Finite set

Definition · applies to sets

High school

A set of items is finite if you could count its elements completely, finishing at some number.

Undergraduate

Consider a set $S$. The set $S$ is finite if $S$ is empty or in bijection with $\setdef{1, 2, \dots, n}$ for some $n \in \nats$.

Examples

The empty set $\emptyset$ (definition) High school
The empty set is finite, because it has $0$ elements, and $0$ is a finite number.
$\setdef{1, 2, 3}$ High school
$\setdef{1, 2, 3}$ is finite because it has $3$ elements.

Things that may be mistaken for examples.

$[0, 1]$ High school
$[0, 1]$ is not finite, because it has infinitely many elements.
$\nats$ High school
$\nats$ is not finite, because the list does not end.
$(0, 1)$ High school
$(0, 1)$ is not finite, because it contains $1/n$ for every $n \geq 2$.
$\mathbb{Q} \cap [0, 1]$ High school
$\mathbb{Q} \cap [0, 1]$ is not finite, because it contains $1/n$ for every $n \geq 1$.
The set of real numbers, $\reals$ High school
The set of all subsets of $\nats$ Undergraduate
The set of all subsets of $\nats$ is not finite, because it is uncountable.
The Cantor set Undergraduate
The Cantor set is not finite, because it is uncountable.

For a number, “finite” means “a real number, not $\pm \infty$,” which is unrelated.