# Definition:Real Interval/Notation

## Contents

## Definition

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

### Unbounded Intervals

Some authors (sensibly, perhaps) prefer not to use the $\infty$ symbol and instead use $\to$ and $\gets$ for $+\infty$ and $-\infty$ repectively.

In Wirth interval notation, such intervals are written as follows:

\(\displaystyle \hointr a \to\) | \(:=\) | \(\displaystyle \set {x \in \R: a \le x}\) | |||||||||||

\(\displaystyle \hointl \gets a\) | \(:=\) | \(\displaystyle \set {x \in \R: x \le a}\) | |||||||||||

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

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

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

### Conventional Notation

These are the notations usually seen for real intervals:

\(\displaystyle \left ({a, b}\right)\) | \(:=\) | \(\displaystyle \set {x \in \R: a < x < b}\) | Open real interval | ||||||||||

\(\displaystyle \left [{a, b}\right)\) | \(:=\) | \(\displaystyle \set {x \in \R: a \le x < b}\) | Half-open real interval | ||||||||||

\(\displaystyle \left ({a, b}\right]\) | \(:=\) | \(\displaystyle \set {x \in \R: a < x \le b}\) | Half-open real interval | ||||||||||

\(\displaystyle \left [{a, b}\right]\) | \(:=\) | \(\displaystyle \set {x \in \R: a \le x \le b}\) | Closed real interval |

but they can be confused with other usages for this notation.

In particular, there exists the danger of taking $\paren {a, b}$ to mean an ordered pair.

### Reverse-Bracket Notation

In order to avoid the ambiguity problem arising from the conventional notation for intervals where an open real interval can be confused with an ordered pair, some authors use the **reverse-bracket notation** for open and half-open intervals:

\(\displaystyle \left ] {\, a, b} \right [\) | \(:=\) | \(\displaystyle \set {x \in \R: a < x < b}\) | Open real interval | ||||||||||

\(\displaystyle \left [ {a, b} \right [\) | \(:=\) | \(\displaystyle \set {x \in \R: a \le x < b}\) | Half-open on the right | ||||||||||

\(\displaystyle \left ] {\, a, b} \right ]\) | \(:=\) | \(\displaystyle \set {x \in \R: a < x \le b}\) | Half-open on the left |

These are often considered to be both ugly *and* confusing, and hence are limited in popularity.

Some sources use a deliberately explicit notation, along the lines:

- $I: a < x < b$ to denote $\openint a b$

- $I: a \le x \le b$ to denote $\closedint a b$

- $I: a \le x < b$ to denote $\hointr a b$

For the sake of convenience, the $I$ is often omitted, for example:

- $a < x < b$ to denote $\openint a b$

However, this notation is discouraged on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Sources

- 1963: Morris Tenenbaum and Harry Pollard:
*Ordinary Differential Equations*... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $2 \text A$: The Meaning of the Term*Set*