Definition:Real Interval/Unbounded Closed

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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 $2 \in I$.


Also see


Sources