Definition:Standard Discrete Metric

Definition

The standard discrete metric on a set $S$ is the metric satisfying:

$\map d {x, y} = \begin{cases} 0 & : x = y \\ 1 & : x \ne y \end{cases}$

This can be expressed using the Kronecker delta notation as:

$\map d {x, y} = 1 - \delta_{x y}$

The resulting metric space $M = \struct {S, d}$ is the standard discrete metric space on $S$.

The discrete metric on $\R^2$ is defined as:

$\displaystyle d_0 \left({x, y}\right) := \begin{cases} 0 & : x = y \\ 1 & : \exists i \in \left\{{1, 2}\right\}: x_i \ne y_i \end{cases}$

where $x = \left({x_1, x_2}\right), y = \left({y_1, y_2}\right) \in \R^2$.

This metric is also reported in some texts as the discrete metric.

Be careful with the word discrete.

A common homophone horror is to use the word discreet instead.

However, discreet means cautious or tactful, and describes somebody who is able to keep silent for political or delicate social reasons.