Definition:Real Interval Types
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:
| \(\ds \hointr a \to\) | \(:=\) | \(\ds \set {x \in \R: a \le x}\) | ||||||||||||
| \(\ds \hointl \gets a\) | \(:=\) | \(\ds \set {x \in \R: x \le a}\) |
Unbounded Open Interval
There are two unbounded open intervals involving a real number $a \in \R$, defined as:
| \(\ds \openint a \to\) | \(:=\) | \(\ds \set {x \in \R: a < x}\) | ||||||||||||
| \(\ds \openint \gets a\) | \(:=\) | \(\ds \set {x \in \R: x < a}\) |
Unbounded Interval without Endpoints
The unbounded interval without endpoints is equal to the set of real numbers:
- $\openint \gets \to = \R$
Other Intervals
Empty Interval
When $a > b$:
| \(\ds \closedint a b\) | \(=\) | \(\, \ds \set {x \in \R: a \le x \le b} \, \) | \(\, \ds = \, \) | \(\ds \O\) | ||||||||||
| \(\ds \hointr a b\) | \(=\) | \(\, \ds \set {x \in \R: a \le x < b} \, \) | \(\, \ds = \, \) | \(\ds \O\) | ||||||||||
| \(\ds \hointl a b\) | \(=\) | \(\, \ds \set {x \in \R: a < x \le b} \, \) | \(\, \ds = \, \) | \(\ds \O\) | ||||||||||
| \(\ds \openint a b\) | \(=\) | \(\, \ds \set {x \in \R: a < x < b} \, \) | \(\, \ds = \, \) | \(\ds \O\) |
When $a = b$:
| \(\ds \hointr a b \ \ \) | \(\, \ds = \, \) | \(\ds \hointr a a\) | \(=\) | \(\, \ds \set {x \in \R: a \le x < a} \, \) | \(\, \ds = \, \) | \(\ds \O\) | ||||||||
| \(\ds \hointl a b \ \ \) | \(\, \ds = \, \) | \(\ds \hointl a a\) | \(=\) | \(\, \ds \set {x \in \R: a < x \le a} \, \) | \(\, \ds = \, \) | \(\ds \O\) | ||||||||
| \(\ds \openint a b \ \ \) | \(\, \ds = \, \) | \(\ds \openint a a\) | \(=\) | \(\, \ds \set {x \in \R: a < x < a} \, \) | \(\, \ds = \, \) | \(\ds \O\) |
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 $1$:
| \(\ds \openint 0 1\) | \(:=\) | \(\ds \set {x \in \R: 0 < x < 1}\) | ||||||||||||
| \(\ds \hointr 0 1\) | \(:=\) | \(\ds \set {x \in \R: 0 \le x < 1}\) | ||||||||||||
| \(\ds \hointl 0 1\) | \(:=\) | \(\ds \set {x \in \R: 0 < x \le 1}\) | ||||||||||||
| \(\ds \closedint 0 1\) | \(:=\) | \(\ds \set {x \in \R: 0 \le x \le 1}\) |