Definition:Real Interval/Unbounded Closed

Definition

There are two unbounded closed intervals involving a real number $a \in \R$, defined as:

 $\displaystyle \hointr a \to$ $:=$ $\displaystyle \set {x \in \R: a \le x}$ $\displaystyle \hointl \gets a$ $:=$ $\displaystyle \set {x \in \R: x \le a}$

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 $I$.

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 denoted as

The notation using $\infty$ is usual:

 $\displaystyle \hointr a \infty$ $:=$ $\displaystyle \set {x \in \R: a \le x}$ $\displaystyle \hointl {-\infty} a$ $:=$ $\displaystyle \set {x \in \R: x \le a}$

On $\mathsf{Pr} \infty \mathsf{fWiki}$ the $\gets \cdots \to$ notation is preferred.

Also known as

Some sources refer to these as half-open infinite (real) intervals.

Examples

Example $1$

Let $I$ be the unbounded closed real interval defined as:

$I := \hointl \gets 3$

Then $2 \in I$.