Can you give an example of a relation that is symmetric and transitive, but not reflexive?
By definition,
$R$, a relation in a set $X$, is reflexive if and only if $\forall x\in X$, $x\,R\,x$.
$R$ is symmetric if and only if $\forall x, y\in X$, $x\,R\,y\implies y\,R\,x$.
$R$ is transitive if and only if $\forall x, y, z\in X$, $x\,R\,y\land y\,R\,z\implies x\,R\,z$.
I can give a relation $\leqslant$, in a set of real numbers, as an example of reflexive and transitive, but not symmetric. But I can't think of a relation that is symmetric and transitive, but not reflexive.