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

  1. Nov 17, 2011 #1
    If (X, d) and (X, r) are metric space, is {X, max(d, r)} necessary a metric space? what about (X, min(d, r))?
     
  2. jcsd
  3. Nov 17, 2011 #2

    micromass

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

    Well, what do you think?? What have you tried?
     
  4. Nov 17, 2011 #3
    Re: If (X, d) and (X, r) are metric space, is {X, max(d, r)} necessary a metric space

    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...
     
  5. Nov 17, 2011 #4

    micromass

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

    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: Nov 17, 2011
  6. Nov 17, 2011 #5
    Re: If (X, d) and (X, r) are metric space, is {X, max(d, r)} necessary a metric space

    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?
     
  7. Nov 17, 2011 #6

    micromass

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

    Sorry, I made a typo, check the post again.
     
  8. Nov 17, 2011 #7
    Re: If (X, d) and (X, r) are metric space, is {X, max(d, r)} necessary a metric space

    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?
     
  9. Nov 17, 2011 #8

    micromass

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

    right.
     
  10. Nov 17, 2011 #9
    Re: If (X, d) and (X, r) are metric space, is {X, max(d, r)} necessary a metric space

    Thanks..you are really helpful
     
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