Definition:Real Interval/Closed

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Definition

Let $a, b \in \R$.

The closed (real) interval from $a$ to $b$ is defined as:

$\closedint a b = \set {x \in \R: a \le x \le b}$


Notation

An arbitrary (real) interval is variously denoted:

$\mathbb I$
$\mathbf I$, often by sources which use the $\textbf {boldface}$ font $\mathbf N, \mathbf Z, \mathbf Q, \mathbf R, \mathbf C$ for the number sets $\N$, $\Z$, and so on
$I$, as used by $\mathsf{Pr} \infty \mathsf{fWiki}$ by preference, although other letters are often found

When two intervals are being discussed at the same time, $J$ is a popular choice for the second to come under the spotlight.


Note that $\mathsf{Pr} \infty \mathsf{fWiki}$ prefers to reserve $\I$ for the specific case of the closed unit interval $\closedint 0 1$.


Wirth Interval Notation

The notation used on this site to denote a real interval is a fairly recent innovation, and was introduced by Niklaus Emil Wirth:

\(\ds \openint a b\) \(:=\) \(\ds \set {x \in \R: a < x < b}\) Open Real Interval
\(\ds \hointr a b\) \(:=\) \(\ds \set {x \in \R: a \le x < b}\) Half-Open (to the right) Real Interval
\(\ds \hointl a b\) \(:=\) \(\ds \set {x \in \R: a < x \le b}\) Half-Open (to the left) Real Interval
\(\ds \closedint a b\) \(:=\) \(\ds \set {x \in \R: a \le x \le b}\) Closed Real Interval


The term Wirth interval notation has consequently been coined by $\mathsf{Pr} \infty \mathsf{fWiki}$.


Also known as

A closed real interval can also be referred to as compact.


Sources whose subject matter is restricted to analysis tend to refer to a closed real interval merely as a closed interval.

However, as $\mathsf{Pr} \infty \mathsf{fWiki}$ defines such a concept on a general ordered set, the full form is required here.


Some sources do not explicitly define an open interval, and refer to a closed real interval merely as an interval.

Such imprecise practice is usually discouraged.


Examples

Example $1$

Let $I$ be the closed real interval defined as:

$I := \closedint 1 3$

Then $3 \in I$.


Also see


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

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