Editing since this seems to be unclear: I ONLY have the data listed below. I do not have any information on the underlying elements. For example, I know A&B = 14, but I don't know which 14 elements match, or the values of those 14 elements.
I am trying to determine if the following is possible.
You have 3 sets A, B, C
You have the Size of A, Size of B, and Size of C.
You also have the intersect of A&B, intersect of A&C, and the intersect of B&C.
Is it possible to determine the intersect A&B&C?
Example:
size A = 25
size B = 30
size C = 22
A&B = 14
A&C = 10
B&C = 9
What is A&B&C?
Also, if this is possible, can it also be done on greater number of sets. For example, A,B,C,D given you have all intersect pairs?
Side note: I have been able to prove the following, but haven't been able to apply it to >2 sets.
Size A = 10
Size B = 20
AUB = 25
A&B = (size A + size B) - AUB
No. Consider the following examples:
$A = \{1,2\}$, $B = \{2,3\}$, $C = \{1,3\}$.
$A = \{1,2\}$, $B = \{1,3\}$, $C = \{1,4\}$.
In each, $|A| = |B| = |C| = 2$ and $|A\cap B| = |B\cap C| = |A\cap C| = 1$. But in the first example, $|A\cap B\cap C| = 0$, while in the second example, $|A\cap B\cap C| = 1$.