Relation on $\mathbb{N}\setminus\{1\}$: $$A_{1}=\{(x,y): x\text{ and }y \text{ are relatively prime}\}.$$ Determine which one of the three properties are satisfied:
$i)$ $(2,2) \notin A_{1}$. So it is not reflexive.
$ii)$ $(2,3) \rightarrow (3,2) \in A_{1}$. A_{1} is symmetric.
$iii)$ $(2,3)$ and $(3,4)$ $\rightarrow$ $(2,4) \notin A_{1}$. Is not transitive.
That means, $A_{1}$ is not an equivalence relation. Is it okay? Thanks in advance!
For point (i) your argument is correct.
For point (ii) You have not shown that the relation is symmetric, you have only given an example. Also your notation is a bit off; you want to show that if $(x,y)\in A_1$ then $(y,x)\in A_1$.
For point (iii) your notation is again a bit off; you want to give a counterexample by showing that $(2,3),(3,4)\in A_1$, but $(2,4)\notin A_1$.