Determine the number of pairs $(A, B)$ of subsets (possibly empty) of $\{1,2, \ldots, 10\}$ such that $A \cap B=\emptyset$.

N.B. If $A \neq B$ then $(A, B) \neq(B, A)$.
Answer: 59049. Determining a pair $(A, B)$ is equivalent to coloring each integer in blue, red, or green (the elements of $A$ in blue, those of $B$ in red, and the others in green). Therefore, there are $3^{10}$ ways to choose the pairs $(A, B)$.