Can someone explain why is $R=\{(1,1), (1,2)\}$ transitive?

Question

Using a digraph I understand transitive relation to be a loop, but $R=\{(1,1), (1,2)\}$ is not a loop.

Thank you for your time!

Answer 1

For a relation $R$ to be transitive, if you have $aRb$ and $bRc$, you also need to have $aRc$

In this case, we have $1R1$ and $1R2$. Hence, we need to have $1R2$, which we do have.

Also, we have $1R1$ and $1R1$, hence we need to have $1R1$, which he have as well.

thus, we have exhausted all combinations of $aRb$ and $bRc$ and we have checked that the corresponding $aRc$ exists.

Hence, the relation $R$ is transitive