Consider the set $[2n] = \{1,2,\dots,2n\}$ for some positive integer $n$. Let $A_1,\dots A_k$ and $B_1,\dots,B_k$ be subsets of $[2n]$ such that every subset of $[2n]$ of cardinality $2$ or more is equal to $B_i\cup A_j$ for some $i,j$. What is the minimal $k$ possible?
First of all, if $n=1$, then $k=1$ is sufficient, so suppose $n>1$.
A first bound is $k^2\ge 4^n -2n-1$, so $k\ge 2^n$.
Notice that combining all $a$-subsets with all $b$-subsets, where $b\ge a$, you obtain all the $c$-subsets where $b\le c\le a+b$, so an upper bound would be $$ k\le \binom{2n}{2} + \binom{2n}{5} + \binom{2n}{11} + \binom{2n}{23} +\dots $$ choosing $A_i=B_i$ all subsets of cardinality $2,5,11,23,\dots$.
If $A_i$ are the subsets of $\{1,\dots,n\}$ and $B_j$ are the subsets of $\{n+1,\dots,2n\}$, then the hypotheses are satisfied, so the minimum $k$ is $2^n$.