Weak formulations: where should the $\forall$ be placed? Shoud I write (with the appropriate assumptions on $b$ and $l$)


*

*Find $u\in U$ such that $\forall v \in V$, $b(u,v)=l(v)$


or


*

*Find $u\in U$ such that $b(u,v)=l(v)$, $\forall v \in V$


I tend to go for the first formulation but I often see the second version.
 A: A quantified statement would read: $$\exists u \in V\ \forall v \in V\ b(u,v) = l(v).$$
In plain English it would read $$\text{there exists $u \in U$ such that $b(u,v) = l(v)$ for all $v \in V$}.$$
One often sees the quantifiers $\forall$ and $\exists$ used informally in writing to stand for "for every" and "there exists" whenever these expressions occur, as with your second version.
tl;dr the first is more along the lines of a properly formed mathematical sentence but the meaning of the second is perfectly clear.
A: Unless you're really writing something about logic, you should avoid using so many logic symbols, it looks a bit ugly. I suggest you pick up any math book, and you'll see very rarely a $\forall, \exists, $ etc.
I would just write, in most cases:

Find $u\in B$ such that for every $v\in V$, it holds that $b(u,v)=l(v)$.

Sure, you can vary in style, or be a bit less verbose, but you should reserve logic symbols for shorthand (writing theorems or proofs in some lecture you're giving, so you can write faster(in the cases that you happen to write this symbols, the $\forall$ at the end or in the middle, is perfectly fine (provided you're clear with the order of the quantifiers, etc))).

If you really want to write formulas in a true formal manner, see this.
