How is the following relation transitive?

Question

Let $A$ be a set and let $\omega$ be a subset of $P(A)$ (power set). Suppose that the elements of $\omega$ are pairwise disjoint. Then, the relation $\sim \omega$ associated with $\omega$ is transitive. I have tried taking example sets and drawing things around but just can't understand the theorem. Could anyone please explain with a simple example? This is what I am learning in high school right now

Answer 1

Suppose $a\sim_\omega b$ and $b\sim_\omega c$, so that there are subsets $S,T\in\omega$ such that $a,b\in S$ and $b,c\in T$. If $S\ne T$, then since $S\cap T\ni b$ is nonempty, this contradicts the assumption that elements of $\omega$ are pairwise disjoint.

Thus, $S=T$ and $a\sim_\omega c$.