Q:

Quick Set Builder Notation Question (Universal Quantifier)

https://en.wikipedia.org/wiki/Linear_equation
I have a question about notation of the set $S=\{(x, mx+y_o)|\forall x\in \mathbb{R}\}$ in the link above.
Why do they put the universal quantifier here? Shouldn't the set $S$ instead be $S=\{(x, mx+y_o)|x\in \mathbb{R}\}$? 

A:

At best, it seems a bit redundant. The way you would read that statement is, 
"the set of solutions is all the ordered pairs such that $x\in\mathbb{R}$ and $y=mx+y_0$"
So there is already a universal quantifier in the statement (for all ordered pairs such that). At worst I'd say it's grammatically incorrect (not a well-formed formula).

