Exercise · ZW-METRIC-SPACE-009 · ★★★ Advanced
For \(P=(0,0)\) and \(Q=(3,4)\) in \(\mathbb{R}^2\), which ordering of the distances \(d_\infty\), \(d_2\) (Euclidean) and \(d_1\) (taxicab) is correct?
Choose one option.
A \(d_\infty<d_2<d_1\) B \(d_1<d_2<d_\infty\) C \(d_2<d_\infty<d_1\) D \(d_\infty<d_1<d_2\) Show answer ← Back to the question
Explanation They are \(d_\infty=4\), \(d_2=5\) and \(d_1=7\). The straight path is shorter than the taxicab one, and the coordinate maximum is smaller still.
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.
\[\begin{aligned}d(x,y)&=0\iff x=y,\qquad d(x,y)=d(y,x),\\ d(x,z)&\le d(x,y)+d(y,z)\end{aligned}\]
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