Relations proof

• April 2nd 2010, 01:22 PM
james121515
Relations proof
Hi guys,

Can you check my relations proof? It feels correct but at the same it almost seems too easy so I'm wondering if perhaps I missed something.

$\text{\textbf{Theorem.}}$ Suppose $R$ is a relation on a set $A$ with the property that for every $a \in A,\,aRx$ for some $x \in A$, that is, every element is related to at least one other element in $A$. Prove that if $R$is symmetric and transitive, then $R$ is reflexive.
$\text{\emph{Proof.}}$ Let $a \in A$. By hypothesis, $aRb$ for some $b \in A$. Since $R$ is symmetric, $bRa$. Since we have that $aRb$ and $bRa$, and we know that $R$ is transitive, $aRa$ so that $R$ is reflexive.