Given $n$ sets $A_1 \ldots A_n$, and $1 \lt k \lt n$, is it true that the union of the intersections of all the combinations of $k$ sets is equal to the intersection of the unions of the same combinations? If yes, how to prove it?
For example, for $n = 3$ and $k = 2$, I have verified that:
$$(A_1 \cap A_2) \cup (A_1 \cap A_3) \cup (A_2 \cap A_3) = (A_1 \cup A_2) \cap (A_1 \cup A_3) \cap (A_2 \cup A_3)$$
The first set consists of all the elements that are in at least $k$ of the $A_i$
The second set consists of all the elements that are in every union of $k$ $A_i$. So an element is not in the set if we can find $k$ $A_i$ that do not contain it, that is, the second set consists of all the elements that are in at least $n-k+1$ $A_i$