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When I was just walking through the hallway of my department I found an exercise sheet asking the student to examine the following proof.

Assume [itex]R\subset M\times M[/itex] is a binary, symmetric, transitive relation. Then for any [itex]a,b \in M[/itex] with [itex]a\sim _R b[/itex] it follows by symmetry that [itex]b\sim _R a[/itex] and thus by transitivity that [itex]a\sim _R a[/itex] i.e. R is also reflexive and therefore an equivalence relation.

The exercise then asks to find the flaw in this argument (and give a counter example). To me the argument makes perfect sense...I am really ashamed, after all this is for first year students

Can someone give a hint?

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# Equivalence relation

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