Problem

Source: Science ON 2021 grade X/2

Tags: algebra, combinatorics, functional equation



Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that $$|f(A)\cap f(B)|=|A\cap B|$$whenever $A$ and $B$ are two distinct subsets of $X$. (Sergiu Novac)