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I've been learning of relations and I'm having trouble on how to proceed with this problem:

$$ \begin{align} \text{On any set } A: a\sim b \enspace\enspace\forall \enspace a,b \in A \end{align} $$

It's pretty obvious to see that $a\sim a$ as $(a,a) \in R \Rightarrow (a,a) \in R$, but how can this be shown true for symmetry and transitivity?

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Symmetry

It follows that $\enspace a \sim b \implies b \sim a \enspace\enspace\forall \enspace a,b \in A $, since anything implies something true: formally, $Q \implies (P \implies Q) $.

The same applies to transitivity.

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If $a\sim b$ holds for all elements $a,b\in A$, then the claims needed for symmetry/transitivity are trivially true.

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  • $\begingroup$ How would I formally show this? $\endgroup$ – Frank Vel Nov 12 '14 at 12:04

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