Representing a Set — Roster Form and Set-Builder Form

A set can be written down in two standard ways.

Roster form (tabular form) lists every element of the set explicitly, separated by commas, inside curly braces. For example, the set of even numbers between 1 and 11 in roster form is

Roster form is simple and direct, but it becomes impractical (or impossible) for a very large or infinite set.

Set-builder form (rule form) describes a set by stating a rule or property that every element must satisfy, instead of listing them. The general pattern is

read as "$A$ is the set of all $x$ such that $x$ satisfies property $P$." The same set $A$ above can be written in set-builder form as

Set-builder form is especially useful when the elements follow a clear rule but are too many (or infinite) to list — for instance, $\{x : x \in \mathbb{N}\}$ describes all of $\mathbb{N}$ in one line, something roster form could never do completely. …