Is it true that one should not write the universal quantifier behind a statement? We all know that there are different ways to say that e.g. an element $x$ belongs to each member of a family of sets $(A_j)_{j \in J}$ for some index set $J$. The most common ways I know are the following:


*

*$\forall j \in J \colon x \in A_j$

*$x \in A_j$ for all $j \in J$

*$x \in A_j \ \forall j \in J$


I think I heared that from some people's point of view the last notation is not considered as "nice". Now my question to the pros outside there is: Is it true that one should not write the universal quantifier behind a statement? Do you recommend using one of the first two notations only? I think there are even some textbooks using the last notation so I am not sure if it is seen as "bad" my most mathematicians. Maybe I am just influenced by some special opinions.

It depends whether you are writing on a blackboard or in a formal article. In a formal article, I would write

$x$ belongs to $A_j$ for all $j \in J$

to have a fluent sentence. See Halmos' recommendations on How to write mathematics.
On a blackboard, I may simply write

$x \in \bigcap_{j\in J} A_j$

