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Equivalent distances questions

  1. Jun 11, 2010 #1
    Hello,
    two distances [itex]d_1[/itex] and [itex]d_2[/itex] are said to be equivalent if for any two pairs (a,b) and (c,d)

    [tex]d_1(a,b)=d_1(c,d) \Leftrightarrow d_2(a,b)=d_2(c,d)[/tex]

    How can I (dis)prove that:

    [tex]d_1(a,b)<d_1(c,d) \Rightarrow d_2(a,b)<d_2(c,d)[/tex]
     
  2. jcsd
  3. Jun 11, 2010 #2

    HallsofIvy

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    Use "trichotomy". For any two real numbers, x and y, one and only one of these must apply:
    1) x= y
    2) x< y
    3) y< x.

    If [itex]d_1(a, b)< d_1(c, d)[/itex] then it is NOT possible that [itex]d_1(a,b)= d_1(c, d)[/itex] nor that [itex]d_1(c, d)> d_1(a, b)[/itex] which implies the same for [itex]d_2(a, b)[/itex] and [itex]d_2(c, d)[/itex].
     
  4. Jun 11, 2010 #3
    There is still something bugging my mind.
    I think the part of your proof that is giving me troubles is this:
    How did you prove that "it implies the same for d2(a,b) and d2(c,d)"

    Thanks in advance.
     
    Last edited: Jun 11, 2010
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