Representing a Set — Roster Form and Set-Builder Form

A set can be written down in two standard, fully interchangeable ways.

Roster form (tabular form) lists every element explicitly, separated by commas, inside curly braces. The set of odd numbers between 1 and 15 in roster form is

It is direct and easy to read, but becomes clumsy — or outright impossible — once a set grows very large or infinite.

Set-builder form (rule form) instead states the property every member must satisfy:

read "$A$ is the set of all $x$ such that $x$ satisfies $P$." The same set $A$ above can be written

Set-builder form earns its keep whenever a rule is clear but the list of elements is long or infinite — $\{x : x \in \mathbb{N}\}$ names the whole of $\mathbb{N}$ in a single line, something roster form could never manage completely. …