I've been wondering recently the following:
Let $\sim $ be a symmetric and transitive relation defined on $S $. Let $a \sim b $, which implies $b \sim a $ by symmetry, and by transitivity, $a \sim a $. Hence, $\sim $ is reflexive.
I can't think of any counterexamples, although any are welcome. Also, if there is some counterexample, where could be the flaw in my reasoning be?