${\displaystyle \{n\mid \exists k\in \mathbb {Z} ,n=2k\}}$

In mathematics and more specifically in set theory, set-builder notation is a notation for specifying a set by a property that characterizes its members.

Specifying sets by member properties is allowed by the axiom schema of specification. This is also known as set comprehension and set abstraction.

Set-builder notation can be used to describe a set that is defined by a predicate, that is, a logical formula that evaluates to true for an element of the set, and false otherwise. In this form, set-builder notation has three parts: a variable, a colon or vertical bar separator, and a predicate. Thus there is a variable on the left of the separator, and a rule on the right of it. These three parts are contained in curly brackets:

or

The vertical bar (or colon) is a separator that can be read as "such that", "for which", or "with the property that". The formula Φ(x) is said to be the rule or the predicate. All values of x for which the predicate holds (is true) belong to the set being defined. All values of x for which the predicate does not hold do not belong to the set. Thus ${\displaystyle \{x\mid \Phi (x)\}}$ is the set of all values of x that satisfy the formula Φ. It may be the empty set, if no value of x satisfies the formula.

A domain E can appear on the left of the vertical bar:

or by adjoining it to the predicate: