Question 8
Let $S = \{1, 2, \dots, 6\}$. Let $\mathcal{S}$ be the set of all subsets of $S$. How many mappings $F: \mathcal{S} \to \mathcal{S}$ are there that satisfy the following condition:
$$
F(F(A) \cup B) = A \cap F(B) \text{ for any pair of subsets of } S.
$$
$$
F(F(A) \cup B) = A \cap F(B) \text{ for any pair of subsets of } S.
$$