Definition:Enumeration/Notation

Notation for Enumeration

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.