Difference of two sets and distributivity If $A,B,C$ are sets, then we all know that $A\setminus (B\cap C)= (A\setminus B)\cup (A\setminus C)$. So by induction 
$$A\setminus\bigcap_{i=1}^nB_i=\bigcup_{i=1}^n (A\setminus B_i)$$
for all $n\in\mathbb N$.
Now if $I$ is an uncountable set and $\{B_i\}_{i\in I}$ is a family of sets, is it true that:
$$A\setminus\bigcap_{i\in I}B_i=\bigcup_{i\in I} (A\setminus B_i)\,\,\,?$$
If the answer to the above question will be "NO", what can we say if $I$ is countable?

Note that
$$
A\setminus \bigcap_{i\in I}B_i=A\cap \left(\bigcap_{i\in I}B_i\right)^c,
$$
where $^c$ denotes the complement. Using De Morgan's laws (which holds for a general $I$) we get
$$
A\cap \left(\bigcap_{i\in I}B_i\right)^c =A\cap \left(\bigcup_{i\in I}B_i^c\right)=\bigcup_{i \in I}\left(A\cap B_i^c\right).
$$
Now note that $A\cap B_i^c =A\setminus B_i$ and we are done.
