TITLE: Set builder notation, left or right of :| convention
QUESTION [19 upvotes]: Set builder notation which specify a subset such as $Z$ or $R$ tend to put this condition on the left, whereas other conditions go on the right.
$$\{ x ∈ Z : x > 0 \}$$
Why is this preferred over, say:
$$\{ x : x > 0  \textrm{ and } x ∈ Z\}$$
I notice that if the left-side has more than a simple term it tends not to have the contains expression. For example:
$$\{ x + 1 : x > 0 \}$$
In such a form could a $Z$ appear on the left, or must it appear on the right if it needs to be specified?
In what situations can the conditions go on the left, as opposed to the right of the $:$ operator?

REPLY [2 votes]: When you use set builder notation, use
$$\{x\in \hbox{Universe of discourse or subset threof }| {\rm TestForMembership}(x)\}.$$
The stuff to the left of the $|$ bounds the discussion and establishes context.