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If R is a binary relation in a set X ≠∅ that is symmetric, transitive, then R is reflexive.

This is false and I have to change the argument to make it true. How can I do this? Thanks!

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  • $\begingroup$ What do you mean by "change the argument to make it true"? Change what argument? $\endgroup$ – Ian Jun 4 '16 at 21:41
  • $\begingroup$ Thanks for replying. I have to change the statement of the problem to make it true. $\endgroup$ – Rodrigo Pazos Jun 4 '16 at 21:45
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If for every $x\in X$ there is some $y\in X$ such that $xRy$, then $yRx$ because $R$ is symmetric. Then, $xRy$ and $yRx$, so $xRx$ since $R$· is transitive.

But the condition "for every $x\in X$ there is some $y\in X$ such that $xRy$" is essential.

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    $\begingroup$ The first thing should be "because $R$ is symmetric". $\endgroup$ – Ian Jun 4 '16 at 21:46
  • $\begingroup$ I think is related to how you define the set. Similar to the answer from "ajotatxe" $\endgroup$ – Rodrigo Pazos Jun 4 '16 at 21:51

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