Is relation between an even numbers and an odd number transitive?

Question

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).

Answer 1

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.