Set containing another set How can I state using Set Notation that each element $n$ in the set $A$ contains a unique set $B$ ?

If you mean that $B$ is an element of each $n\in A$, you can write $B\in\bigcap A$: for any collection $A$ of sets, by definition $x\in\bigcap A$ if and only if $x\in n$ for each $n\in A$. If you mean that $B$ is a subset of each $n\in A$, then each element of $B$ is an element of $\bigcap A$, and you can write $B\subseteq\bigcap A$.
There are of course more complicated ways to express the same things.
