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A symmetric, transitive relation on a set that is not reflexive

  1. Sep 5, 2010 #1
    Can someone give an example of one? I can't think of one...
     
  2. jcsd
  3. Sep 5, 2010 #2

    Landau

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    Let X={a,b} (a and b distinct). Define the relation R on X by R={(a,a)}. Then R is symmetric and transitive, but not reflexive on X since (b,b) is not in R.

    The point is that reflexivity involves a set ("reflexive on X": FOR ALL x in X we must have (x,x) in R), but symmetry and transivity are defined by means of an implication (IF ... is in R, THEN ... is in R).
     
  4. Sep 6, 2010 #3
    Perfect. I think I understand now. Thank you.
     
  5. Sep 6, 2010 #4
    You could also take the empty relation on a nonempty set.
     
  6. Dec 15, 2010 #5
    I think a good practical example of a relation with these properties is floating point equality - all floating point numbers equal themselves, but NaN != NaN, so the relation is not truly reflexive.
     
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