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Math Help - Equivalence Classes

  1. #1
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    Equivalence Classes

    How do you find the distinct equivalence classes of R?

    A = {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5}. R is defined on A as follows:
    For all x, y elements of A, x R y <=> 3|(x-y).

    I have the answer, but I do not understand the work leading up to the answer. I appreciate any help.

    I am having the same issue with this one:
    A = {-4, -3, -2, -1, 0, 1, 2, 3, 4}. R is defined on A as follows:
    For all (m,n) elements of A, m R n <=> 5|(m^2 - n^2).
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  2. #2
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    Question Re: Equivalence Classes

    Quote Originally Posted by lovesmath View Post
    How do you find the distinct equivalence classes of R?

    A = {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5}. R is defined on A as follows:
    For all x, y elements of A, x R y <=> 3|(x-y).

    I have the answer, but I do not understand the work leading up to the answer. I appreciate any help.

    I am having the same issue with this one:
    A = {-4, -3, -2, -1, 0, 1, 2, 3, 4}. R is defined on A as follows:
    For all (m,n) elements of A, m R n <=> 5|(m^2 - n^2).
    In the first case you want x-y to a multiple of 3.
    So -4~\&~2 are in the same class because -4-2=-6 a multiple of 3.

    You do the next one: multiples of 5.
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    Re: Equivalence Classes

    So for 3|(x - y), do you find the combinations of numbers that give you remainders 0, 1 and 2, and each of those combinations will be its own equivalence class?
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    Re: Equivalence Classes

    Quote Originally Posted by lovesmath View Post
    So for 3|(x - y), do you find the combinations of numbers that give you remainders 0, 1 and 2, and each of those combinations will be its own equivalence class?
    All you do its look at each pair of numbers.
    Is their difference a multiple of 3?
    If yes, those two numbers belong to the same class.
    If no, they are not in the same class.

    So \{-4,-1,2,5\} is one class.
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    Re: Equivalence Classes

    For the second relation, 5|(m^2 - n^2), I got the following equivalence classes:
    {0}, {-4, -1, 1, 4}, {-3, -2, 2, 3}

    Are these correct?
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  6. #6
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    Re: Equivalence Classes

    Quote Originally Posted by lovesmath View Post
    For the second relation, 5|(m^2 - n^2), I got the following equivalence classes:
    {0}, {-4, -1, 1, 4}, {-3, -2, 2, 3} Are these correct?
    Yes they are correct. Good for you.
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