Set-Builder Notation

Set-builder notation describes sets using rules instead of listing elements. For example, $\{x \mid x > 0\}$ means "all $x$ such that $x$ is greater than $0$."

Another example: $\{n^2 \mid n \in \mathbb{N}\}$ is the set of all perfect squares ($0, 1, 4, 9, 16, \ldots$). The vertical bar $\mid$ means "such that."

Set-builder notation is useful when you cannot list all elements. For infinite sets or sets defined by conditions, this notation is cleaner than trying to write out everything.