For relation $$\left\{(1, 1),(2, 2)\right\}$$ decide whether it is symmetric, whether it is antisymmetric, and whether it is transitive?
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And what is the relation that we're talking about? – Sudarsan Dec 17 '14 at 19:10
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i know the definitons,it must be symmetric but if it is symmetric, then it cannot be antisymmetric right? i am confused. – Wardruna Dec 17 '14 at 19:13
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Not necessarily. Just apply the definitions to find your answers. – Arthur Dec 17 '14 at 19:14
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then it is both symmetric and antisymmetric? – Wardruna Dec 17 '14 at 19:16
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Absolutely correct. – Arthur Dec 17 '14 at 19:17
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i thought so.Someone in good math website said " if it is symmetric, then it cannot be antisymmetric".That is why i was confused.Anyway i got it know.Thanks – Wardruna Dec 17 '14 at 19:20
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A relation $R$ can be both symmetric and antisymmetric.
For any such relation, suppose $a R b$. Then by symmetry, $bRa$. By antisymmetry, since $a R b$ and $b R a$, then $ a = b$. Hence, for any such relation, it must be that $ a R b \implies a = b$. You can check that any relation which satisfies $a R b \implies a = b$ is both symmetric and antisymmetric, so that these are equivalent statements. Your relation can easily be checked to satisfy this property (though you haven't stated on what set the relation is, it doesn't matter here), so it is both symmetric and antisymmetric.
Logan M
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