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Reflexivity Implies Symmetry?

  1. May 20, 2014 #1
    1. The problem statement, all variables and given/known data
    Is this relation, R, on ## S= \{ 1, 2, 3 \} \\ R = \{ (1,1), (2,2) , (3,3) \}##

    Symmetric?
    It is obvious that it is reflexive.
     
  2. jcsd
  3. May 20, 2014 #2
    Nevermind. I just read somewhere that reflexive statements don't count towards symmetry. Apparently, it involves something like a diagonal class; I guess they pair this combinations in a matrix like form.

    Anyway. Thanks.
     
  4. May 20, 2014 #3

    HallsofIvy

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    The relation R= {(1, 1), (2, 2), (3, 3), (1, 3)} is "reflexive" but not "symmetric" so reflexive does not "imply" symmetry. However, in this case there is no (x, y) in the relation without a corresponding (y, x) so this particular example is both reflexive and symmetric.
     
  5. May 20, 2014 #4

    pasmith

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    The relation R can be described as "xRy if and only if x = y". Thus R is an equivalence relation because equality is an equivalence relation. Hence R is reflexive, symmetric and transitive.
     
  6. May 20, 2014 #5
    Wait, so my R is an equivalence relation then? Supposedly, it partitions the set into disjoint classes. I guess that my classes would be [1], [2], [3]?

    HallsofIvy, thank you for the clarification. I should have stated that in this case, it means the same.
     
  7. May 20, 2014 #6

    micromass

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    Yes.
     
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