Definition:Real Interval/Open

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Definition

Let $a, b \in \R$.

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

$\openint a b := \set {x \in \R: a < x < b}$


Notation

An arbitrary (real) interval is frequently denoted $\mathbb I$.

Sources which use the $\textbf {boldface}$ font for the number sets $\N, \Z, \Q, \R, \C$ tend also to use $\mathbf I$ for this entity.

Some sources merely use the ordinary $\textit {italic}$ font $I$.

Some sources prefer to use $J$.


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 $\mathsf{Pr} \infty \mathsf{fWiki}$.


Examples

Example $1$

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

$I := \openint 1 2$

Then $3 \notin I$.


Example $2$

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

$I := \openint 0 2$

Then $2 \notin I$.


Also see


Technical Note

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

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


Sources

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