12. Let $A, B$ be subsets of the set $X=\{1,2, \cdots, n\}$. If every number in $A$ is strictly greater than all the numbers in $B$, then the ordered subset pair $(A, B)$ is called "good". Find the number of "good" subset pairs of $X$.
12. Let $|A \cup B|=k$. The number of subsets with $k$ elements is $C_{n}^{k}$. For any $k$-element subset $\left\{a_{1}, a_{2}, \cdots, a_{k}\right\} \subseteq X, a_{1}<a_{2}<\cdots<a_{k}$, set $B$ can be $\emptyset, \left\{a_{1}\right\}, \left\{a_{1}, a_{2}\right\}, \cdots, \left\{a_{1}, a_{2}, \cdots, a_{k}\right\}$, for a total of $k+1$ possibilities. Therefore, the number of good subset pairs is
$$
\sum_{k=0}^{n}(k+1) C_{n}^{k}=2^{n}+n \cdot 2^{n-1} .
$$