Definition:Real Interval

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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:

$\forall x, y \in I : \forall z \in \R : (x < z < y \implies z \in I)$


Definition 2

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


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:

\(\displaystyle \hointr a \to\) \(:=\) \(\displaystyle \set {x \in \R: a \le x}\) $\quad$ $\quad$
\(\displaystyle \hointl \gets a\) \(:=\) \(\displaystyle \set {x \in \R: x \le a}\) $\quad$ $\quad$


Unbounded Open Interval

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

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

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 $a > b$:

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

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 $a = b$:

$\left [{a \,.\,.\, b} \right] = \left [{a \,.\,.\, a} \right] = \left\{{x \in \R: a \le x \le a}\right\} = \left\{{a}\right\}$


Unit Interval

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

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


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:

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


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


Also see


Generalizations


Sources