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


*

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

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


Translate the two statements into English:


*

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

*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.
