How this calculator works
Set counts include every member of that set, including members shared with other sets. Pairwise intersections in three-set mode also include the triple intersection. Inclusion-exclusion first gives the union; subtracting shared counts then gives disjoint regions. Every region must be a non-negative whole number, and the union cannot exceed the universe. Missing-count modes solve the same identity using a supplied union. The region table identifies where each count belongs in a Venn diagram. Three-set symmetric difference counts members of an odd number of sets, including the triple region.
Formula
Two sets: union = A + B - AB. Three sets: union = A + B + C - AB - AC - BC + ABC. A only = A - AB - AC + ABC. Outside = U - union.
Worked example
With U=30, A=10, B=12, C=9, AB=4, AC=3, BC=5 and ABC=2, the union is 21. The A-only, B-only and C-only regions are 5, 5 and 3. Pair-only regions are 2, 1 and 3; the triple region is 2 and outside is 9.
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Frequently asked questions
Do pair intersections include the triple overlap?
Yes. Enter inclusive intersections. The tool subtracts the triple count to obtain pair-only regions.
Why is my input rejected?
A negative derived region or union larger than the universe is inconsistent with finite set counts.
Can I recover one missing intersection?
Choose its missing-count mode and enter the known union along with the other inclusive counts. For three-set-only quantities, select three sets first.