Derive and prove a general formula for the number of elements which are in an odd number of the sets $A_1,A_2,...,A_n$, written in terms of $|A1|$, $|A2\cap A7|$, $|A3\cap A4\cap A9|$, etc., possibly multiplied by coefficients.
I do not really understand this question, does it mean number of elements in total that are included in either $1$ set, $3$ sets, $5$ set, etc of any of the $n$ sets? Or does it mean we need to calculate the cardinality of a set like $|A_1|$ or $|A_1\cup A_2\cup A_3|$?
The later one is much easier to calculate, we just use the inclusion-exclusion principle. But if it was the first case, how can we possibly calculate number of elemnets without using the union symbol?
To be found is:$$\left|\left\{x:\sum_{i=1}^n\mathbf1_{A_i}(x)\text{ is odd}\right\}\right|$$
It can be verified with induction that:$$\left\{x:\sum_{i=1}^n\mathbf1_{A_i}(x)\text{ is odd}\right\}=\Delta_{i=1}^n A_i$$For the induction step observe that: $$\sum_{i=1}^{n+1}\mathbf1_{A_i}(x)\text{ is odd}\iff $$$$\sum_{i=1}^{n}\mathbf1_{A_i}(x)\text{ is odd and }\mathbf1_{A_{n+1}}(x)=0\vee\sum_{i=1}^{n}\mathbf1_{A_i}(x)\text{ is even and }\mathbf1_{A_{n+1}}(x)=1$$
Also with induction it can be shown that:
$$1_{\Delta_{i=1}^n A_i}=\sum_{i=1}^n1_{A_i}-2\sum_{1\leq i<j\leq n}1_{A_i\cap A_j}+\cdots+(-2)^{n-1}1_{A_1\cap\cdots\cap A_n}$$
If $\mu$ denotes a measure and on both sides we integrate then we find:$$\mu(\Delta_{i=1}^n A_i)=\sum_{i=1}^n\mu(A_i)-2\sum_{1\leq i<j\leq n}\mu(A_i\cap A_j)+\cdots+(-2)^{n-1}\mu(A_1\cap\cdots\cap A_n)$$
Taking for $\mu$ the counting measure we find:$$|\Delta_{i=1}^n A_i|=\sum_{i=1}^n|A_i|-2\sum_{1\leq i<j\leq n}|A_i\cap A_j|+\cdots+(-2)^{n-1}|A_1\cap\cdots\cap A_n|$$