$A= \{a,b,c\}$ and $R=\{(a,b),(b,c),(c,c)\}$. Find the transitive closure

Question

Please explain it to me how one can calculate $R^2$ and $R^3$. The solution says $R^2 = \{(a,c), (b,c),(c,c)\}$ $R^3= \{(a,c),(b,c),(c,c)\}$.

Answer 1

It seems like that $R^2=R$. Look at $(a,c)\notin R$, otherwise, there must be some $x\in \{a,b,c\}$ such that $(a,x)\in R$ and $(x,c)\in R$. It looks impossible.

Please see the definition of the copmosition of relations.