Let set $A$ be ${1,2,3}$, and let a relation $R$ on $A$ be
$\left \{ (1,1),(1,2),(2,1),(2,2) \right \}$
$R$ is both symmetric and transitive, but not reflexive. The key point here is that there may be some element in set A which is not related to any of the element in $R$, but to be reflexive, all elements must be related to themselves.