Question 121
For a finite set $S$ denote by $P(S)$ the set of all subsets of $S$ (including the empty set and the set $S$). The function $f : P(S) \to \mathbb{R}$ is such that
$$
f(X \cap Y) = \min(f(X), f(Y))
$$
for any two $X, Y \in P(S)$. Find the largest number of distinct values that $f$ can take.
$$
f(X \cap Y) = \min(f(X), f(Y))
$$
for any two $X, Y \in P(S)$. Find the largest number of distinct values that $f$ can take.