Let's imagine that we have a finite set S and finite subsets S_1, S_2,.....S_k. We also have 2 numbers: lower_i and higher_i for each subset. We want to answer: Is there a subset $$T \subseteq S $$ such that for each i runs: $$lower_i <= |T \cap S_i| <= higher_i$$
So, we want to prove that this problem is NP-complete by reduction 3-SAT NP-complete problem to this.
Suppose you have an instance of 3-SAT with variables $p_1,\ldots,p_n$. Then we make the set $S = \{p_1,\ldots,p_n,\neg p_1,\ldots,\neg p_n\}$ of all literals. For $1 \leq i \leq n$ we make a subset $S_i = \{p_i,\neg p_i\}$, with $\mathrm{lower}_i = \mathrm{higher}_i = 1$. Then for each clause $C = (l_1 \lor l_2 \lor l_3)$, where $l_1,l_2,l_3 \in S$, we make a set $S_C = \{l_1,l_2,l_3\}$ with $\mathrm{lower}_C = 1$ and $\mathrm{higher}_C = 3$.
An assignment of variables which satisfies the 3-SAT instance is the same as a set of literals $T$ which satisfies the subset problem.
