Why do we have to include the pairs $(b, b)$ and $(c, c)$ in the transitive closure?

Question

The problem is:

Find the reflexive, symmetric and the transitive closure of the following relation:

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

on the set of elements $A = \{a, b, c\}$

Finding the reflexive and the symmetric closure for $R$ was quite trivial, but I am a little bit confused with why we have to include the pairs $(b, b)$ and $(c, c)$ in the following transitive closure:

$T_R = \{(a, a), (a, b), (b, c), (c, b), (a, c), (b, b), (c, c)\}$

Answer 1

We have $(b,c)\in R$ and $(c, b)\in R,\;$ so for transitive closure, we need $\,(b, b)\in T_R.$

Similarly, for transitive closure, $(c, b)\in R$ and $(b, c)\in R\,$ require that $(c, c) \in T_R$.