RO-NMO 2019 Problem 104

Let be two nonempty finite sets $A,B$ consisting of nonegative integers. Prove that $$\left\{ f:\mathcal{P} (A)\longrightarrow B| f\left( X\cap Y \right) =\min\left( \left| f(X) \right| ,\left| f(Y) \right| \right) \right\} =\left\{ g:\mathcal{P} (A)\longrightarrow B| f\left( X\cup Y \right) =\max\left( \left| g(X) \right| ,\left| g(Y) \right| \right) \right\}$$ and find the cardinal of these sets.

$\mathcal{P}$ denotes the power of sets.