# Thread: [SOLVED] Equivalence relation proof?

1. ## [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.

2. 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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# composition of two equivalenceequivalence relation is an equivalence relation

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