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Math Help - Important Equivalence Relation Problem

  1. #1
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    Important Equivalence Relation Problem

    Hi! I really need help on this one problem:

    Suppose that R1 and R2 are equivalence relations on the set S. Determine whether each of these combinations of R1 and R2 must be an equivalence relation.

    a) R1 U R2
    b) R1 ^ R2 (^ = intersection)

    Please respond ASAP. This assignment is due tomorrow morning. I'm really sorry for making it so last minute. Thank you so much for being super awesome!

    Edit: I guess I should add that I think a) is not an equivalence relation and b) is a equivalence relation though I don't know how to prove it.
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  2. #2
    Super Member Gamma's Avatar
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    a) Consider X=\{1,2,3\}
    R_1=\{(1,1), (2,2), (1,2), (2,1),(3,3)\}
    R_2=\{(1,1),(2,2),(3,3),(2,3),(3,2)\}

    The union fails transitivity. (1,2),(2,3)\in R_1 \cup R_2; however, (1,3)\not \in R_1 \cup R_2

    b) Let S,T be equivalence relations and let R=S\cap T.

    If both S and T are equivalence relations we know (a,a) is in both so it is in the intersection for all a.

    If (a,b)\in R then (a,b) is in S and T, by symmetry (b,a) is in both S and T, so it is in R.

    If (a,b) and (b,c) are in R they are in both S and T, so (a,c) must be in S and T by transitivity of each separately, so (a,c) is in R as well.
    Last edited by Gamma; August 5th 2009 at 11:23 PM. Reason: missing comma
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  3. #3
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    Thanks for you help, Gamma!
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