Definition:Closed Interval/Integer Interval

Definition

Let $S$ be the set $\N$ of natural numbers or $\Z$ of integers.

Let $\struct {S, \le}$ be the totally ordered set formed from $S$ and the usual ordering $\le$ on numbers.

Let $m, n \in S$.

The integer interval between $m$ and $n$ is denoted and defined as:

$\quad \closedint m n = \begin{cases} \set {x \in S: m \le x \le n} & : m \le n \\ \O & : n < m \end{cases}$ where $\O$ is the empty set.

Also denoted as

Some authorities consider $\closedint m n$ to be an abuse of notation, as there is nothing in it intrinsically to distinguish it from the closed real interval $\set {x \in \R: m \le x \le n}$.

Such sources prefer to use the more conventional $\set {m, m + 1, \ldots, n}$, but then again it relies upon the implicit understanding that the domain is the set of integers.

The context will frequently be sufficient to allow the reader to determine whether $\closedint m n$ is to be interpreted as $\set {x \in \Z: m \le x \le n}$ or $\set {x \in \R: m \le x \le n}$, but it is recommended that the convention be specifically defined when it is used.

The compact and sturdy $\overline {a, b}$ has just appeared before the author of this edit, who wonders whether to promote it as a standard notation on $\mathsf{Pr} \infty \mathsf{fWiki}$.

Technical Note

The $\LaTeX$ code for \(\closedint {a} {b}\) is \closedint {a} {b} .

This is a custom $\mathsf{Pr} \infty \mathsf{fWiki}$ command designed to implement Wirth interval notation.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers