We know that for $n>2$ events $\{A_1, \ldots, A_n\}$, the concepts of mutually independence and pairwise independence are different. If we take some subsets of the indices $[n]$, say $\mathcal{I} \subseteq 2^{[n]}$, can we always find an example such that
$$P(\cap_{i \in S} A_i) = \prod_{i \in S} P(A_i), \forall S\in \mathcal{I}$$ and $$P(\cap_{i \in S} A_i) \neq \prod_{i \in S} P(A_i), \forall S\not\in \mathcal{I}?$$
If such examples always exist, can we further assign the direction of inequality arbitarily? That is to say, given a disjoint union $\mathcal{I} \sqcup \mathcal{J} \sqcup \mathcal{K} = 2^{[n]}$, find an example such that
$$P(\cap_{i \in S} A_i) = \prod_{i \in S} P(A_i), \forall S\in \mathcal{I}$$ and $$P(\cap_{i \in S} A_i) > \prod_{i \in S} P(A_i), \forall S\in \mathcal{J}$$ and $$P(\cap_{i \in S} A_i) < \prod_{i \in S} P(A_i), \forall S\in \mathcal{K}.$$