# Definition:Real Interval

## Contents

## Definition

Informally, the set of all real numbers between any two given real numbers $a$ and $b$ is called a **(real) interval**.

There are many kinds of **intervals**, each more-or-less consistent with this informal definition.

### Definition 1

A **(real) interval** is a subset $I$ of the real numbers such that:

- $\forall x, y \in I: \forall z \in \R : \paren {x \le z \le y \implies z \in I}$

### Definition 2

A **real interval** is a subset of $\R$ that is one of the following real interval types:

- closed bounded interval
- open bounded interval
- left half-open bounded interval
- right half-open bounded interval
- closed and bounded on the right, also known as a closed unbounded below real interval
- open and bounded on the right, also known as an open unbounded below real interval
- closed and bounded on the left, also known as a closed unbounded above real interval
- open and bounded on the left, also known as an open unbounded above real interval
- unbounded interval without endpoints

## Terminology

### Endpoints

The numbers $a, b \in \R$ are known as the **endpoints** of the interval.

$a$ is sometimes called the **left hand endpoint** and $b$ the **right hand endpoint** of the interval.

### Length

The difference $b - a$ between the endpoints is called the **length** of the interval.

### Midpoint

The **midpoint** of a real interval is the number:

- $\dfrac {a + b} 2$

## Interval Types

It is usual to define intervals in terms of inequalities.

These are in the form of a pair of brackets, either round or square, enclosing the two endpoints of the interval separated by two dots.

Whether the bracket at either end is round or square depends on whether the end point is **inside** or **outside** the interval, as specified in the following.

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

### Bounded Intervals

#### Open Interval

The **open (real) interval** from $a$ to $b$ is defined as:

- $\openint a b := \set {x \in \R: a < x < b}$

#### Half-Open Interval

There are two **half-open (real) intervals** from $a$ to $b$.

#### Right half-open

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

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

#### Left half-open

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

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

#### Closed Interval

The **closed (real) interval** from $a$ to $b$ is defined as:

- $\closedint a b = \set {x \in \R: a \le x \le b}$

#### Bounded Interval

Let $I$ be an interval.

Let $I$ be either open, half-open or closed.

Then $I$ is said to be a **bounded (real) interval**.

### Unbounded Intervals

#### Unbounded Closed Interval

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

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

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

#### Unbounded Open Interval

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:

- $\openint \gets \to = \R$

### Other Intervals

#### Empty Interval

When $a > b$:

\(\displaystyle \left [{a \,.\,.\, b} \right]\) | \(=\) | \(\, \displaystyle \left\{ {x \in \R: a \le x \le b}\right\} \, \) | \(\, \displaystyle =\, \) | \(\displaystyle \varnothing\) | |||||||||

\(\displaystyle \left [{a \,.\,.\, b} \right)\) | \(=\) | \(\, \displaystyle \left\{ {x \in \R: a \le x < b}\right\} \, \) | \(\, \displaystyle =\, \) | \(\displaystyle \varnothing\) | |||||||||

\(\displaystyle \left ({a \,.\,.\, b} \right]\) | \(=\) | \(\, \displaystyle \left\{ {x \in \R: a < x \le b}\right\} \, \) | \(\, \displaystyle =\, \) | \(\displaystyle \varnothing\) | |||||||||

\(\displaystyle \left ({a \,.\,.\, b} \right)\) | \(=\) | \(\, \displaystyle \left\{ {x \in \R: a < x < b}\right\} \, \) | \(\, \displaystyle =\, \) | \(\displaystyle \varnothing\) |

When $a = b$:

- $\left [{a \,.\,.\, b} \right) = \left [{a \,.\,.\, a} \right) = \left\{{x \in \R: a \le x < a}\right\} = \varnothing$

- $\left ({a \,.\,.\, b} \right] = \left ({a \,.\,.\, a} \right] = \left\{{x \in \R: a < x \le a}\right\} = \varnothing$

- $\left ({a \,.\,.\, b} \right) = \left ({a \,.\,.\, a} \right) = \left\{{x \in \R: a < x < a}\right\} = \varnothing$

Such empty sets are referred to as **empty intervals**.

#### Singleton Interval

When $a = b$:

- $\closedint a b = \closedint a a = \set {x \in \R: a \le x \le a} = \set a$

#### Unit Interval

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

\(\displaystyle \openint 0 1\) | \(:=\) | \(\displaystyle \set {x \in \R: 0 < x < 1}\) | |||||||||||

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

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

\(\displaystyle \closedint 0 1\) | \(:=\) | \(\displaystyle \set {x \in \R: 0 \le x \le 1}\) |

## 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 $I$.

### 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 see

- Equivalence of Definitions of Real Interval
- Definition:Integer Interval
- Real Number Line is Metric Space

- Results about
**real intervals**can be found here.

### Generalizations

- Definition:Open Rectangle, a generalization to higher dimensional spaces
- Definition:Half-Open Rectangle
- Definition:Interval of Ordered Set

## 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*: Definition $2.1$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 6$: Subsets - 1991: Felix Hausdorff:
*Set Theory*(4th ed.) ... (next): Preliminary Remarks - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.9$: Graphing Functions - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**interval**