Definition:Real Interval/Half-Open/Left

From ProofWiki
Jump to navigation Jump to search

Definition

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


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

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


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 $I$.

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 left half-open interval is also called:

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


Also see


Technical Note

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


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Definition:Real_Interval/Half-Open/Left&oldid=690448"