Describing sets of real numbers

We will often be working with sets of real numbers in this chapter. The set-builder notation allows us to use inequalities to describe sets. For example, ${x \in ℝ : 0 < x \leq 1}$ describes the set of real numbers between 0 (exclusive) and 1 (inclusive).

The interval notation

The interval notation is often a more convenient way to describe these sets of real number.

interval

We can rewrite the previous example of ${x \in ℝ : 0 < x \leq 1}$ in interval notation as $(0,1] .$ Round parenthesis, ( or ), indicate that the end value is not included while square brackets, [ or ], indicate that we include the end value.

Examples

Inequality	Interval Notation
$x < - 3$	$(- \infty, - 3)$
$x \leq 5$	$(- \infty, 5]$
$1 < x < 4$	$(1, 4)$
$1 \leq x \leq 4$	$[1, 4]$
$1 \leq x < 4$	$[1, 4)$
$1 < x \leq 4$	$(1, 4]$
$x \geq 2$	$[2, \infty)$
$x > 0$	$(0, \infty)$
$x \in ℝ, x \neq 7$	$(- \infty, 7) \cup (7, \infty)$
$x \leq - 1$ or $x > 3$	$(- \infty, - 1] \cup (3, \infty)$
$x \in ℝ$	$(- \infty, \infty)$