Consider the following relation $\mathfrak{R}$: in a plane $\pi$, fixed a point $O$, for every pair of points $A$ and $B$, we say that $A \mathfrak{R} B$ if and only if $A,B,O$ are collinear. Is $\mathfrak{R}$ an equivalence relation? If yes, find the equivalence classes.
I have estabilished that $\mathfrak{R}$ is an equivalence relation and I have found the equivalence classes: they are the lines passing through the point $O$.
But now I have a problem: the point $O$ belongs to every equivalence class, because it belongs to every line passing through the point $O$. This is a problem because the equivalence classes have to be disjoint.
What am I doing wrong? How is it possible?