I am confused here. For the set $\{ a, b, c\}$ how is the relation $\{(a, b), (b, c), (a, c)\}$ transitive ?
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Hint: Write out the definition of transitive. – Kyle Gannon May 13 '16 at 21:07
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You only have two pairs of the form $\;(a,b),\,\,(b,c)\in R\;$ , and also $\;(a,c)\in R\;$ , so it is true that whenever $\;(x,y),\,(y,z)\in R\;$ , also $\;(x,z)\in R\;$, and that's transitivity
DonAntonio
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Wont we look for transitivity for (b,c) and (a,c) , like for (b,c) there is no (c,a ) or (c,b) ? – abhay May 13 '16 at 21:11
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@abhay Exactly. In those cases, by definition, transitivity doesn't apply. only in the cases when it applies we can check, and in this case it happens in the only case we can apply it. – DonAntonio May 13 '16 at 21:14
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Because no matter how you select elements $x,y,z \in \{a, b, c\}$ such that $xRy$ and $yRz$ (hint: there's only one way), you have $xRz$, which is what it means to be transitive.
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So what about the relation { (a,b) } on the set { a,b,c} ? Is that transitive ? – abhay May 13 '16 at 21:14
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Look at the definition of transitivity, and realise that there's no way to select elements that satisfy the requirements, that means there's no elements that have to satisfy anything for transitivity to be true. - It's just like the empty relation, that's transitive too, neither case is particularly interesting though. – Henrik supports the community May 13 '16 at 21:21
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