Venn Diagrams and Inclusion-Exclusion: Counting Overlaps in Surveys
October 29, 2026
When groups overlap, adding their sizes counts the overlap twice. The inclusion-exclusion principle fixes this. For two sets: |A ∪ B| = |A| + |B| − |A ∩ B|. The union is the number in at least one set; the intersection is the number in both.
In a survey of 100 students, 60 like pizza (A), 45 like tacos (B), and 20 like both. At least one: 60 + 45 − 20 = 85. Neither: 100 − 85 = 15. Only pizza: 60 − 20 = 40. Only tacos: 45 − 20 = 25.
Skip the math and get your answer instantly:
Open the Percentage Calculator →Regions of the Venn Diagram
The diagram has four regions for two sets: only A (40), both (20), only B (25), and neither (15), which sum to 100. Always check that all regions add up to the total. A common mistake is to report 'pizza 60, tacos 45' as separate groups that total 105, which exceeds the 100 students; the overlap is the reason.
| Region | Calculation | Students |
|---|---|---|
| Only pizza | 60 - 20 | 40 |
| Both | given | 20 |
| Only tacos | 45 - 20 | 25 |
| Neither | 100 - 85 | 15 |
| Total | 100 |
Three Sets
For three sets: |A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|. Suppose 30 people use app A, 25 use B, and 20 use C. Overlaps: A and B 10, A and C 8, B and C 6, and all three 3. The union is 30 + 25 + 20 − 10 − 8 − 6 + 3 = 54. The triple overlap is added back because it was subtracted three times in the pairwise overlaps after being counted three times initially, leaving zero. The number using exactly one app = 54 − (people in exactly two) − 3; exactly two = (10 − 3) + (8 − 3) + (6 − 3) = 15, so exactly one = 54 − 15 − 3 = 36.
Applications
Customer overlap among product lines, audience reach across media (unique viewers), probability: P(A or B) = P(A) + P(B) − P(A and B), counting numbers divisible by 2 or 3 up to 100 (50 + 33 − 16 = 67), and database queries combining conditions. In marketing, the unduplicated reach of two campaigns is less than the sum of their audiences.
Problem-Solving Tips
Start from the innermost region (triple overlap) and work outward by subtracting. Draw the diagram, fill in the numbers, and use the total to find 'neither'. Read the wording: 'at least one', 'exactly one', 'both', and 'only' mean different regions.
Frequently Asked Questions
What is the inclusion-exclusion formula for two sets?
|A ∪ B| = |A| + |B| − |A ∩ B|.
How do I find those in neither set?
Subtract the union from the total.
Why add back the triple overlap for three sets?
Because it was subtracted too many times in the pairwise overlaps.
What does 'only A' mean?
In A but not in B or C.
Where is it used?
Surveys, probability, marketing reach, and counting problems.