Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 47 / 4 / b

A metric dissimilarity must satisfy, for all $x,y,z$, the four propertiesThey are nonnegativity, identity of indiscernibles, symmetry and the triangle inequality. The following two transformations of the simple matching coefficient behave differently at the identity.

For fixed $p\ge 1$, the complement of the simple matching coefficient is the fraction of coordinates that disagree:This is the normalized Hamming distance. It is nonnegative, symmetric, and is zero exactly when every coordinate agrees. At each coordinate,Summing and dividing by $p$ proves the triangle inequality. Therefore $d_1$ is a metric on the binary vectors.

For every binary vector $x$, $S_1(x,x)=1$, and henceThus $d_2$ is not a metric, because it fails identity of indiscernibles. It is nonnegative and symmetric. In fact it also satisfies the triangle inequality trivially: every value lies in $[1/2,1]$, so $d_2(x,z)\le 1\le d_2(x,y)+d_2(y,z)$. That does not repair its nonzero self-distance.