If the binary relations T and S are both transitive, then the intersection of these binary relations ( the binary relation $T \cap S$ is transitive.
My proof is as follows:
Let $\{ (x,y), (y,z)\} \in T$ and $\{ (x,y), (y,z)\} \in S$
Since T is transitive, $\{ (x,z)\} \in T$ as well.
Since S is transitive, $\{ (x,z)\} \in S$ as well.
So, $\{ (x,y), (y,z), (x,z)\} \in T$ and $\{ (x,y), (y,z), (x,z)\} \in S$
By definition of intersection
$T\cap S=\{ (x,y), (y,z), (x,z)\}$
That is, $T\cap S$ is also transitive.
Is this proof sufficient to show this statement?
Or how can improve my proof?
Thank you.