Venn Diagram Problems made clear.
Fill the most specific overlaps first. For counting questions, every person belongs to exactly one region of the finished diagram, even if they belong to several sets.
Key concepts in Venn Diagram Problems
Two sets
n(A∪B) = n(A)+n(B)−n(A∩B). The common region was counted twice.
Only A
n(A only) = n(A)−n(A∩B). “Both” is usually included in each set total.
Neither
Neither = universe total − union.
Three sets
Union = A+B+C−AB−AC−BC+ABC, where pairwise intersections include the triple intersection.
Four non-overlapping regions
| Region | Count |
|---|---|
| Tea only | 20 |
| Both tea and coffee | 10 |
| Coffee only | 15 |
| Neither | 5 |
| All students | 50 |
A reliable solving method
- Identify the universe and whether set totals include overlaps.
- For three sets, put the triple intersection in first.
- Subtract it from pair intersections before filling “exactly two” regions.
- Add all exclusive regions and check against the universe total.
Step-by-step solved examples
Of 50 students, 30 like tea, 25 coffee and 10 both. How many like neither?
- Union = 30+25−10 = 45.
- Neither = 50−45.
- Five students like neither.
Using that data, count tea only and coffee only.
- Tea only = 30−10 = 20.
- Coffee only = 25−10 = 15.
- Check: 20+10+15+5 = 50.
A,B,C have sizes 20,18,15. Pair intersections are 8,6,5; triple is 3. Find union.
- Add set sizes: 20+18+15 = 53.
- Subtract pair intersections: 53−8−6−5 = 34.
- Add triple once: union = 37. Pair counts here include the triple.
Venn Diagram Problems: practice questions
Choose one answer before opening the explanation. Write down your reasoning as well as the option; a correct guess is different from a method you can repeat.
A=20, B=15, both=5. Union?
Show answer and explanation — question 1
Answer B: 30
20+15−5 = 30.
In the same data, A only?
Show answer and explanation — question 2
Answer C: 15
20−5 = 15.
Universe=40 and union=30. Neither?
Show answer and explanation — question 3
Answer B: 10
40−30 = 10.
Both=7, A only=12. Total in A?
Show answer and explanation — question 4
Answer D: 19
A includes its exclusive 12 plus the shared 7.
AB=9 including triple=4. In A and B but not C?
Show answer and explanation — question 5
Answer B: 5
9−4 = 5.
Disjoint sets A=8, B=12. Union?
Show answer and explanation — question 6
Answer D: 20
No overlap, so union is 8+12 = 20.
Frequently asked questions
Can the union exceed the population?
No, if all sets are drawn from that population. Such a result signals inconsistent data or a counting error.
Must two circles overlap?
No. Draw disjoint, intersecting or nested sets according to the information, not a fixed template.
Identify the universe and whether set totals include overlaps. Add all exclusive regions and check against the universe total.