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$\def\R{\mathrel R}$A relation $R$ is transitive if $a\R b$ and $b\R a$ implies $a\R c$

because it is never $a\R b$ and $b\R a$, it is always true because no matter what $a\R c$ is, if the LHS is false, the statement is always true.

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    Your definition of 'transitive' has a typo.2012-11-22
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    Transitivity of a relation $R$ on a set $S$ means that for all $a, b, c \in S$, IF $\,a\,R\,b\,$ AND $\,b\,R\,c$, then necessarily $\,a\,R\,c$.2012-11-22

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