Question 91
Let $A, B \subset \mathbb{N}$ be two finite and nonempty sets. Denote by $\mathcal{F}$ the set of all functions $f : \mathcal{P}(A) \to B$ such that
$$
f(X \cap Y) = \min(f(X), f(Y)), \forall X, Y \subset A,
$$
and by $\mathcal{G}$ the set of all functions $g : \mathcal{P}(A) \to B$ such that
$$
g(X \cup Y) = \max(g(X), g(Y)), \forall X, Y \subset A.
$$
Prove that $\mathcal{F}$ and $\mathcal{G}$ have the same number of elements and find that number.
$$
f(X \cap Y) = \min(f(X), f(Y)), \forall X, Y \subset A,
$$
and by $\mathcal{G}$ the set of all functions $g : \mathcal{P}(A) \to B$ such that
$$
g(X \cup Y) = \max(g(X), g(Y)), \forall X, Y \subset A.
$$
Prove that $\mathcal{F}$ and $\mathcal{G}$ have the same number of elements and find that number.