Suppose that I have a set $A$ and I want to compute the probability that this set contains at least one of the elements $x_1,\dots, x_n$. In other words,
$p(x_1 \in A \text{ or } \dots \text{ or } x_n \in A)$
I want to apply the inclusion-exclusion formula to compute this probability: $p(x_1 \in A \text{ or } \dots \text{ or } x_n \in A) $
$ = \sum_{k_1=1}^n p( \{x_{k_1}\} \subset A) - \sum_{k_1=1}^n \sum_{k_2=k_1}^n p(\{x_{k_1},x_{k_2}\} \subset A) + ... \\ + (-1)^{n-1} \sum_{k_1=1}^n \dots \sum_{k_n=k_{n-1}}^n p(\{x_{k_1},\dots x_{k_n}\} \subset A)$
Is this formula correct? Is there a better notation to show this that is easier to follow for the reader?
It's a matter of taste, and what you perceive your audience's comfort level with notation to be.
If I were writing it, I'd first define $A_i$ to be the event that $x_i\in A$, and then write something like $$P(\bigcup_i A_i) = \sum_i P(A_i) - \sum_{i<j} P(A_i\cap A_j) + \cdots $$ and maybe say that $\bigcup A_i$ is the event that $x_1\in A$ or $x_2\in A$, etc. The reason I prefer this notation is because it does not involve all those $\subset$ or $\in$ signs that your formula does.