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Thread: Relations proof

  1. #1
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    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.
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  2. #2
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    Your proof is correct.
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