Decide whether this Problem NPC or P?

Question

Input: A finite set A, subsets S1, . . . , Sn ⊆ A, and a number k ∈ N. Question: Does there exist a set R ⊆ A with |R| = k such that |R ∩ Si| = |Si| for all 1 ≤ i ≤ n? I read somewhere (without mentioning why) that it's an NP-complete problem, but I think it can be solved simply by taking the union of all subsets and comparing it to k. If it isn't equal to k, then we can't form a subset R ⊆ A with |R| = k such that |R ∩ Si| = |Si| for all 1 ≤ i ≤ n. is it an NPC problem or P problem ?

Answer 1

Right, it is in $\text{PTIME}$. The condition $|R\cap S_i| = |S_i|$ is equivalent to $S_i\subseteq R$. So the problem is to find a subset $R$ whose size is $k$ and contains all sets in $\{S_i\}_{i\in [n]}$. In other words, $ \bigcup\limits_{i\in [n]} S_i \subseteq R$. So you can simply find the smallest such set $\bigcup\limits_{i\in [n]} S_i$ and then compare $|\bigcup\limits_{i\in [n]} S_i|$ to $k$. If $|\bigcup\limits_{i\in [n]} S_i| > k$ or $k > |A|$, then there is no such set $R$. Otherwise, there is such a set.

As a conclusion, the problem is not known to be $\text{NP-complete}$, since we do not know yet whether $\text{PTIME} = \text{NP}$.