If (X, d) and (X, r) are metric space, is {X, max(d, r)} necessary a metric space?

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Main Question or Discussion Point

If (X, d) and (X, r) are metric space, is {X, max(d, r)} necessary a metric space? what about (X, min(d, r))?
 

Answers and Replies

  • #2
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Well, what do you think?? What have you tried?
 
  • #3
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It's easy to verify d(x,y)=0 iff x=y and d(x, y)=d(y,x),
but I don't know how to prove triangle Inequality...
 
  • #4
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Well, is the following trye

[tex]d(x,z)\leq \max\{d(x,y),r(x,y)\}+\max\{d(y,z),r(y,z)\}[/tex]

???
 
Last edited:
  • #5
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yes..but if r(x,z) is greater than d(x,z), and r(y,z) is smaller than d(y,z), then this inequality is not necessary true...right?
 
  • #6
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Sorry, I made a typo, check the post again.
 
  • #7
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I see...if r(x,y) is greater than d( x,y), then d(x,z)≤max{d(x,y),r(x,y)}+max{d(y,z),r(y,z)} is also true...

Thanks ...and I think (X, min(d, r)) is not a metric space..right?
 
  • #8
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Thanks ...and I think (X, min(d, r)) is not a metric space..right?
right.
 
  • #9
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Thanks..you are really helpful
 

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