Equivalence Relations of the Circle $x^2+y^2=r^2$ I have been studying Set Theory and this question came up:

Let us assume the set $C_r$, consisting of the points of the circumference of the circle with a center at $(0,0)$ and a radius $r>0$, as a collection of ordered pairs of elements characterized by a binary relation at $\mathbb{R}$ defined as:
  \begin{equation}
(x,y) \in C_r, \quad \text{if and only if} \quad  x^2+y^2=r^2
\end{equation}



*

*How is it possible to define the domain or the image of this relation in order for it to be a function of one variable, that being $x$ or $y$?

*Considering now that $r$ is a variable, define an equivalence relation in $\mathbb{R}^2$ such that the sets $C_r$ and the unit set $\{ (0,0) \}$ to be equivalence classes of the equivalence relation.Could this procedure be generalized for any $\phi: \mathbb{R}^2 \to \mathbb{R}$ and not just for $\phi(x,y)=x^2+y^2$? If yes, what would the correspoding equivalence classes be?


For the first one I think that solving for one of $x$ or $y$ would do the trick, e.g: $x=\pm \sqrt{r^2-y^2}$, but then, this is not a function. I would have to constraint the domain to the semicircle but that will not cover the whole circle. For the second question I am not really sure how to proceed. 
Thank you.
 A: The answer to the first question is "it's not possible."  For every $y$ there are two $x$'s that satisfy the equation.  A function allows only one occurrence of each value of $x$.
(And mutatis mutandis.)
I think the answer to the second question is "yes" (if I understand it.)  You're really taking the level curves of the function $\phi$ as equivalence classes, and that should work just fine.
A: You are correct for the first question and B. Goddard elaborates on that.
The second question is "yes".  Note that $C_r = \phi^{-1}(r^2)$, the inverse image of $r^2$ with respect to $\phi$.  This is what B. Goddard is calling "level curves", but that's an overly geometrical picture.  It's simply the set $\{(x,y)\in\mathbb{R}^2\mid\ \phi(x,y)=r^2\}$. This is an equivalence class of the equivalence relation $$(x_1,y_1)\sim(x_2,y_2) \iff \phi(x_1,y_1)=\phi(x_2,y_2)$$ which you can easily prove is an equivalence relation, i.e. that it is reflexive, symmetric, and transitive.  Every function will give rise to such an equivalence relation on its domain.
