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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.

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  • $\begingroup$ You must show that $\sim$ is reflexive, symmetric, and transitive. Which of these are you stuck on and what have you tried? $\endgroup$ – grndl Sep 18 '16 at 7:28
  • $\begingroup$ I know I must show he 3 properties. That's the definition of equivalence relation. What I meant was I don't quite understand what the question wants and how I can use the given information to determine the equivalence relation holds. $\endgroup$ – Mathematicing Sep 18 '16 at 7:30
  • $\begingroup$ Isn't the problem just to prove that the relation is an equivalence relation (you said you understand he geometric interpretation, so I leave that aside)? If you know the definition of an equivalence relation, then what part of the question do you not understand? $\endgroup$ – grndl Sep 18 '16 at 7:35

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