I've been looking through some exercises in set theory and came upon this one.
Let $R$ be a relation on $P(\mathbb Z)$ (=power set of integers) defined as follows:$$ARB(\text{A is in relation with B}) \iff A \cap B \not= \emptyset$$
Find transitive closure of $R$.
We defined transitive closure of a relation $R$ as $$\displaystyle R^T=\bigcup_{n\ge1}R^n$$ and the composition of relations as $$(x,z) \in RS \iff \exists y, xSy \text{ and } yRz $$
I have no idea how to find transitive closure, as we have only done exercises in finite cases which is relatively easy. It's a bit hard to visualize what would even $R^2$ look like, let alone all the other powers of $R$.
Any hint would be very helpful.
It would be helpful to observe that if $A$ and $B$ are not empty, then $C=A\cup B$ satisfies that both $A\cap C$ and $B\cap C$ are not empty.
So in reality, $R^T$ is just $R^2$. You should be able to get a better idea of what's going on now.