Let $S = {1,2,3,4}. X, Y in mathcal{P}(S)$ and R be a relation $R(X,Y): |X cap Y| = 1$. Is the relation R transitive?

Question

I'm reading up on Relations and Functions with the book Discrete Mathematics by G. Chartrand, P. Zhang and one of the exercises there proposes that the relation R over $mathcal{P}(S)$ where $R(X,Y): |X cap Y| = 1$ is symmetric only — which I can make sense with since my mental model with this is that $(a, b) in R$ never occurs because $|X cap Y| = 1$ is only true when $|X|$ or $|Y|$ is 1, therefore only $(a,a) in R$, thus the implication for symmetry is vacuously true.
My concern is that with my thinking of how the relation R is symmetric, then this can also mean that it's transitive as well, since $(a,b) in R land (b,c) in R$ never occurs and therefore is vacuously true also.
Is my thinking correct? How is relation R symmetric only and why can't it be transitive as well?
Edit
Okay. I had $(a,a) in R(X cap Y, X cap Y)$ in mind for whatever reason when I was thinking $(a, a) in R$.

fleablood · Accepted Answer

I have absolutely no idea why you think $|Xcap Y|$ only if $|X|$ or $|Y|$ is $1$.
Simple case $X={1,2}$ and $Y={2,3}$ then $Xcap Y = {2}|$ and $|Xcap Y| = 1$ and $R(X,Y)$.
Further more if $Z = {3,4}$ then $R(Y,Z)$ but $Xcap Z =emptyset$ so $not R(X,Z)$ even though $R(X,Y)$ and $R(Y,Z)$.  So not transitive.
....
Any way  $X cap X = X$ so $R(X,X)$ only if $|X| = 1$.  So if $|X| ne 1$ we do not have $R(X,X)$ so it is not reflexive.
$Xcap Y=Ycap X$ so $|Xcap Y| = |Ycap X|$ so $R(X,Y) iff R(Y,X)$ so symmetry holds.
Transitivity:  There no way to determine $Xcap Z$ from $Xcap Y$ and $Xcap Z$.  It is certainly possible for $X$ and $Y$ to have only one thing in common and $Y$ and $Z$ to have one thing in common but $X$ and $Z$ to have nothing or many things in common.
Let $X = $ the prime natural numbers.  Let $Y = $ the even natural numbers.  And let $Z=$ then numbers between $11$ through $13$ inclusively.
$X cap Y= {2}$.  $Ycap Z={12}$ and $Xcap Z = {11,13}$.  Not transitive.

Let $S = {1,2,3,4}. X, Y in mathcal{P}(S)$ and R be a relation $R(X,Y): |X cap Y| = 1$. Is the relation R transitive?

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