Three-Set Venn Diagram Counting

Quick answer: For three finite sets, n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C). Pairwise intersections normally include the triple overlap.

Worked example

In a class of 40, three clubs have 20, 18 and 15 members. Pairwise overlaps are 8, 6 and 5; three students join all three. How many join at least one?

1. Add individual counts: 20 + 18 + 15 = 53.
2. Subtract pair overlaps: 53 − 8 − 6 − 5 = 34.
3. Restore the triple overlap once: 34 + 3 = 37.
4. Thus 40 − 37 = 3 join none.

Check the result

Exactly two clubs: (8 − 3) + (6 − 3) + (5 − 3) = 10. Exactly one: 37 − 10 − 3 = 24.

Common mistake to avoid

Do not subtract the triple overlap from a pair count unless calculating a pair-only region.

Try it yourself

1. In this example, how many are in A only? 2. How many are in at least two clubs?

Answers and reasoning

1. 20 − 8 − 6 + 3 = 9. 2. Exactly two plus all three = 10 + 3 = 13.

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