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 interval is frequently denoted $\mathbb I$, although some sources use just $I$. Others use $\mathbf I$.


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:

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


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


Also known as

Such an interval can also be referred to as compact.


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