Definition 2 (Inclusion criterion). Where C is a condition, {x : C} denotes the set whose elements are exactly the objects satisfying C, and C is the inclusion criterion for that set.
6Inclusion criterion notation
The other common way to write down a set is to give a condition which determines whether or not any given element is a member of that set. This generally involves writing a variable (often x), followed by a colon, followed by the condition that the variable needs to satisfy in order to be a member of the set. (Some authors prefer to write a vertical line, ‘∣’, in place of the colon.)
For example, we might write down {x : x is an even number}, which denotes the set which contains the even numbers and nothing else. This is particularly useful when we need to write down sets with infinitely many members, which is more common than you might think.
The condition itself—what an object has to satisfy in order to be an element of the set—is what I will call the inclusion criterion for that set:
So the inclusion criterion for {x : x is an even number} is that x is an even number.