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