Definition:Real Interval/Unit Interval

From ProofWiki
Jump to navigation Jump to search

Definition

A unit interval is a real interval whose endpoints are $0$ and $1$:

\(\displaystyle \openint 0 1\) \(:=\) \(\displaystyle \set {x \in \R: 0 < x < 1}\)
\(\displaystyle \hointr 0 1\) \(:=\) \(\displaystyle \set {x \in \R: 0 \le x < 1}\)
\(\displaystyle \hointl 0 1\) \(:=\) \(\displaystyle \set {x \in \R: 0 < x \le 1}\)
\(\displaystyle \closedint 0 1\) \(:=\) \(\displaystyle \set {x \in \R: 0 \le x \le 1}\)


Closed

The closed interval from $0$ to $1$ is denoted $\mathbb I$ (or a variant) by some authors:

$\mathbb I := \left [{0 \,.\,.\, 1} \right] = \left\{{x \in \R: 0 \le x \le 1}\right\}$

This is often referred to as the closed unit interval.


Open

The open interval between $0$ and $1$ is referred to as the open unit interval.

$\left({0 \,.\,.\, 1}\right) = \left\{ {x \in \R: 0 < x < 1}\right\}$


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