Definition:Real Interval/Unbounded Open

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Definition

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

\(\displaystyle \openint a \to\) \(:=\) \(\displaystyle \set {x \in \R: a < x}\)
\(\displaystyle \openint \gets a\) \(:=\) \(\displaystyle \set {x \in \R: x < a}\)

Using the same symbology, the set $\R$ can be represented as an unbounded open interval with no endpoints:

$\openint \gets \to = \R$


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


Also denoted as

The notation using $\infty$ is usual:

\(\displaystyle \openint a \infty\) \(:=\) \(\displaystyle \set {x \in \R: a < x}\)
\(\displaystyle \openint {-\infty} a\) \(:=\) \(\displaystyle \set {x \in \R: x < a}\)
\(\displaystyle \openint {-\infty} \infty\) \(:=\) \(\displaystyle \R\)

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


Also see


Sources