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Symmetric Property in Metric Spaces Implied by Other Conditions of a Metric Space

  1. May 28, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that the following three conditions of a metric space imply that d(x, y)=d(y, x):

    (1) d(x, y)>=0 for all x, y in R
    (2) d(x, y)=0 iff x=y
    (3) d(x, y)=<d(x, z)+d(z, y) for all x, y, z in R

    (Essentially, we can deduce a reduced-form definition of a metric space, one without explicitly stating the reflexivity condition because the other 3 conditions imply it)

    2. Relevant equations

    The three conditions above.

    3. The attempt at a solution

    I've gone in circles, getting nowhere.
  2. jcsd
  3. May 28, 2012 #2
    I would start looking for a counterexample of what you're trying to prove.
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