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Math Help - another equivalence relation..

  1. #1
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    another equivalence relation..

    Define the equivalence relation S by S = {(x,y)ͼAxA | 5 divides (x^2-y^2)}.
    Determine the partition of A that is induced by S.

    I am not exactly sure how to do this one but from what I gather from my textbook it seems to sorta start like this..

    Given
    <br />
 S \cdot S = \{(x,y) \in AxA | 5 divides (x^2-y^2)\}<br />

    <br />
=\{(x,y) \in A^2 | (x^2-y^2) = 5k, \forall k \in Z\}<br />

    <br />
=\{(x,y) \in A^2 | (x^2 = 5k+y^2, \forall k \in Z\}<br />

    <br />
=\{(x,y) \in A^2 | y^2 = x^2-5k, \forall k \in Z\}<br />

    Am I doing this correctly? On the right track?
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  2. #2
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    i got the final answers of..

    X = {xͼA | All real numbers }
    Y = {yͼA | -sqrt(5k) >= y sqrt(5k), For some integer of k}

    dunno if that is correct..
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  3. #3
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    Grandad's Avatar
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    Equivalence Classes

    Hello Kitizhi
    Quote Originally Posted by Kitizhi View Post
    Define the equivalence relation S by S = {(x,y)ͼAxA | 5 divides (x^2-y^2)}.
    Determine the partition of A that is induced by S.

    I am not exactly sure how to do this one but from what I gather from my textbook it seems to sorta start like this..

    Given
    <br />
 S \cdot S = \{(x,y) \in AxA | 5 divides (x^2-y^2)\}<br />

    <br />
=\{(x,y) \in A^2 | (x^2-y^2) = 5k, \forall k \in Z\}<br />

    <br />
=\{(x,y) \in A^2 | (x^2 = 5k+y^2, \forall k \in Z\}<br />

    <br />
=\{(x,y) \in A^2 | y^2 = x^2-5k, \forall k \in Z\}<br />

    Am I doing this correctly? On the right track?
    Are you familiar with the modulo notation, to show the remainder when one integer is divided by another? If x^2-y^2 is a multiple of 5, then x^2-y^2 \equiv 0 \mod 5

    Now look at the various possible values of x^2, for x \equiv 0, 1,2 , 3, 4 \mod 5:

    x \equiv 0 \Rightarrow x^2 \equiv 0

    x \equiv 1 \Rightarrow x^2 \equiv 1

    x \equiv 2 \Rightarrow x^2 \equiv 4

    x \equiv 3 \Rightarrow x^2 \equiv 4

    x \equiv 4 \Rightarrow x^2 \equiv 1


    So (x \equiv 0) \land (x^2-y^2 \equiv 0) \Rightarrow y \equiv 0 \mod 5

    And \Big((x \equiv 1) \lor (x \equiv 4)\Big) \land (x^2-y^2 \equiv 0) \Rightarrow (y \equiv 1) \lor (y \equiv 4) \mod 5

    Do you understand the notation I've used here? Can you complete it, starting with \Big((x \equiv 2) \lor (x \equiv 3)\Big) \land (x^2-y^2 \equiv 0) \Rightarrow ... and then using this to describe the equivalence classes?

    Grandad
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