Set-Builder Notation
Set-builder notation describes a set by stating the property or condition its elements must satisfy, written as {x | condition} or {x : condition}.
Formula
\{x \mid \text{condition on } x\}
Definition
Set-builder notation is a way to describe a group of numbers by writing a rule that all the numbers follow. It uses braces $\{ \}$ and a vertical bar $\mid$ (or a colon) that means "such that," taking the form $\{\text{variable} \mid \text{condition}\}$; it precisely describes solution sets without listing every element and is equivalent to interval notation for simple inequalities, though it is more flexible because it can describe complex conditions. Formally, $\{x \in D \mid P(x)\}$ defines the set of elements of domain $D$ satisfying predicate $P$; in axiomatic set theory (ZFC), the axiom schema of separation guarantees this set exists when $P$ is a first-order formula, and the notation generalizes to indexed families, $\{f(x) \mid x \in D\}$, which defines the image of a function.
Example
$\{x \mid x > 5\}$ is read "the set of all $x$ such that $x$ is greater than $5$," meaning every number bigger than $5$. $\{x \mid -3 \le x < 7\}$ is equivalent to $[-3, 7)$ in interval notation, and $\{x \mid x^2 < 4 \text{ and } x > 0\} = \{x \mid 0 < x < 2\} = (0, 2)$ shows how it handles more complex conditions. $\{n \in \mathbb{Z} \mid n^2 - 5n + 6 = 0\} = \{2, 3\}$, and $\{(x, y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1\}$ defines the unit circle.
Key Insight
Set-builder notation is like giving a membership rule for a club: any number that passes the rule gets in. Russell's paradox, the set of all sets that do not contain themselves, shows that unrestricted set comprehension is inconsistent; the axiom of separation in ZFC restricts set-builder notation to subsets of an existing set, resolving the paradox.