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Math Help - [SOLVED] Equivalence relation proof?

  1. #1
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    [SOLVED] Equivalence relation proof?

    If R1 and R2 are equivalence relations on A, then must the composition R1oR2 also be an equivalence relation on A?Prove or give a counterexample.

    I know that an equivalence relation is symmetric, reflexive and transitive , but I don't know how to prove whether or not the composition relation has the same properties. (I think it does because I tried to find a counterexample , but failed). How do I prove it if it's true or go about finding an appropriate counterexample if it's false?
    Any input would be appreciated.

    Thanks in advance
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  2. #2
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    If I didn't miscalculate, the following is a counterexample:

    A = {0 1 2}

    R1 = {<0 0> <0 1> <1 0> <1 1> <2 2>}

    R2 = {<0 0> <1 1> <1 2> <2 1> <2 2>}

    I found this by first trying to prove that R1oR2 is an equivalence relation. Reflexivity was okay, but symmetry didn't seem to follow, and as I tried to prove symmetry the above example kind of suggested itself from the variables I had working in the failed attempt at proving symmetry.
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