From inputs to Inclusion-Exclusion output
The sample counts give 40+35+30−12−9−8+4=80 members in at least one set.
Why the Inclusion-Exclusion relation holds
Adding three set sizes counts pairwise overlaps twice and the triple overlap three times. Subtracting pairwise intersections and restoring the triple intersection once leaves each union member counted exactly once.
Start the Inclusion-Exclusion review with Size of set A. Compare Size of set A with its source, then test Size of set B in a second Inclusion-Exclusion run without changing the first Inclusion-Exclusion case.
Add the individual set sizes, subtract all three pairwise intersections, and add the triple intersection back once. Inclusion-Exclusion also leads to zero-overlap case.
Failure points in a Inclusion-Exclusion setup
Intersection counts must be mutually consistent: a pairwise overlap cannot exceed either parent set, and a triple overlap cannot exceed any pairwise overlap.
How to label a Inclusion-Exclusion result
Increasing an exclusive portion raises the union equally; increasing a stated overlap can lower the union because more members were counted repeatedly. That behavior gives the inclusion-exclusion output a built-in reasonableness test.
Mutually exclusive events have zero intersections and therefore reduce the formula to simple addition. Writing “Union size” beside the output prevents that mix-up.
The principle combines survey categories, database audiences, divisibility sets, eligibility groups, and probability events with overlap. A related application of Inclusion-Exclusion is union by complement.
Interpreting the Inclusion-Exclusion output
Before accepting Union size, restore the Inclusion-Exclusion inputs Size of set A and Size of set B. Estimate Union size independently. Then vary Size of set C alone and observe the new Inclusion-Exclusion output. This isolates the changed part of Inclusion-Exclusion.
The number from Inclusion-Exclusion is incomplete without its convention. Record the roles Size of set A, Size of set B, Size of set C, Size of A∩B, Size of A∩C, Size of B∩C, Size of A∩B∩C, together with whether positions are distinguishable, events may overlap, or trials are replaced. Those facts define the sample space and cannot be recovered reliably from a copied result alone.
For Inclusion-Exclusion, use a boundary case such as zero events, one object, certainty, or impossibility when it belongs to the model. A correct boundary result is a useful defense against an unnoticed indexing or probability-scale error.