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Math Help - Still need help with relation problem

  1. #1
    CPR
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    Please help with equivalence relations problem

    Here is my problem:
    Let H= {1,2,3,4}, A = HxH, and define a relation on A by (s,t) R(u,v) if and only if st = uv.

    1. Show R is an equivalence relation on A.
    2. Compute the partition A/R that corresponds to the equivalent relation.


    This is what I have:
    R= (1,1), (1,2), (1,3),(1,4), (2,1), (2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1), (4,2), (4,3),(4,4).
    It is reflexive, symmetric, and transitive. Therefore it is an equivalence relation.

    R(1) = {1,2}, R(3) {3,4}

    However, I am not sure about the partition A/R. Please help.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by CPR View Post
    Here is my problem:
    Let H= {1,2,3,4}, A = HxH, and define a relation on A by (s,t) R(u,v) if and only if st = uv.

    1. Show R is an equivalence relation on A.
    2. Compute the partition A/R that corresponds to the equivalent relation.


    This is what I have:
    R= (1,1), (1,2), (1,3),(1,4), (2,1), (2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1), (4,2), (4,3),(4,4).
    It is reflexive, symmetric, and transitive. Therefore it is an equivalence relation.

    R(1) = {1,2}, R(3) {3,4}

    However, I am not sure about the partition A/R. Please help.
    You've got your set for R incorrect. R will be a set of pairs of ordered pairs. Ie. it will take the form
    R = \{ [ (s,t),~(u,v) ] ~ | (s,t), (u,v) \in A ~\text{ and } st = uv \}

    So, for example (2, 1) R (1, 2) implies that the ordered pair [(2, 1), (1, 2)] belongs to the set R. etc.

    -Dan
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  3. #3
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    Quote Originally Posted by CPR View Post
    Here is my problem:
    Let H= {1,2,3,4}, A = HxH, and define a relation on A by (s,t) R(u,v) if and only if st = uv.
    1. Show R is an equivalence relation on A.
    2. Compute the partition A/R that corresponds to the equivalent relation.
    OK. Show us what you have done for yourself.
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  4. #4
    CPR
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    This is what I have:
    R= (1,1), (1,2), (1,3),(1,4), (2,1), (2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1), (4,2), (4,3),(4,4).
    It is reflexive, symmetric, and transitive. Therefore it is an equivalence relation.



    However, I do no know exactly how to do the partition A/R. Please help.
    This what I think
    A/R= {(1,2), (3,4)}
    Please explain to me what to do! Ive been struggling with this for 2 days. Im getting really frustrated.
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  5. #5
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    That is not R, that is AxA.
    R relates pairs in AxA to themselves.
    (1,4)R(2,2) because (1)(4)=(2)(2).
    Now begin again.
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  6. #6
    CPR
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    I would if I knew what to do. I see what you have written and suggested, but I dont know how to do what you have suggested. how is it that (1,4) R
    (2,2) or how does (1)(4) = (2)(2). Are you saying 1x4 = 2x2? How did you get these ordered pairs?
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  7. #7
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    R relates pairs to pairs.
    Can you tell which of the are true?
    (2,2)R(1,4) T F
    (2,3)R(2,4) T F
    (2,2)R(2,2) T F
    (2,3)R(3,2) T F
    (1,3)R(1,4) T F
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  8. #8
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by CPR View Post
    Here is my problem:
    Let H= {1,2,3,4}, A = HxH, and define a relation on A by (s,t) R(u,v) if and only if st = uv.

    1. Show R is an equivalence relation on A.
    2. Compute the partition A/R that corresponds to the equivalent relation.
    Quote Originally Posted by topsquark View Post
    You've got your set for R incorrect. R will be a set of pairs of ordered pairs. Ie. it will take the form
    R = \{ [ (s,t),~(u,v) ] ~ | (s,t), (u,v) \in A ~\text{ and } st = uv \}

    So, for example (2, 1) R (1, 2) implies that the ordered pair [(2, 1), (1, 2)] belongs to the set R. etc.

    -Dan
    You asked for some extra help, but there isn't really that much more to give.

    Showing that R is an equivalence relation on A is easy. To compute the partition, simply find the equivalence classes:
    (1, 1)
    (1, 2) ~ (2, 1)
    (1, 3) ~ (3, 1)
    (2, 2) ~ (1, 4) ~ (4, 1)
    etc.

    -Dan
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  9. #9
    CPR
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    T
    F
    T
    T
    F

    I am know the first one bc you told me. How is (2,2)R (1,4)
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  10. #10
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by CPR View Post
    I am know the first one bc you told me. How is (2,2)R (1,4)
    Quote Originally Posted by CPR View Post
    Here is my problem:
    Let H= {1,2,3,4}, A = HxH, and define a relation on A by (s,t) R(u,v) if and only if st = uv.
    s = 2, t = 2
    u = 1, v = 4

    Since 2*2 = 1*4 we know that (2, 2) R (1, 4). Hence (2, 2) and (1, 4) are in the same equivalence class.

    -Dan
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