Writing $I=\left [ 0,1 \right ]$ and $I^{2}=\left \{ \left ( x,y \right ):x,y \in I \right \}$,
define for $x,y \in I$
- $\left ( x,y \right )\sim \left ( x,y \right )$
- $\left ( 0,y \right )\sim \left ( 1,y \right )\sim \left ( 0,y \right )$
- $\left ( x,0 \right )\sim \left ( x,1 \right )\sim \left ( x,0 \right )$
- $\left ( 0,0 \right ) \sim \left ( 1,1 \right ),\left ( 1,1 \right )\sim \left ( 0,0 \right ),\left ( 1,0 \right ) \sim \left ( 0,1 \right ),\left ( 0,1 \right )\sim \left ( 1,0 \right )$
Question: Show that $\sim$ is an equivalence relation and give a geometrical representation of the set $I^{2}/\sim$ of equivalence class.
I have an idea of what is going on geometrically but I do not quite understand how I should go about determining the equivalence relation in this question.
Any help is appreciated.
Thanks in advance.