Set-Builder Notation
Pre-AlgebraSet-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 | that means "such that."
Example
$\{x \mid x > 5\}$ is read "the set of all $x$ such that $x$ is greater than $5$." It means every number bigger than $5$.
Key Insight
Set-builder notation is like giving a membership rule for a club: any number that passes the rule gets in.
Definition
Set-builder notation has the form $\{\text{variable} \mid \text{condition}\}$, where the vertical bar ($\mid$) or colon ($:$) means "such that." It precisely describes solution sets without listing every element. It is equivalent to interval notation for simple inequalities.
Example
$\{x \mid -3 \le x < 7\}$ is equivalent to $[-3, 7)$ in interval notation. For equations: $\{x \mid x = 4\} = \{4\}$. For "no solution": the empty set is written as $\{\}$ or the empty set symbol $\emptyset$.
Key Insight
Set-builder notation is more flexible than interval notation because it can describe complex conditions: $\{x \mid x^2 < 4 \text{ and } x > 0\} = \{x \mid 0 < x < 2\} = (0, 2)$.
Definition
Set-builder notation $\{x \in D \mid P(x)\}$ 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. The notation generalizes to indexed families: $\{f(x) \mid x \in D\}$ defines the image of a function.
Example
$\{n \in \mathbb{Z} \mid n^2 - 5n + 6 = 0\} = \{2, 3\}$. $\{(x, y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1\}$ defines the unit circle. $\{p \in \mathbb{Z}^+ \mid p \text{ is prime and } p < 10\} = \{2, 3, 5, 7\}$.
Key Insight
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.