Express inclusion-exclusion principle in terms of matrix operations

Question

First of all i denote $\{1,2,3,...,m\}$ by $[m]$

Let there be a collection of sets $\alpha=\{A_{1},A_{2},...,A_{m}\}$ such $\bigcup_{i\in[m]}A_{i}\subseteq [n]$ Consider any function $f:\mathcal{P}([n])\rightarrow \mathbb{C}$

It is well known that $$f(\bigcup_{i\in[m]}A_{i})=\sum_{i=1}^{m}(-1)^{i-1}\sum_{I\subseteq[m],|I|=i}f(\bigcap_{l\in I}A_{l})$$

That formula is called a inclusion-exclusion principle

But what if we associate to $\alpha$ a matrix $\beta=(x_{A_{i}}(j))_{i\in[m],j\in[n]}$

Where $x_{A_{i}}(j)=\begin{cases} 1 &\text{when } j\in A_{i}\\0 &\text{otherwise } \end{cases}$

How to express inclusion-exclusion principle using operations on matrix $\beta$?

By operation i mean for example adding, multiplying rows, columns, taking determinant etc.

In particular i am intrested in case when $f$ is probability measure P.

Here are the things i done in this case: Consider binomiall and independent distribution over all the numbers from the set $[n]$. Probability that we pick an element $i$ is equal to $p$.

I noticed that $f(A\cap B)=P(A\cap B)=p^{|A\cup B|}$ (where $A$ stands both for set and an event to include set , same with $B$)

But how to use a matrix there?

Answer 1

As darij grinberg noted, the inclusion-exclusion principle holds only if $f$ is additive over $[n]$ or, equivalently, is a (signed) measure over $[n]$, and then $$f(A)=\int x_A\,df$$ for any $A\subseteq[n]$.

Now, to get the inclusion-exclusion principle from properties of indicators $x_A$ of sets $A$, write \begin{align}x_{\bigcup_1^m A_j}&=1-\prod_1^m(1-x_{A_j}) \\ &=\sum_{i=1}^m(-1)^{i-1}\sum_{J\in\binom{[n]}i}\prod_{j\in J} x_{A_j} \\ &=\sum_{i=1}^m(-1)^{i-1}\sum_{J\in\binom{[n]}i}x_{\bigcap_1^m A_j}, \end{align} where $\binom{[n]}i$ is the set of all sets $J$ such that $J\subseteq[n]$ and $|J|=i$.
Integrating now with respect to $df$, we get the inclusion-exclusion principle: \begin{align}f(\bigcup_1^m A_j)&= \sum_{i=1}^m(-1)^{i-1}\sum_{J\in\binom{[n]}j}f(\bigcap_1^m A_j). \end{align}