The Inclusion-Exclusion Principle is usually expressed as a way of determining unions from intersections, i.e.
$$\mathbb{P}(A_1\cup A_2)=\mathbb{P}(A_1)+\mathbb{P}(A_2)-\mathbb{P}(A_1\cap A_2)$$
$$\mathbb{P}(A_1\cup A_2\cup A_3)=\mathbb{P}(A_1)+\mathbb{P}(A_2)+\mathbb{P}(A_3)-\mathbb{P}(A_1\cap A_2)-\mathbb{P}(A_1\cap A_3)-\mathbb{P}(A_2\cap A_3)+\mathbb{P}(A_1\cap A_2\cap A_3)$$
I was wondering if the dual ('reverse?') (proper terminology?) was also possible, getting intersections from unions.
The first was trivial
$$\mathbb{P}(A_1\cap A_2)=\mathbb{P}(A_1)+\mathbb{P}(A_2)-\mathbb{P}(A_1\cup A_2)$$
and applying that to the $n=3$ also showed
$$\mathbb{P}(A_1\cap A_2\cap A_3)=\mathbb{P}(A_1)+\mathbb{P}(A_2)+\mathbb{P}(A_3)-\mathbb{P}(A_1\cup A_2)-\mathbb{P}(A_1\cup A_3)-\mathbb{P}(A_2\cup A_3)+\mathbb{P}(A_1\cup A_2\cup A_3)$$
Exact same formula as above with interchanged of $\cap \leftrightarrow \cup$
Questions
- Does that always hold true? The same formula of Inclusion-Exclusion can be used with $\cap \leftrightarrow \cup$ $$\mathbb{P}\left(\bigcup_{i=1}^n A_i\right)=\sum_{i=1}^n \mathbb{P}(A_i) -\sum_{i<j}\mathbb{P}(A_i\cap A_j)+\sum_{i<j<k}\mathbb{P}(A_i\cap A_j\cap A_k)- \cdots +(-1)^{n-1} \mathbb{P}\left(\bigcap_{i=1}^n A_i\right)$$
$$\mathbb{P}\left(\bigcap_{i=1}^n A_i\right)=\sum_{i=1}^n \mathbb{P}(A_i) -\sum_{i<j}\mathbb{P}(A_i\cup A_j)+\sum_{i<j<k}\mathbb{P}(A_i\cup A_j\cup A_k)- \cdots +(-1)^{n-1} \mathbb{P}\left(\bigcup_{i=1}^n A_i\right)$$
- Does this have a proper terminology or name to use for reference?