This question already has an answer here:
Pretty much what the title asks. But here's some context:
Suppose $X$ is finite and $R$ is a relation on $X$ that is not reflexive but it is symmetric. Also, suppose we can rule out $xRy$ and $yRz$ both occurring $\forall x,y,z\in X$ s.t. $x\neq y$ and $y\neq z$ and $x\neq z$. So we can rule out that $xRy$ and $yRz$ when $x,y,z$ are distinct. At this point, it seems that $R$ might be vacuously transitive. However, since $R$ is symmetric, $xRy$ and $yRx$ are both fine. But since $R$ is not reflexive, we cannot have $xRx$, which seems to be a violation of transitivity if $xRy$ and $yRx$. Is this a "legitimate" counter-example?