# Describe the equivalence classes for each equivalence relation

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.

• When one of your questions has a good answer you should vote it up and accept it. (You can only accept one answer, but can upvote several if you like more than one.) Jun 7, 2015 at 21:44

## 1 Answer

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.