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Homework Help: Equivalence relation help

  1. Sep 22, 2005 #1
    Hey guys, wasn't sure what forum to post this in. So if this is the wrong forum, I apologize. Anyway, I have a problem in Real Analysis that I can't quite get. Here it is:

    Let f:A->B and R is a relation on A such that xRy iff f(x) = f(y).
    a.) Prove R is an equivalence relation
    b.) Show g:A->E is surjective
    c.) Show h:E->B is injective
    d.) Prove f(x) = h(g(x)).

    I solved parts a, b, and c. My problem is part d... I don't even know where to begin. It just doesn't make sense to me when I think about it. Thanks for any help.

    EDIT: I just realized I didn't put what E is. E is the equivalence classes on any particular element. So, it's the set of all equivalence classes for this function.
    Last edited: Sep 22, 2005
  2. jcsd
  3. Sep 22, 2005 #2


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    For two functions to be equal, they have to send the same element to the same image.
  4. Sep 22, 2005 #3
    So are we showing that if f(x) = h(g(x)), then g(x) = x? Here is exactly what I have written so far:

    "Proof: In order to show that two functions are equal, we must show that for any x in the domain, we will get the same output y in the codomain. So, if f(x) = x, then h(g(x)) = x as well. By the definition if being injective, x = g(x)."

    I'm lost from there.
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