Definition – Intervals of real numbers.

Definition

Let $x,a,b\in{\mathbb{R}}$ with $a\leq{b}$ . We define the following intervals to be sets as follows,

$\begin{aligned} &{[a, b]} = \{x\in{\mathbb{R}}~|~a\leq{x}\leq{b}\}, ~~~ {(a, b]} = \{x\in{\mathbb{R}}~|~a<x\leq{b}\} \\\\ &{(a, b)} = \{x\in{\mathbb{R}}~|~a < x < b\},~~~{[a, b)} = \{x\in{\mathbb{R}}~|~a \le{x}<b\}\end{aligned}$

Notes

Note how we require $a\leq{b}$ . This is because $a>b$ would cause a contradiction in the conditions for $x$ to be in any of the above intervals.

The intervals [a, b] and (a, b)

We call the interval ${[a, b]}$ a closed interval, and the interval ${(a, b)}$ a open interval. The sets ${[a, b)}$ and ${(a, b]}$ are called half open, or half closed intervals.