Does $\{(a,b), (b,a), (a,d), (b,d)\}$ hold transitive property?

Question

I have been working on one of the problem from Velleman's How to prove book and there is a relation $R$ like this:

$R = \{(a,b), (b,a), (a,d), (b,d)\}$

We have to find out if $R$ is transitive.

Now, looking at the example, I concluded that it isn't transitive since $(a,b) \land (b,a) \implies (a,a)$ doesn't hold for relation $R$. But the answer given here says that it is transitive. Am I missing something ?

Answer 1

For a relation $R$ to be transitive, we require that if $(x,y) \in R$ and $(y,z) \in R$, then it must also be the case that $(x,z) \in R$. As you have correctly observed, for your example, $(a,b) \in R$ and $(b,a) \in R$, but $(a,a) \notin R$.

So your relation is not transitive and the answer on the website you found is wrong. If we add $(a,a)$ and $(b,b)$ to the relation, it will become transitive, however.