Metric problems

Question

I am taking the GRE in less than 10 days, and I have never taken analysis. And I would like to tackle metric problems and I was wondering if anyone could show me a certain strategy to solve problems as the following.

For every set $S$ and every metric $d$ on $S$, which of the following is a metric on $S$ ?

(A)$4 + d$

(B)$e^d-1$

(C)$d-|d|$

(D)$d^2$

(E)$\sqrt d$

I studied on my own and this is what I know about metrics.

If $\delta$ is a metric on $S$ and $x,y,z \in S$

a), $\delta (x,y) \ge 0 ,\forall x,y \in S$

b), $\delta (x,y) + \delta (y,z) \ge \delta (x,z) , \forall x,y,z \in S$

c), $\delta (x,x) = 0 ,\forall x \in S$

I think that (A) is not an answer because $4+d (x,x) = 4 \neq 0$.

For the rest, I have no idea how to proceed...

is this GRE question? – dato datuashvili Apr 12 '13 at 08:25 — dato datuashvili, Apr 12 '13 at 08:25
Yes, this is a GRE question.
d is undefined.
– hyg17 Apr 12 '13 at 19:46 — hyg17, Apr 12 '13 at 19:46

Paul · Accepted Answer · 2013-04-13T00:30:14.500

1

Note the metric $\delta$ must satisfies: $\delta(x,y)=\delta(y,x).$

C is. As $0 \le d$, then $d-|d|=d-d=0$. It satisfies a),b),c).

edited Apr 13 '13 at 00:30

answered Apr 12 '13 at 09:20

Paul

20,553

Thanks for the help, but I cannot explain why the others are not. – hyg17 Apr 12 '13 at 19:48
The others don't satisfy condition b). – Paul Apr 13 '13 at 00:32
Thank you, you've been so helpful. – hyg17 Apr 13 '13 at 06:14
YOU are welcome. – Paul Apr 13 '13 at 06:16

Metric problems

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