Definition:Real Interval/Empty
Definition
Let $a, b \in \R$.
Let $\left [{a \,.\,.\, b} \right]$, $\left [{a \,.\,.\, b} \right)$, $\left ({a \,.\,.\, b} \right)$ and $\left ({a \,.\,.\, b} \right)$ be real intervals: closed, half-open and open as defined.
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When $a > b$:
| \(\ds \closedint a b\) | \(=\) | \(\, \ds \set {x \in \R: a \le x \le b} \, \) | \(\, \ds = \, \) | \(\ds \O\) | ||||||||||
| \(\ds \hointr a b\) | \(=\) | \(\, \ds \set {x \in \R: a \le x < b} \, \) | \(\, \ds = \, \) | \(\ds \O\) | ||||||||||
| \(\ds \hointl a b\) | \(=\) | \(\, \ds \set {x \in \R: a < x \le b} \, \) | \(\, \ds = \, \) | \(\ds \O\) | ||||||||||
| \(\ds \openint a b\) | \(=\) | \(\, \ds \set {x \in \R: a < x < b} \, \) | \(\, \ds = \, \) | \(\ds \O\) |
When $a = b$:
| \(\ds \hointr a b \ \ \) | \(\, \ds = \, \) | \(\ds \hointr a a\) | \(=\) | \(\, \ds \set {x \in \R: a \le x < a} \, \) | \(\, \ds = \, \) | \(\ds \O\) | ||||||||
| \(\ds \hointl a b \ \ \) | \(\, \ds = \, \) | \(\ds \hointl a a\) | \(=\) | \(\, \ds \set {x \in \R: a < x \le a} \, \) | \(\, \ds = \, \) | \(\ds \O\) | ||||||||
| \(\ds \openint a b \ \ \) | \(\, \ds = \, \) | \(\ds \openint a a\) | \(=\) | \(\, \ds \set {x \in \R: a < x < a} \, \) | \(\, \ds = \, \) | \(\ds \O\) |
Such empty sets are referred to as empty intervals.
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 $\mathsf{Pr} \infty \mathsf{fWiki}$.