 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-47/4/b/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

For fixed

p \geq 1

, the complement of the simple matching coefficient is the fraction of coordinates that disagree:

1 - S_{1} (x, y) = \frac{1}{p} \sum_{r = 1}^{p} 1 {x_{r} \neq = y_{r}} .

This is the normalized Hamming distance. It is nonnegative, symmetric, and is zero exactly when every coordinate agrees. At each coordinate,

1 {x_{r} \neq = z_{r}} \leq 1 {x_{r} \neq = y_{r}} + 1 {y_{r} \neq = z_{r}} .

Summing and dividing by

p

proves the triangle inequality. Therefore $d_{1}$ is a metric on the binary vectors.

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