Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    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?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook