Definition:Real Interval/Half-Open/Right

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Let $a, b \in \R$ be real numbers.

The right half-open (real) interval from $a$ to $b$ is the subset:

$\hointr a b := \set {x \in \R: a \le x < b}$


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 $I$.

Some sources prefer to use $J$.

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:

\(\ds \openint a b\) \(:=\) \(\ds \set {x \in \R: a < x < b}\) Open Real Interval
\(\ds \hointr a b\) \(:=\) \(\ds \set {x \in \R: a \le x < b}\) Half-Open (to the right) Real Interval
\(\ds \hointl a b\) \(:=\) \(\ds \set {x \in \R: a < x \le b}\) Half-Open (to the left) Real Interval
\(\ds \closedint a b\) \(:=\) \(\ds \set {x \in \R: a \le x \le b}\) Closed Real Interval

The term Wirth interval notation has consequently been coined by $\mathsf{Pr} \infty \mathsf{fWiki}$.

Also defined as

Some sources, when defining a half-open real interval, require that $a < b$.

This is to eliminate the degenerate case where the interval is the empty set.

Also known as

A right half-open interval is also called:

a half-open interval on the right
a left half-closed interval
a half-closed interval on the left.

Also see

Technical Note

The $\LaTeX$ code for \(\hointr {a} {b}\) is \hointr {a} {b} .

This is a custom $\mathsf{Pr} \infty \mathsf{fWiki}$ command designed to implement Wirth interval notation.

The name is derived from half-open interval on the right.