We have a bunch of intersecting regions: $$X_1,\dots, X_n,$$
all with non-negative volume, and we know $V(X_i)$ and $V\left((\cup_{a\in A}X_a)\cap (\cup_{b\in B}X_b)\right)$ for any disjoint $A,B\subset \{1,\dots,N\}.$ What is the simplest way to get $V(\cup_i X_i)$ in terms of these values?
I think there should be an easy formula for $V(\cup_i X_i)$ in terms of these values for general $n$, and it probably doesn't involve more than $2n-1$ terms.
For $n=2$ it's easy:
$$V(X_1\cup X_2)=V(X_1)+V(X_2)-V(X_1\cap X_2)$$
For $n=3$ it's still straightforward:
$$V(X_1\cup X_2 \cup X_3)= V(X_1)+V(X_2)+V(X_3)-V(X_1\cap X_2) - V((X_1\cup X_2)\cap X_3)$$
Thanks!
Enthdegree gave a recursive solution in the comments. This can be used to derive the explicit form
$$ V\left(\bigcup_iX_i\right)=\sum_iV(X_i)-\sum_iV\left(\left(\bigcup_{j\lt i}X_j\right)\cap X_i\right)\;, $$
of which your equations for $n=2$ and $n=3$ are special cases.