Which of the following relations on $\{1,2,3\}$ is an equivalence relation? $$\begin{array}{l}{R_{1}=\{(1,1),(2,2),(3,3),(1,2),(2,1)\}} \\ {R_{2}=\{(1,1),(2,2)\}} \\ {R_{3}=\{(1,2),(2,3),(3,1)\}}\end{array}$$
$$\begin{array}{l}{R_{4}=\{(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)\}} \\ {R_{5}=\{(1,1),(2,2),(3,3),(1,2),(2,3),(3,1)\}} \\ {R_{6}=\{(1,2),(2,1)\}}\end{array}$$
My answer
In order for $R_i$ to be an equivalence relation it should meet four criteria


*

*$R_i$ should be a subset of $\{1,2,3\} \times \{1,2,3\}$

*$R_i$ should be reflexive. symmetric and transitive.


My conclusion is that none of the relations above are an equivalence relation. But the answer key is $R_1$. The problem is that $3$ is not related to any of the others and one cannot see that $R_1$ is transitive.
Q1: Why am I wrong?
Q2: Why isn't $R_5$ an equivalence relation?

$R_2$ is not reflexive since you don't have $(3,3)$.
$R_3$ is not reflexive.
$R_4$ is not reflexive since we haven't $(1,1)$.
$R_5$ is not transitive because you have $(1,2),(2,3)$ but not $ (1,3)$.
$R_6$ is not reflexive.
$R_1$  is an  equivalence relation.
