# Complement of Interval Defined by Absolute Value

## Theorem

Let $\xi, \delta \in \R$ be real numbers.

Let $\delta > 0$.

Then:

### Complement of Open Interval Defined by Absolute Value

$\set {x \in \R: \size {\xi - x} \ge \delta} = \R \setminus \openint {\xi - \delta} {\xi + \delta}$

where:

$\openint {\xi - \delta} {\xi + \delta}$ is the open real interval between $\xi - \delta$ and $\xi + \delta$
$\setminus$ denotes the set difference operator.

### Complement of Closed Interval Defined by Absolute Value

$\set {x \in \R: \size {\xi - x} > \delta} = \R \setminus \closedint {\xi - \delta} {\xi + \delta}$

where:

$\closedint {\xi - \delta} {\xi + \delta}$ is the closed real interval between $\xi - \delta$ and $\xi + \delta$
$\setminus$ denotes the set difference operator.