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I'm working on this problem:

Ift(P,Q)=max(|x1 - x2|,|y1 - y2|),show that t is a metric for the set of all ordered pairs of real numbers.

I have proved the first three parts of the definition of a metric

1) t(P,Q) >0

2) t(P,Q) =0 IFF P=Q

3) t(P,Q) = t(Q,P)

all not so hard.

I'm having trouble getting started on the 4th part:

4) t(P,Q) <= t(P,R) + t(R,Q)

My confusion lies in where, exactly, I'm trying to go.

I wrote:Let R=|z1-z2|

It appears that I need to go for something that looks like this:

max(|x1 - x2|,|y1 -y2|)<=max(|x1 - x2|,|z1-z2|)+max(|y1 -y2|,|z1-z2|)

So I did this:

t(P,Q)=max(|x1 - x2|,|y1 -y2|)=max(|x1-z1+z1-x2|,|y1+z2-z2+y2|)

now, I could group x1-z1 and z1-x2, together and do the same thing with the second part, but I don't see how this is going to get me any closer to proving that this is a metric. I know I'm just having a block on something. What I have done doesn't seem right, and I don't know which way to go.

Any nudges in the right direction will be greatly appreciated.

CC

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# Homework Help: Metric spaces

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