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.

 A part of this page has to be extracted as a theorem:this needs to be proved

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