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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 Summary

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

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 ...

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