# Definition:Enumeration

## Definition

### Finite Sets

Let $X$ be a finite set of cardinality $n \in \N$.

An **enumeration** of $X$ is a bijection $x: \N_n \to X$, where $\N_n = \left\{{1, \ldots, n}\right\}$.

It is common to define a **(finite) enumeration** by writing, for example:

- Let $X = \left\{{x_1, \ldots, x_n}\right\}$
*[...]*

### Countably Infinite Sets

Let $X$ be a countably infinite set.

An **enumeration** of $X$ is a bijection $x: \N \to X$.

It is common to denote an **(infinite) enumeration** by writing, for example:

- Let $X = \left\{{x_1, x_2, \ldots}\right\}$
*[...]*

or:

- Let $x_1, x_2, \ldots$ be an enumeration of $X$.
*[...]*

## Notation

In order to avoid tedious case distinctions between finite and countably infinite sets, many authors write:

- Let $X = \left\{{x_1, x_2, \ldots}\right\}$
*[...]*

implying that $X$ be countable, but not excluding the possibility that $X$ is actually finite.

Some authors use the abbreviated notation $\left\{{x_1}\right\}$ for both.

## Also see

- Definition:Sequence, of which
**enumerations**are specific examples - Definition:Countable