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?