1.16 A set with $n$ elements has how many different pairs of disjoint subsets?
[Solution] Let $A$ and $B$ be two disjoint subsets. For any element $x$ in an $n$-element set $M$, either $x \in A$, or $x \in B$, or $x \in M-(A \cup B)$, and exactly one of these three conditions holds. Therefore, there are $3^n$ different ways to partition, which means there are $3^n$ different ordered pairs of disjoint subsets.

If the pairs of subsets are unordered, i.e., $\{A, B\}$ and $\{B, A\}$ are considered the same pair, then because $\{\varnothing, \emptyset\}$ remains the same after swapping the order, the total number of such pairs of disjoint subsets is
$$
\frac{3^n-1}{2}+1=\frac{3^n+1}{2} \text{. }
$$