Proving if a relation is or is not transitive.

Let S be a set that contains at least two different elements. Let R be the
relation on $P(S)$, the set of all subsets of S, defined by $(X, Y ) in R$ if and only if
$X cap Y = emptyset$.

I am trying to prove if this is transitive or not. I from what I am understanding about transitivity I don’t think it is.

Here is my answer right now:

$R$ is not transitive because if $X={1,2}$ and $Y={3,4}$ and $Z={2, 6}$ then $X cap Y = emptyset$ and $Ycap Z = emptyset$ but $X cap Z = {2}$.

I am not sure if this works as a counter example.