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Homework Help: Prove that d is a METRIC

  1. Mar 8, 2010 #1
    1. The problem statement, all variables and given/known data
    Let (X,ρ) and (Y,σ) be metric spaces.
    Define a metric d on X x Y by d((x1,y1),(x2,y2))=max(ρ(x1,x2),σ(y1,y2)).
    Verify that d is a metric.

    2. Relevant equations

    3. The attempt at a solution
    I proved positive definiteness and symmetry, but I am not sure how to prove the "triangle inequality" property of a metric. How many cases do we need in total, and how can we prove it?

    Any help is appreciated!
     
  2. jcsd
  3. Mar 8, 2010 #2
    So to verify the triangle inequality, we need to prove that
    max(ρ(x1,x2),σ(y1,y2))≤ max(ρ(x1,x3),σ(y1,y3)) + max(ρ(x3,x2),σ(y3,y2)) for ANY (x1,y1),(x2,y2),(x3,y3) in X x Y.

    How many separate cases do we need? I have trouble counting them without missing any...Is there a systematic way to count?

    Case 1: max(ρ(x1,x2),σ(y1,y2))=ρ(x1,x2), max(ρ(x1,x3),σ(y1,y3))=ρ(x1,x3), max(ρ(x3,x2),σ(y3,y2)) =ρ(x3,x2)

    This case is simple, the above inequality is true since ρ is a metric.


    Case 2: max(ρ(x1,x2),σ(y1,y2))=ρ(x1,x2), max(ρ(x1,x3),σ(y1,y3))=σ(y1,y3), max(ρ(x3,x2),σ(y3,y2)) =ρ(x3,x2)

    For example, how can we prove case 2?


    Any help is appreciated!
     
    Last edited: Mar 8, 2010
  4. Mar 8, 2010 #3
    Suppose that [tex]\rho(x_1,x_2)\ge\sigma(y_1,y_2)[/tex]. What do you know about [tex]\rho(x_1,x_3)+\rho(x_3,x_2)[/tex]? Can you infer anything about the right-hand side of your inequality based on that?
     
  5. Mar 8, 2010 #4
    Case 2: max(ρ(x1,x2),σ(y1,y2))=ρ(x1,x2), max(ρ(x1,x3),σ(y1,y3))=σ(y1,y3), max(ρ(x3,x2),σ(y3,y2)) =ρ(x3,x2)

    We'll have ρ(x1,x3)+ρ(x3,x2) ≥ σ(y1,y2)).

    But I think for case 2, we need to prove that ρ(x1,x2)≤σ(y1,y3)+ρ(x3,x2) instead?? How can we prove it?

    Thanks!
     
  6. Mar 13, 2010 #5
    Still confused...please help...
     
  7. Mar 13, 2010 #6
    Okay, you need to show, for all points, that

    [tex]\max(\rho(x_1,x_2),\sigma(y_1,y_2))\le\max(\rho(x_1,x_3),\sigma(y_1,y_3))+\max(\rho(x_3,x_2),\sigma(y_3,y_2))[/tex].

    Suppose [tex]\rho(x_1,x_2)\ge\sigma(y_1,y_2)[/tex]. Since [tex]\rho[/tex] is a metric, you know that [tex]\rho(x_1,x_3)+\rho(x_3,x_2)\ge\rho(x_1,x_2)[/tex].

    So what do you know about [tex]\max(\rho(x_1,x_3),\square)+\max(\rho(x_3,x_2),\square)[/tex], regardless of what's in the squares? You know it's at least as big as [tex]\rho(x_1,x_2)[/tex].
     
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