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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 $\cap \leftrightarrow \cup$ $$\mathbb{P}\left(\bigcup_{i=1}^n A_i\right)=\sum_{i=1}^n \mathbb{P}(A_i) -\sum_{i<j}\mathbb{P}(A_i\cap A_j)+\sum_{i<j<k}\mathbb{P}(A_i\cap A_j\cap A_k)- \cdots +(-1)^{n-1} \mathbb{P}\left(\bigcap_{i=1}^n A_i\right)$$

$$\mathbb{P}\left(\bigcap_{i=1}^n A_i\right)=\sum_{i=1}^n \mathbb{P}(A_i) -\sum_{i<j}\mathbb{P}(A_i\cup A_j)+\sum_{i<j<k}\mathbb{P}(A_i\cup A_j\cup A_k)- \cdots +(-1)^{n-1} \mathbb{P}\left(\bigcup_{i=1}^n A_i\right)$$

  1. Does this have a proper terminology or name to use for reference?
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  • $\begingroup$ 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. $\endgroup$ – Don Thousand Mar 5 at 16:02
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1: Yes

it always holds true, this is called:

2: The General form of the Inclusion-Exclusion Principle

See Wikipedia

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