What is wrong with the following "proof" that $\sim$ is reflexive? 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.