Counting disjoint Venn regions

Place each object in exactly one region. Start with the overlap when it is known. If \(n(A)=a\), \(n(B)=b\) and \(n(A\cap B)=c\), the A-only and B-only counts are \(a-c\) and \(b-c\).

\(n(A\cup B)\)\({}=n(A)+n(B)-n(A\cap B)\)\(n((A\cup B)')\)\({}=n(\xi)-n(A\cup B)\)

Adding the two set totals counts the overlap twice; subtract it once. “Exactly one” means the sum of the two non-overlap regions. “At least one” includes the overlap.

Check that every region count is a nonnegative integer and that the four disjoint region counts sum to the universal total. A negative region count signals inconsistent data or an incorrect calculation.