Let ~ be an equivalence relation on $\mathbb R^2$ such that $\left( x_1, y_1 \right)$ ~ $\left(x_2, y_2 \right)$ iff $y_1=y_2$.

Let ~ be an equivalence relation on $\mathbb R^2$ such that $\left(x_1, y_1 \right)$ ~ $\left(x_2, y_2 \right)$ iff $x^2_1 + y^2_1$ = $x^2_2 + y^2_2$.

Describe the equivalence classes for each equivalence relation above.

I am having difficulties understanding how to go about this problem. For the first relation, I think a geometric description may be the set of all horizontal lines on the Cartesian xy-plane, but I'm not sure if that's what I'm supposed to write. The second relation might be the set of circles, but I don't know.

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1 Answer 1


Your hunches are correct. For the first one, it's horizontal lines in the plane. For the second, it's circles centered at the origin.


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