What is a complete set of representatives for an equivalence relation on a set?

[geskekj]

Homework Statement

Definition: let R be an equivalence relation on a set X. A subset of X containing exactly one element from each equivalence class is called a complete set of representatives. now define a relation R on RxR by (x,y)R(u,v) <---> x^2 + y^2 = u^2 + v^2. You don't have to prove that R is an equivalence relation. Find a complete set of representatives. Carefully justify the answer.

Homework Equations

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The Attempt at a Solution

I am not sure where to go with this. I know that x^2+y^2 is a circle. I am working with a few other people and this is all we could come up with!

 

[morphism]

Evidently (x,y)R(u,v) iff (x,y) and (u,v) lie on the same circle centered at the origin. So this tells you precisely what the equivalence classes are.