Let ~ be a symmetric and transitive relation on a set A. What is wrong with the folloing "proof" that $\sim$ is reflexive?
Proof: $a\sim b$ implies $b\sim a$ by symmetry; then $a\sim b$ and $b\sim a$ imply that $a\sim a$ by transitivity, thus $a\sim a$.)
We must have $\forall a \in A$, $a \sim a$.
You just proved that if $\exists b$ s.t. $a \sim b$, then $a \sim a$. Why is there such $b$?
For example, the empty relation satisfies symmetry and transitiveness, but the lack of existence of something to "apply" them yields non-reflexivity.