Definition:Real Interval Types

From ProofWiki
Jump to: navigation, search

Definition

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 $a = b$:

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


Unit Interval

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

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


Also see