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In mathematics and other areas, a binary relation R over a set X is symmetric if it holds for all a and b in X that a is related to b if and only if b is related to a. - Wikipedia

I do understand that a relation is symmetric if for all aRb there is bRa. What i can't understand is what the bold part in above definition means.

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It just means that $aRb \iff bRa$ for all $a,b\in X$ and not just for some of them. ("Just for some of them" means that for one pair $a,b$ it holds but for other two elements $c,d$ it doesn't. You don't want that to happen)

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Relation $R\subseteq X\times X$ is symmetric if:$$\forall a,b\in X\;[aRb\implies bRa]$$In words: $$\text{ for all } a,b\text{ in }X\text{ from }aRb\text{ it follows that }bRa$$

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