I am confused here. For the set $\{ a, b, c\}$ how is the relation $\{(a, b), (b, c), (a, c)\}$ transitive ?
2 Answers
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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
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$\begingroup$ Wont we look for transitivity for (b,c) and (a,c) , like for (b,c) there is no (c,a ) or (c,b) ? $\endgroup$– abhayMay 13, 2016 at 21:11
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$\begingroup$ @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. $\endgroup$ May 13, 2016 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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$\begingroup$ So what about the relation { (a,b) } on the set { a,b,c} ? Is that transitive ? $\endgroup$– abhayMay 13, 2016 at 21:14
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$\begingroup$ 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. $\endgroup$ May 13, 2016 at 21:21
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