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Composition of two equivalence relations

  1. Dec 10, 2012 #1
    1. The problem statement, all variables and given/known data
    The question is let E1 and E2 be equivalence relations on set X. A new relation R is defined as the E1 o E2, the composition of the two relations. We must prove or disprove that R is an equivalence relation.


    2. Relevant equations



    3. The attempt at a solution
    I know that we must prove
    1) reflexive - this is easy just E1 = E1
    2) symmetric
    3)transitive

    but I am unsure of how to prove the last two.
    Thanks for any help in advance and if you need more information I will try to provide.
     
  2. jcsd
  3. Dec 10, 2012 #2

    haruspex

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    I'm unfamiliar with the concept of composition of equivalence relations. Does it mean that if xE1y and yE2z then xE1oE2z?
     
  4. Dec 11, 2012 #3
    Let E1 and E2 be equivalence relations on a non-empty set X. Define a new relationRonXbyxRyifthereexistsaz∈XsuchthatxE1 zandzE2 y. TherelationR is often denoted as E1 ◦ E2 and is called the composition of the relations E1 and E2. Prove or disprove: R is an equivalence relation on X, which in words is that the composition of equivalence relations is an equivalence relation.

    This is the rest of the information
     
  5. Dec 11, 2012 #4

    haruspex

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    OK, that's what I guessed.
    I don't understand your proof there. What do you mean by 'E1=E1'? It's not the equivalence of equivalence relations that's at issue.
    Write those last two out in terms of what you would need to prove re E1oE2.
     
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