Transitive closure relation

Question

I have a following relation on the set {A,B,C,D}
R = {(a,a);(a,c);(b,d);(c,d);(d,c)}
What is the smallest number of tuples that has to be added in order for the relation to become transitive?
It is a bit confusing for me.
I am sure that I need to add (a,d) since there is a path (a,c) ^ (c,d) and I need to add (b,c) since there is a path (b,d) ^ (d,c), but do I need to add (c,c) and (d,d) since there is a path (c,d) ^ (d,c) ?

Answer 1

You are correct that we need to add $(a, d), (b, c)$. And yes, for the reason you give, for the relation to be transitive requires that since $(c, d), (d, c) \in R$, we also need to add $(c, c)$ and $(d,d)$.