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Math Help - equivalence relation

  1. #1
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    equivalence relation

    question



    An equivalence relation R defined on a set contains
    the pairs (1,1 ), (1,2 ), ( 2,3). Find R, given that R A x A.

    have to do this for a assignment but have no idea where to start I missed the class covering this. could anyone tell me the basic technique how to solve this so i know where to start at least!
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  2. #2
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    Re: equivalence relation

    An equivalence relation on A is a subset of A x A with certain properties. The three pairs you show do not currently satisfy those properties.

    1. You need reflexivity: For all a \in A,(a,a) \in R.
    2. You need symmetry: If (a,b) \in R, then (b,a) \in R.
    3. You need transitivity: If (a,b)\in R and (b,c) \in R then (a,c) \in R.
    Last edited by SlipEternal; October 22nd 2013 at 11:44 AM.
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  3. #3
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    Re: equivalence relation

    sorry wrote question wrong

    An equivalence relation R defined on a set A ={1,2,3,4}contains
    the pairs (1,1 ), (1,2 ), ( 2,3). Find R, given that R ≠ A x A.
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  4. #4
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    Re: equivalence relation

    What I am saying is, add pairs until you satisfy those three properties.
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  5. #5
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    Re: equivalence relation

    im sorry i dont understand
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  6. #6
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    Re: equivalence relation

    Check: for any a \in A, do you have (a,a) \in R? No. So, add those pairs that you are missing. (1,2) \in R. Do you have (2,1) \in R? No. So add that pair. Etc.

    Edit: By add those pairs, I mean add the pairs to the list of pairs you know must be in R. You started with the three given pairs. Then, from those three, you use the properties to figure out which other pairs must also be in R. In the end, you should find 10 pairs in R. If another equivalence relation R' has R \subseteq R' and you know R' \setminus R \neq \emptyset, then it is easy to show that R' = A \times A. In other words, R is the unique equivalence relation on A with those three pairs.
    Last edited by SlipEternal; October 22nd 2013 at 12:59 PM.
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  7. #7
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    Re: equivalence relation

    Quote Originally Posted by ronanbrowne88 View Post
    question
    An equivalence relation R defined on a set contains
    the pairs (1,1 ), (1,2 ), ( 2,3). Find R, given that R A x A.
    have to do this for a assignment but have no idea where to start I missed the class covering this. could anyone tell me the basic technique how to solve this so i know where to start at least!
    Define F=\{(1,1),(1,2),(2,3)\}. The diagonal is \Delta_A=\{(1,1),(2,2),(3,3),(4,4)\}

    Define G=F\cup\Delta_A and H=(G\circ G)\cup G

    Define R=H\cup H^{-1}.

    Show that R is an equivalence relation.
    Thanks from topsquark
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