Suppose I have (a,b) where a is even numbers and b is odd numbers. Is it transitive? I think it is because there is no instance where (a,b) and (b,c) but not (a,c).
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Yes, exactly. There are no counter examples because there is no integer $b$ that is both odd and even.
That is, $(a,b)\wedge(b,c)$ is necessarily false, so $\forall a\,\forall b\,\forall c~\Big(\big((a,b)\wedge(b,c)\big)\to(a,c)\Big)$ is a vacuous truth.
Graham Kemp
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