Describing equivalence classes and finding the quotient for a relation.

Question

The set $A = \mathbb{R} \times \mathbb{R}. (x_1,y_1) \sim (x_2,y_2)$ if $x^2_1+y^2_1 = x^2_2+y^2_2$.

I need some help describing the equivalence classes and describing all elements in $A/{\sim}$. Any help in describing how to figure out this would be really helpful. Thanks :)

Answer 1

Note that $x_1^2+y_1^2 = x_2^2+y_2^2$ if and only if $\sqrt{x_1^2+y_1^2} = \sqrt{x_2^2+y_2^2},$ and $\sqrt{x_1^2+y_1^2}$ is the distance from $(0,0)$ to $(x_1,y_1).$ Thus two points are equivalent precisely if they're at the same distance from the origin. From that you can see what the equivalence classes look like.

Answer 2

Equivalence class of a point $(x_1,y_1)$ under the relation given here is the set of all points on the circle in the plane with the centre at the origin and radius equal to $\sqrt{(x_1^2+y_1^2)}$. I hope you are satisfied with this answer