Set Theory: Transitive?

I have a question regarding Relations on Sets. Here is the problem:

Let $S=left{ a, b, cright}$. Then $R=left{ (a,a), (a,b), (a,c)right}$. Which of the properties (reflexive, symmetric, or transitive) does the relation, $R$, possess?

Here is what I was thinking:

Not reflexive. Since $(b,b)$ and $(c,c)$ are not in $S$, we have that $(b not R b) , (c not R c)$– therefore $R$ is not reflexive.

Not symmetric. Given $(a,b)$, then we would also need $(b,a)$ to show that $aRb$ and $bRa$.

Not transitive. To show the transitive property, we need to show if $aRb$ and $bRc$, then $aRc$. This would require $(a,b), (b,c)$ — which we do not have.

So, to me it would appear as though this relation, $R$, is not reflexive, symmetric, nor transitive. Does this seem like I have the right mindset?

Thanks for looking!

Mia