NEC Research Institute Technical Report | 1991/1992/2002 |

**Abstract: **
We show that symmetric set difference and Euclidian distance
have particular $[0-1]$ {\em normalized} forms that remain
metrics.

The first of these $|A \triangle B| / |A \cup B|$ is easily established and generalizes to measure spaces.

The second applies to vectors in $\bbR^n$ and is given by $\|X-Y\|/(\|X\|+\|Y\|)$. That this is a metric is more difficult to demonstrate and is true for Euclidian distance (the $L_2$ norm) but for no other integral Minkowski metric. The short and elegant proof we give is due to David Robbins and Marshall Buck \cite{RB93}.

We also explore a number of variations.

**Keywords: **
Metric Space,
Distance Function,
Similarity Function/Coefficient,
Euclidian Distance,
Association,
Clustering,
Vector Quantization,
Pattern Recognition,
Statistical Methods.

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