Complement of Interval Defined by Absolute Value
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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.