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    Equivalence Relations

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    Verify that each of the following are equivalence relations on the plane R^2 (where R are real numbers) and describe the equivalence classes geometrically.

    1) (x1,y1)R(x2,y2) if and only if x1 = x2

    2) (x1,y1)R(x2,y2) if and only if x1 + y1 = x2+y2

    3) (x1,y1)R(x2,y2) if and only if
    x1^2 + y1^2 = x2^2 + y2^2.

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    Solution Preview

    To show R is an equivalence relation, we need to show:
    (a) aRa (b) if aRb, then bRa (c) if aRb and bRc, then aRc.

    1) (x1,y1)R(x2,y2) if and only if x1 = x2
    (a) (x1,y1)R(x1,y1) since x1=x1
    (b) if (x1,y1)R(x2,y2), then x1=x2, so x2=x1, thus (x2,y2)R(x1,y1)
    (c) if (x1,y1)R(x2,y2) and (x2,y2)R(x3,y3), then x1=x2 and x2=x3. Thus we have ...

    Solution Summary

    Equivalence realtions are verified and the equivalence class is described geometrically.