Give an example to show that when the symmetric closure of the reflexive closure of the transitive closure of a relation is formed, the result is not necessarily an equivalence relation.
My attempt at a solution:
$R = \{(2,1),(2,3)\}$.
Transitive closure: $\{(2,1),(2,3)\}$.
Reflexive closure: $\{(1,1),(2,1),(2,2),(2,3),(3,3)\}$.
Symmetric closure: $\{(1,1),(1,2),(2,1),(2,2),(2,3),(3,2),(3,3)\}$.
Since the set is missing $(1,3)$ and $(3,1)$ to be transitive, it is not an equivalence relation.
I am not sure if this is correct.