Find the smallest relation containing the relation
$$R=\{ (1,2),(2,1),(2,3),(3,4),(4,1) \}$$ that is
Reflexive and transitive
Reflexive, transitive and symmetric
Well this seems easy to do. However, I'm not sure whether the question is meant to find the (for the first part) the reflexive and transitive closures, or is it something else?
If it's a closure case, then the first part would be: $$R=\{ (1,2),(2,1),(2,3),(3,4),(4,1),(1,1),(2,2),(3,3),(4,4),(1,3),(2,4),(3,1),(4,2)\}$$
But this doesn't seem right for some reason and just wanted to clarify what the question is asking.
You’re missing $\langle 1,4\rangle,\langle 3,2\rangle$, and $\langle 4,3\rangle$; the first is required by $\langle 1,2\rangle$ and $\langle 2,4\rangle$, the second by $\langle 3,4\rangle$ and $\langle 4,2\rangle$, and the last by $\langle 4,1\rangle$ and $\langle 1,3\rangle$, for instance. The reflexive, transitive closure of $R$ is in fact $\{1,2,3,4\}\times\{1,2,3,4\}$. (And you should not call it $R$, as that name is already in use for the original relation.)