Does $R\circ R=R$ imply that R is a diagonal?

Question

Let $\Delta$ be the diagonal in $A\times A$.

If the relation, R, contains $\Delta \subset R$ and had the property $R\circ R=R$, then $\Delta$ must be R.

proof

Because for all $(a,b)$ in the set $A\times A$ where $aRb$ there exists another pair $(b,c)$ such that $bRc$; but because of $R\circ R=R$, $c$ must be equal to $b$ and $b$ must be equal to $a$. Therefore the only values of $A\times A$ that hold true for $R\circ R=R$ are $(\forall x \in A), (x,x)\in R$ or $R=\Delta$.

Does this make sense? Could someone help me better explain this, or show me a better way of saying this?

Answer 1

Any transitive and reflexive relation will obey $\Delta \subset R$ and $R \circ R = R$. Many are not equal to $\Delta$...