Exercise · ZW-METRIC-SPACE-007 · ★★☆ Intermediate
Let \(d(x,y)=(x-y)^2\) on \(\mathbb{R}\). For the points \(0,1,2\), what is \(d(0,2)-\big(d(0,1)+d(1,2)\big)\)?
Choose one option.
Show answer
- D. 2
Explanation
It is \(4-(1+1)=2>0\): the triangle inequality is violated, so \(d\) is not a metric.
Review the concept: Metric space
A metric space is a set \(X\) together with a distance \(d\) between its points: a nonnegative number for each pair, symmetric, zero only between a point and itself, and satisfying the triangle inequality.
\[\begin{aligned}d(x,y)&=0\iff x=y,\qquad d(x,y)=d(y,x),\\ d(x,z)&\le d(x,y)+d(y,z)\end{aligned}\]
More practice problems: metric space
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