Example 3. Let $A=A_{1} \cup A_{2} \cup \cdots \cup A_{k}$. (i) The intersection of any $k$ sets is not empty; (ii) The intersection of any $k+1$ sets is an empty set. Find: (1) What is the minimum value of $|A|$; (2) What is the minimum value of $\left|A_{1}\right|$ when $|A|$ is the smallest?
(1) Consider any $r$-element subset $\left\{i_{1}, i_{2}, \cdots, i_{r}\right\}$ of the index set $\{1,2, \cdots, k\}$. For any element $a$ in $A_{1} \cap A_{1} \cap \cdots \cap A_{1}$, the correspondence $\left\{i_{1}, i_{2}, \cdots, i_{r}\right\} \rightarrow a$ is an injection. This is because if $\left\{j_{1}, j_{2}, \cdots, j_{r}\right\}$ also corresponds to $a$, it would result in the intersection of $\tau+1$ subsets being non-empty. Thus, when $f$ is a surjection, it is a bijection. This implies that $|A|$ is at least the number of $r$-element subsets of the index set, i.e., $|A| \geqslant C_{x}^{\tau}$.
(2) Consider any $A_{1}$. Take any other $r$ sets from $S$, their intersection has at least one element (non-empty). And this intersection is disjoint from $A_{1}$. Since there are $C_{k-1}^{r}$ different ways to choose, we have $\left|A_{1}\right| \leqslant|A|-C_{k-1}^{r}$. When $|A| = C_{k}^{r}$, $\left|A_{1}\right| \leqslant C_{r-1}^{r-1}$. On the other hand, $A_{1}$ intersected with any $r-1$ other sets chosen from $S$ has at least one element, thus $\left|A_{1}\right| \geqslant C_{k-1}^{r-1}$. This shows that when $|A|$ takes the minimum value $C_{k}^{r}$, the order of each $A_{1}$ is $C_{k-1}^{r-1}$.