Question 18
Let $A = \{1, 2, \dots, 2012\}$, $B = \{1, 2, \dots, 19\}$ and $S$ be the set of all subsets of $A$. Determine the number of functions $f : S \to B$ satisfying the condition $f(A_1 \cap A_2) = \min\{f(A_1), f(A_2)\}$ for all $A_1, A_2 \in S$.