Are reflexive relations transitive?

Question

Let $R=\{(2,2),(3,3)\}$. Then $S=R \circ R= \{(2,2),(3,3)\}$

So, $S \subset R$, there is theory that $R$ is transitive. Then every reflexive relation is transitive?

Answer 1

No. Take $R=\bigl\{(1,1),(2,2),(3,3),(1,2),(2,3)\bigr\}$. Then $R\circ R=R\cup\bigl\{(1,3)\bigr\}$ and therefore $R\circ R\not\subset R$.