How is this an equivalence relation?

Question

From my textbook

The smallest equivalence relation $R_1$ in the set {$1,2,3$} containing $(1,2)$ and $(2,1)$ is {$(1,1)(2,2)(3,3)(1,2)(2,1)$}

Again,

In mathematics, an equivalence relation is a binary relation that is at the same time a reflexive relation, a symmetric relation and a transitive relation.

Is {$(1,2)(2,1)(1,1)$} the transitive relation here? If not,how is it an equivalence relation?

Answer 1

$\{(1,2), (2,1), (1,1)\}$ is a transitive relation because it satisfies the definition of transitive relations, which is that for every $x,y, z\in\{1,2\}$, we know that if $(x,y)\in R$ and $(y,z)\in R$, then we also have $(x,z)\in R$.

Answer 2

Since R must be an equivalence relation, it must be reflexive and therefore ${(1,1),(2,2),(3,3)} \subseteq R$. Now, because we need that $\{(1,2),(2,1)\} \subseteq R$, we get $\{(1,1),(2,2),(3,3),(1,2),(2,1)\} \subseteq R$. We've shown why no element of $R$ can be removed , and the previous set is itself a well defined equivalence relation, so this makes $R = \{(1,1),(2,2),(3,3),(1,2),(2,1)\} $ the smallest equivalence relation possible with the imposed condition.