Set-builder notation

Summary

Set-builder notation describes a set by a property of its elements. For a set $X$ and a predicate $P(x)$ on $X$, the set $\{x \in X \mid P(x)\}$ has as elements exactly the $a \in X$ such that $P(a)$ is true.

Remarks

A colon is a common alternative to the bar: $\{x \in X : P(x)\}$. Example: $\{n \in \mathbb{Z} \mid n \text{ is even}\}$ is the set of even integers. The form $\{f(x) \mid x \in X\}$ denotes the set of the values $f(x)$. This is the image of $X$ under $f$. Without the bound $x \in X$, the notation $\{x \mid P(x)\}$ can lead to Russell's paradox.

Explanations

L1 human

Instead of a list of elements, you give a test. $\{x \in \mathbb{R} \mid x^2 = 4\}$ means "all real numbers whose square is $4$". This set is $\{-2, 2\}$.