(5) Let $A_{1}, A_{2}, \cdots, A_{n}$ be $n$ different subsets of the set $S=\{1,2, \cdots, n\}$ $(n \geqslant 4)$. To represent these subsets, construct an $n$ by $n$ array of numbers, where the number in the $i$-th row and $j$-th column is defined as $a_{i j}=\left\{\begin{array}{ll}0, & i \notin A_{j}, \\ 1, & i \in A_{j},\end{array}\right.$ Then, which of the following statements is incorrect? ( ).
(A) All the numbers in the first column of the array are 0 if and only if $A_{1}=\varnothing$
(B) All the numbers in the $n$-th column of the array are 1 if and only if $A_{n}=S$
(C) The sum of the numbers in the $j$-th row of the array indicates how many elements are in the set $A_{j}$
(D) The sum of all $n^{2}$ numbers in the array does not exceed $n^{2}-n+1$
(5) $\mathrm{C}$ Hint: When one of $A_{1}, A_{2}, \cdots, A_{n}$ is $S$ itself, and the remaining $n-1$ subsets are $n-1$ distinct $(n-1)$-element subsets of $S$, the sum of all $n^{2}$ numbers in the array is maximized, being $n^{2}-n+1$, thus, $\mathrm{D}$ is correct. The sum of the numbers in the $j$-th row of the array indicates how many subsets element $j$ belongs to, therefore, $\mathrm{C}$ is incorrect.