# Reverse Inclusion - Exclusion Principle

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

1. 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

$$\mathbb{P}\left(\bigcap_{i=1}^n A_i\right)=\sum_{i=1}^n \mathbb{P}(A_i) -\sum_{i

1. Does this have a proper terminology or name to use for reference?
• Remember that $$\bigcap\limits_{i=1}^n A_i=\bigg(\bigcup\limits_{i=1}^n A_i^c\bigg)^c$$Where $A^c$ is the complement of $c$. This should directly answer your question. – Don Thousand Mar 5 at 16:02