# 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 $\N_n = \set {1, \ldots, n}$.

### Countably Infinite Sets

Let $X$ be a countably infinite set.

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

## Also defined as

Some sources define an **enumeration** as starting from $0$ rather than $1$.

Hence for a **finite enumeration**, one would write:

- Let $X = \set {x_0, x_1, \ldots, x_{n - 1} }$

and for an **infinite enumeration**:

- Let $X = \set {x_0, x_1, x_2, \ldots}$

It is unimportant which convention is used.

## Notation

A **finite enumeration** would usually be denoted as:

- Let $X = \set {x_1, \ldots, x_n}$.

An **infinite enumeration** would usually be denoted either as:

- Let $X = \set {x_1, x_2, \ldots}$

or:

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

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

- Let $X = \set {x_1, x_2, \ldots}$

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

Some authors use the abbreviated notation $\set {x_k}$ for both.

## Also see

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

## Sources

- 1989: George S. Boolos and Richard C. Jeffrey:
*Computability and Logic*(3rd ed.) ... (previous) ... (next): $1$ Enumerability - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**enumeration**