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Define an equivalence relation

  1. Nov 6, 2009 #1
    If I have a subset, how do I define an equivalence relation.
    I understand it has to satisfy three properties:transitive, symmetric and reflexive, but I'm not sure how to give an explicit definition of the equivalence relation, for example on I where
    [itex]I=\{(x,y) : 0 \le x\le 1 \ \& \ 0 \le y \le 1\}[/itex]
     
  2. jcsd
  3. Nov 6, 2009 #2
    Re: equivalence

    Do you know what a cartesian product is? If you don't its a very important topic for anyone learning set theory to know.

    If you do, then an equivalence relation R from A to B is a subset of A X B. In other words an equivalence relation R contains those ordered pairs (a,b) [tex]\in[/tex] A X B such that a is related to b by R.

    In your example that equivalence relation is a subset of [tex]\Re[/tex] X [tex]\Re[/tex] consisting of those (x,y) [tex]\in[/tex] [tex]\Re[/tex] X [tex]\Re[/tex] such that 0 [tex]\leq[/tex] x [tex]\leq[/tex] 1, 0 [tex]\leq[/tex] y [tex]\leq[/tex] 1.

    Hope that makes sense to you.
     
  4. Nov 7, 2009 #3
    Re: equivalence

    HI

    That does makes sense, but I can't see how to define an explicit equivalence relation...?
     
  5. Nov 7, 2009 #4
    Re: equivalence

    I x I has the required properties, right?
     
  6. Nov 7, 2009 #5
    Re: equivalence

    sorry...I don't follow(again)
     
  7. Nov 9, 2009 #6
    Re: equivalence

    The equivalence relation you gave is a relation on the set I. I X I is the cartesian product of I with itself. Since the relation R is from I to I it is a subset of I X I. An equivalence relation is a set and can be written as such.

    Perhaps if you rephrased your question I could be of more help?
     
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