Definition:Real Interval/Empty

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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 $a > b$:

\(\displaystyle \left [{a \,.\,.\, b} \right]\) \(=\) \(\, \displaystyle \left\{ {x \in \R: a \le x \le b}\right\} \, \) \(\, \displaystyle =\, \) \(\displaystyle \varnothing\)
\(\displaystyle \left [{a \,.\,.\, b} \right)\) \(=\) \(\, \displaystyle \left\{ {x \in \R: a \le x < b}\right\} \, \) \(\, \displaystyle =\, \) \(\displaystyle \varnothing\)
\(\displaystyle \left ({a \,.\,.\, b} \right]\) \(=\) \(\, \displaystyle \left\{ {x \in \R: a < x \le b}\right\} \, \) \(\, \displaystyle =\, \) \(\displaystyle \varnothing\)
\(\displaystyle \left ({a \,.\,.\, b} \right)\) \(=\) \(\, \displaystyle \left\{ {x \in \R: a < x < b}\right\} \, \) \(\, \displaystyle =\, \) \(\displaystyle \varnothing\)

When $a = b$:

$\left [{a \,.\,.\, b} \right) = \left [{a \,.\,.\, a} \right) = \left\{{x \in \R: a \le x < a}\right\} = \varnothing$
$\left ({a \,.\,.\, b} \right] = \left ({a \,.\,.\, a} \right] = \left\{{x \in \R: a < x \le a}\right\} = \varnothing$
$\left ({a \,.\,.\, b} \right) = \left ({a \,.\,.\, a} \right) = \left\{{x \in \R: a < x < a}\right\} = \varnothing$


Such empty sets are referred to as empty intervals.


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