Definition:Distance/Sets/Real Numbers
Definition
Let $S, T$ be a subsets of the set of real numbers $\R$.
Let $x \in \R$ be a real number.
The distance between $x$ and $S$ is defined and annotated $\ds \map d {x, S} = \map {\inf_{y \mathop \in S} } {\map d {x, y} }$, where $\map d {x, y}$ is the distance between $x$ and $y$.
The distance between $S$ and $T$ is defined and annotated $\ds \map d {S, T} = \map {\inf_{\substack {x \mathop \in S \\ y \mathop \in T} } } {\map d {x, y} }$.
Also denoted as
Some sources write $\operatorname{dist}$ instead of $d$.
Examples
Example 1
Let $S \subseteq \R$ be the subset of the set of real numbers $\R$ defined as:
- $S := \set {0, 1, 2}$
Then:
- $\map d {3, S} = 1$
Example 2
Let $S \subseteq \R$ be the subset of the set of real numbers $\R$ defined as:
- $S := \openint 0 1$
Then:
- $\map d {3, S} = 1$
Example 3
Let $S \subseteq \R$ be the subset of the set of real numbers $\R$ defined as:
- $S := \closedint 1 2$
Then:
- $\map d {3, S} = 1$
Example 4
Let $S \subseteq \R$ be the subset of the set of real numbers $\R$ defined as:
- $S := \openint 2 3$
Then:
- $\map d {3, S} = 0$
Sources
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 2$: Continuum Property: Exercise $\S 2.13 \ (4)$