Discussion
Let X be a set with at least two elements. Let \(d_1\), \(d_2\) and \(d_3\) be metrics on X. Which one of the following is NOT a metric on X?
Answer: B
The one that is not a metric is \(d(x, y) := \max(d_2(x, y), 3)\).
A metric must give \(d(x, x) = 0\). Here \(d(x, x) = \max(0, 3) = 3\), so it fails at once.
The others are metrics. Truncating a metric with \(\min(d_1, 3)\) keeps the triangle inequality. \(\frac{10 d_3}{1 + d_3}\) is a positive multiple of the standard bounded metric \(\frac{d}{1+d}\), built from an increasing, subadditive function. An average of metrics, \(\frac{1}{3}(d_1 + d_2 + d_3)\), is again a metric, since each property survives addition and scaling.
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