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Equivalnce relation proof

  1. Mar 24, 2012 #1
    I am trying to prove this as I am practicing for a test but I am pretty much clueless on this problem:

    Prove that if ~ is an equivalence relation on a set s and [a] denotes the equivalence class of a in s under ~, then a ~ b if and only if [a] = .


    If anyone can give me some points on how to approach or start this problem it would be great. Thanks.
     
  2. jcsd
  3. Mar 24, 2012 #2


    It falls directly out of the definition of equivalence relation, so it's tricky to think of a hint. But what happens if [a] [itex]\neq[/itex] ? Then there must either be an element in ____ that's not in ______ or vice versa. Then what?
     
  4. Mar 25, 2012 #3
    So would it be safe to assume that an equivalence relation ~ on a set s is a relation satisfying a,b in s. If [a] != , then there must be an element in a that's not in b or vice versa. Therefore, this contradicts that [a] = .


    Is this close? I am pretty much clueless on this proof.
     
  5. Mar 25, 2012 #4
    No. You need to go back to your book and read what an equivalence relation is.

    Aren't a and b assumed to be elements of s? So a,b in s is true of all a and b in s. Has nothing to do with equivalence relations.




    Well yes, if you assume [a] != then that contradicts [a] = . Isn't that always the case no matter what [a] and are?

    I think you need to read your text and/or class notes to understand what an equivalence relation is.
     
    Last edited: Mar 25, 2012
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