Let $X$ be a nonempty set, and $\{A_\alpha : \alpha\in I\}$ be a partition of $X$. If $B\subseteq X$ such that $A_\alpha\cap B\neq\emptyset$ for every $\alpha\in I$, is $\{A_\alpha\cap B : \alpha\in I\}$ a partition of $B$?
I just need a starting point on how to think about this. Is there any logic that says taking an intersection between a partition of $X$ and a subset of $X$ gives you a partition of that subset?
Thanks!