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): enumeration