \ldots, n\} ~$

Answer 1

The second one is simply wrong, that's not how things should be written.

It's unambiguous here because there is only one quantifier, but if had been something like $\exists yP(x,y)\forall x$ you wouldn't know whether $\forall x\exists yP(x,y)$ or $\exists y\forall xP(x,y)$ was meant. In my experience, the second option is what's usually meant. But I've seen it mean the first one too.

I reiterate, the second one is wrong, the fact that it's not really ambiguous (in some sense), doesn't make it right.

I personally find it acceptable to put quantifiers in the end only if you're mixing natural language with mathematical symbols. For example, "$\exists yP(x,y)$ for all $x$" is of the form "$Q(x)$ for all $x$" which should be translated to "$\forall xQ(x)$", that is, "$\forall x\exists yP(x,y)$".
What's the difference between this and the second (candidate) statement in the question? I didn't use '$\forall$', I actually wrote "for all".

Answer 2

Translate the two statements into English:

1. For all v, either $f(x_v) = 0$ or $g(x_v) = 0$.

2. Either $f(x_v) = 0$ or $g(x_v) = 0$, for all v.

The two statements are equivalent. The difference is that the emphasis that this is is true for all $v$ in the set $\{1,\dots,n\}$ has been moved.