Relations proof

**james121515** · April 2nd 2010, 01:22 PM

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

$\text{\textbf{Theorem.}}$ 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.

**MoeBlee** · April 2nd 2010, 01:25 PM

Your proof is correct.

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