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Equivalence Relations

  1. Jun 17, 2012 #1
    1. The problem statement, all variables and given/known data
    Which of these relations on {0, 1, 2, 3} are equivalence relations? Determine the properties of an equivalence relation that the others lack

    a) { (0,0), (0,2), (2,0), (2,2), (2,3), (3,2), (3,3) }

    This one is not reflexive

    2. Relevant equations
    I understand that reflective means a=a, but I don't understand how this one isn't. I think the real issue here is that I obviously don't understand exactly what reflexive really means.

    Any help would be GREAT as I have an exam tomorrow morning and this is proving to be more difficult than I expected.
     
  2. jcsd
  3. Jun 17, 2012 #2

    vela

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    For the relation R to be reflexive, you must have aRa for all a in {0, 1, 2, 3}.
     
  4. Jun 17, 2012 #3
    I apologize, but can you spell it out for me? I guess I don't understand why (1,1) is the problem, but not (1,0) and (0,1).

    Thanks!
     
  5. Jun 18, 2012 #4

    vela

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    Do you understand what the ordered pair (1,0) means in the context of relations?
     
  6. Jun 18, 2012 #5
    I think it means, in order to me an Equivalence Relation, there must also exist (0,1). Correct?
     
  7. Jun 18, 2012 #6

    vela

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    Why would it mean that?
     
  8. Jun 18, 2012 #7
    As an exercise, try finding the smallest set containing the above, which is also an equivalence relation. This idea, the completion of a set, is a pervasive one in advanced maths.
     
  9. Jun 18, 2012 #8

    vela

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    Let ##a, b \in X## and ##R \subset X\times X##. When you say ##(a,b)\in R##, it means aRb, that is, a is related to b.

    For a relation R to be reflexive, you must have that for every element a in X, aRa or, in ordered-pair notation, ##(a,a) \in R##. Do you see now why your problem's R isn't reflexive?
     
  10. Jun 18, 2012 #9
    Reflexive doesn't mean a = a. The equality is a relation of equivalence, but a relation of equivalence need not be "=".
     
  11. Jun 18, 2012 #10

    HallsofIvy

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    Reflexive means "if a is in the set, then (a, a) must be in the relation". 1 is in the set. Is (1, 1) in the relation?
     
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