Definition:Closed Interval

Definition

Let $\left({S, \preceq}\right)$ be a totally ordered set.

Let $m, n \in S$. Then the closed interval between $m$ and $n$ is denoted and defined as:

$\left[{m . . n}\right] = \begin{cases} \left\{{x \in S: m \preceq x \land x \preceq n}\right\} & : m \preceq n \\ \varnothing & : n \prec m \end{cases}$

This notation is a fairly recent innovation, and was introduced by C. A. R. Hoare and Lyle Ramshaw .

The older notation, which is more frequently seen, is $\left[{m, n}\right]$. However, it can easily be confused with other usages of the same or similar notation, so its use is deprecated.

Integer Interval

When $S$ is the set $\N$ of natural numbers or $\Z$ of integers, then $\left[{m . . n}\right]$ is called an integer interval.

Also see

• Closed Real Interval, whose definition is compatible with this.

Sources

• Seth Warner: Modern Algebra (1965): $\S 16$