(16 points) Let $A, B, A, (1 \leqslant i \leqslant k)$ be sets.
(1) How many ordered pairs $(A, B)$ satisfy $A \cup B=\{a, b\}$? Why?
(2) How many ordered pairs $(A, B)$ satisfy $A \cup B=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$? Why?
(3) How many ordered tuples $\left(A_{1}, A_{2}, \cdots, A_{k}\right)$ satisfy $A_{1} \cup A_{2} \cup \cdots \cup A_{k}=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$? Why?
Ni, (1)9 $\quad(2) 3^{n} \quad(3)\left(2^{k}-1\right)^{n}$
(1) and (2) are special cases of (3), so we only need to prove (3).

To determine the number of ordered sets $\left(A_{1}, A_{2}, \cdots, A_{k}\right)$, we can divide it into $n$ steps.

In the first step, consider the possibilities for $a_{1}$ to belong to $A_{1}, A_{2}, \cdots, A_{k}$. For $A_{1}$, there are two possibilities: $a_{1} \in A_{1}$ or $a_{1} \notin A_{1}$. Similarly, for $A_{2}$, there are two possibilities, and so on, for $A_{k}$, there are also two possibilities. This gives a total of $2^{k}$ possibilities.

However, since $A_{1} \cup A_{2} \cup \cdots \cup A_{k}=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$, there are $2^{k}-1$ possibilities.

Similarly, in the second step, consider the possibilities for $a_{2}$ to belong to $A_{1}, A_{2}, \cdots, A_{k}$, which also has $2^{k}-1$ possibilities.

In the $n$-th step, consider the possibilities for $a_{n}$ to belong to $A_{1}, A_{2}, \cdots, A_{k}$, which also has $2^{k}-1$ possibilities.
By the multiplication principle, we get
the number of ordered sets $\left(A_{1}, A_{2}, \cdots, A_{k}\right)$ is $\left(2^{k}-1\right)^{n}$.