Intervals

Quick Reference

Interval	Inequality Notation	Set Representation	Type
$[a,b]$	$a\le x\le b$	$\{x\in ℝ\mid a\le x\le b\}$	Closed
$(a,b)$			Open
$[a,b)$			Half-open
$(a,+\mathrm{\infty })$			Infinite open

Types of Intervals

The closed interval $[a,b]$ is the set of all real numbers x that satisfy the inequalities $a\le x\le b$. That is:

$$[a,b]=\{x\in ℝ\mid a\le x\le b\}$$

(The set of all real numbers is denoted by $ℝ$, and $x\in ℝ$ means that x is a real number.)

The open interval $(a,b)$ is the set of all real numbers x that satisfy the inequalities . That is,

• The numbers a and b are called the endpoints of the above intervals. In interval notation, a square bracket indicates the inclusion of the corresponding endpoint of the interval, and a parenthesis indicates its exclusion. For example, $[a,b)$ means

We can also define infinite intervals such as $(a,+\mathrm{\infty }),(-\mathrm{\infty },b]$, etc.