Exercise · ZW-METRIC-SPACE-006 · ★★☆ Intermediate
Why is \(d(x,y)=|x^2-y^2|\) not a metric on \(\mathbb{R}\)?
Choose one option.
Show answer
- C. It equals \(0\) for \(x=1,\ y=-1\) although \(x\ne y\)
Explanation
For \(x=1\) and \(y=-1\) we get \(|1-1|=0\) but \(x\ne y\), so separation fails. It is symmetric and nonnegative.
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
- ZW-METRIC-SPACE-001 In ℝ^2 with the Euclidean distance, what is the distance between (1,2) and (4,6)?
- ZW-METRIC-SPACE-002 With the taxicab distance d_1(P,Q)=|p_1-q_1|+|p_2-q_2|, what is d_1((1,2),(4,6))?
- ZW-METRIC-SPACE-003 With the max distance d_∞(P,Q)=max\|p_1-q_1|,|p_2-q_2|\, what is d_∞((1,2),(4,6))?
- ZW-METRIC-SPACE-004 Let X carry the discrete metric (d(a,b)=1 if ane b and d(a,a)=0). For three distinct points a,b,c,…
- ZW-METRIC-SPACE-005 In a metric space, if d(x,y)=0, what follows?
- ZW-METRIC-SPACE-007 Let d(x,y)=(x-y)^2 on ℝ. For the points 0,1,2, what is d(0,2)-big(d(0,1)+d(1,2)big)?
- ZW-METRIC-SPACE-008 In ℝ^3 with the Euclidean distance, what is the distance between (3,4,8) and (1,1,2)?
- ZW-METRIC-SPACE-009 For P=(0,0) and Q=(3,4) in ℝ^2, which ordering of the distances d_∞, d_2 (Euclidean) and d_1 (taxic…