Definition:Enumeration

From ProofWiki
Jump to: navigation, search

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