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Math Help - Abelian Group , proving etc.

  1. #1
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    Abelian Group , proving etc.

    /3ℤ

    Show that for all classes:
    _ _
    x,y ∈ ℤ/3ℤ

    The following is true:
    _ _ ______ ______
    ∀x1, x2 ∈ x, y1,y2 ∈ y : x1 + y1 = x2 + y2

    I dont now how to make lines directly above the letters

    Okay so I dont know if I need that, but what does the ℤ/3ℤ say? Am I creating classes depending on Euclidean division?

    divisible by 3: 0:= [0] = {...,-9,-6,-3,0,3,6,9,...}
    remainder 1 1:= [1] = {...,-8,-5,-2,1,4,7,10,13,...}
    remainder 2 2:= [2] = {...,-10,-7,-4,-1,2,5,8,11,14,...}

    How can I combine it with the statement I have to prove?


    Additionally I need to show that (ℤ/3ℤ , +) is an Abelian group.
    In this case I I would draw a matrix and analyze it:

    + 0 1 2
    0 0 1 2
    1 1 2 0
    2 2 0 1

    Okay so 0 doesnt change anything. This means that the neutral element is 0
    The invers is 0, because it is a part of every row.
    It is also symmetrical on the diagonal line.

    This means it is an Abelian group.
    Is this proof enough?
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  2. #2
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    Re: Abelian Group , proving etc.

    x_1 = 3q_1 + r_1, x_2 = 3q_2 + r_1, y_1 = 3q_3+r_2, y_2 = 3q_4 + r_2 for some integers q_1,q_2,q_3,q_4,r_1,r_2 with 0\le r_1 < 3, 0\le r_2 < 3.
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  3. #3
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    Re: Abelian Group , proving etc.

    Thanks for your answer, but I would like to understand the solution instead of getting the solution itself.
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  4. #4
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    Re: Abelian Group , proving etc.

    I did not give you the solution. I used the division algorithm for each variable. That's all I did. I did not tell you what to do with the variables. I did not prove any aspect of the problem. I just gave you a start.
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  5. #5
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    Re: Abelian Group , proving etc.

    Oh okay, it just looks so complicated to me, that I thought it is an actual solution. Sadly, my point stands still. I dont know what to do with it. What are the qs?
    Sorry for being such a dumbo. But I seriously dont know how to approach this problem.
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  6. #6
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    Re: Abelian Group , proving etc.

    As I said, I used the division algorithm. So, x_1 = 3q_1 + r_1 means q_1 is the quotient when x_1 is divided by 3 and r_1 is the remainder. Just go back to the Division Algorithm, and it should all make sense.

    Now, to show that \overline{x_1+y_1} = \overline{x_2+y_2}, you need to show that (x_2+y_2) - (x_1+y_1) is divisible by 3.
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