Partition Refinement (in writing ...)

Recently, i am studying the data structure of Partition Refinement. It is an useful technique to partition set. The main application of the technique is as follows:

$\textbf{Q}$: Given two sets $S$ and $X$, how to find $S \cap X$ and $S \setminus X$?

A general idea is that itemize each element $x \in X$ and test whether $x \in S$. Obviously, the running time is $O(|S| \cdot |X|)$. But the running time is not good. Is there a beautiful algorithm which has better complexity? Fortunately, we can solve the problem in $O(|X|)$ using Partition Refinement technique. In the following, i will describe the data structure of Partition Refinement and design an algorithm to solve the above problem. There are other applications, e.g., LBFS, the connected components of $\overline{G}$ and so on.