# Is the i-e principle for symmetric difference useful?

Let $E$ be a set and $A$ and $B$ two subsets. We may define the symmetric difference of $A$ and $B$ by setting :

$$A\Delta B:=(A\cup B)\cap (A\cap B)^c$$

There are many interesting things about this construction, one is that we have $(\mathcal{P}(E),\Delta)$ is a group. Another one is that we have an analogous of the inclusion-exclusion principle see there :

How I Prove The Cardinality of the Symmetric difference of $n$ sets

We also know that an element is in $A_1\Delta ...\Delta A_n$ if it is in an odd number of $A_i$.

Okay, so we know many things about it. This is easily dealt with and I expect my students to understand it. My concern is that I don't really have any interesting/relevant problem (of course I can artificially construct one) using the i-e principle for symmetric difference, so here is my question :

Do we have meaningful examples of the use of symmetric difference in combinatorics mathematics (maybe in probability) ?

Of course, in my set of exercises we count the number of surjective functions using i-e principle. Likewise, I would like to know if we can count something relevant with the symmetric difference i-e principle.

## 1 Answer

Maybe the following construction of a complete measure space satisfies your needs:

Let $(\Omega,\mathcal{F},\mu)$ be a measure space. Then $(\Omega,\mathcal{F'},\mu')$ is a complete measure space, whereat

$$\mathcal{F'}:=\{A\subseteq\Omega\mid\exists B\in \mathcal{F}\text{ such that }A\triangle B\text{ is a measure-zero set}\}$$

and

$$\mu'(A):=\mu(B).$$