To solve a two-set Venn diagram question, fill the overlap first, then the two “only” regions, then the region outside both circles. Every number must add up to n(ξ).
This lesson is part of SPM Mathematics set operations. If the symbols are unfamiliar, read reading union, intersection and complement notation first.
Worked example: a survey of 40
Here is an original problem. Of 40 students in a class, 25 play football (F), 18 play badminton (B) and 6 play both. Find how many play only football, only badminton, and neither sport.
Step 1, the overlap. Write 6 in the region shared by F and B.
Step 2, the only regions. Football only = 25 − 6 = 19. Badminton only = 18 − 6 = 12.
Step 3, outside. Inside the circles there are 19 + 6 + 12 = 37 students. So neither = 40 − 37 = 3.
The answers are 19 play only football, 12 only badminton and 3 neither.
Check with the union formula
n(F ∪ B) = n(F) + n(B) − n(F ∩ B) = 25 + 18 − 6 = 37. That matches the total inside the circles.
It also tells you how many play at least one sport (37) and exactly one sport (19 + 12 = 31). Different words in the question point to different regions.
| Wording | Region | Value |
|---|---|---|
| Both sports | F ∩ B | 6 |
| Football only | F ∩ B′ | 19 |
| Exactly one sport | Only F plus only B | 31 |
| At least one sport | F ∪ B | 37 |
| Neither | (F ∪ B)′ | 3 |
The mistake that costs marks
The common slip is writing 25 in the football circle and 18 in the badminton circle, then adding 6 for the overlap. The diagram now counts 25 + 18 + 6 = 49 students, which is more than the class of 40.
The correct diagram has 19 in the football-only region, because the 25 football players already include the 6 who play both.
| Step | Wrong | Right |
|---|---|---|
| Football region | 25 | 19 (only) |
| Badminton region | 18 | 12 (only) |
| Total inside circles | 49 | 37 |
A quick test helps: add every number on the diagram and compare with n(ξ). If the sum is wrong, the overlap has been double-counted or the outside region is missing.
When the overlap is not given
Some questions give the union or the neither count instead. In a class of 36, 20 like Milo (M), 15 like tea (T) and 7 like neither. Find how many like both.
n(M ∪ T) = 36 − 7 = 29. Use the formula backwards: 29 = 20 + 15 − n(M ∩ T), so n(M ∩ T) = 35 − 29 = 6. Then Milo only = 14 and tea only = 9, and 14 + 6 + 9 + 7 = 36.
Check yourself
In a group of 50 pupils, 32 own a bicycle (B), 27 own a scooter (S) and 8 own neither. How many own both, and how many own exactly one?
Answer
n(B ∪ S) = 50 − 8 = 42. From 42 = 32 + 27 − n(B ∩ S), n(B ∩ S) = 59 − 42 = 17.
Bicycle only = 32 − 17 = 15, and scooter only = 27 − 17 = 10.
Exactly one = 15 + 10 = 25. Check: 15 + 17 + 10 + 8 = 50.
What to study next
Three sets follow the same centre-out order but have more regions. Continue with solving three-set Venn diagram questions, or test all four skills in the set operations practice set.
For a teacher to watch your diagrams as you build them, see online one-to-one Mathematics tuition.