In a three-set Venn diagram, fill the centre first, then the three regions that belong to exactly two sets, then the three that belong to exactly one, and finally the outside.
This lesson extends two-set Venn diagram questions within SPM Mathematics set operations. Read that lesson first if the two-circle method is still new.
Worked example: three school clubs
A survey asks students about Scouts (S), Robotics (R) and Drama (D). The data are below.
| Item | Count |
|---|---|
| n(S) | 28 |
| n(R) | 25 |
| n(D) | 20 |
| n(S ∩ R) | 10 |
| n(S ∩ D) | 8 |
| n(R ∩ D) | 7 |
| n(S ∩ R ∩ D) | 4 |
| In none of the clubs | 5 |
Find how many students were surveyed and how many belong to exactly two clubs.
Step 1, the centre. S ∩ R ∩ D = 4.
Step 2, pairs only. Each pair total includes the centre, so subtract 4.
S and R only = 10 − 4 = 6, S and D only = 8 − 4 = 4, and R and D only = 7 − 4 = 3.
Step 3, singles only. Take each set total and remove the regions already inside that circle.
Scouts only = 28 − (6 + 4 + 4) = 14, Robotics only = 25 − (6 + 3 + 4) = 12, and Drama only = 20 − (4 + 3 + 4) = 9.
Reading the answers
Inside the circles: 14 + 12 + 9 + 6 + 4 + 3 + 4 = 52 students. Add the 5 in no club, so 57 students were surveyed.
Exactly two clubs: 6 + 4 + 3 = 13 students. The region “all three” is separate and is not included.
Check with the formula: n(S ∪ R ∪ D) = 28 + 25 + 20 − 10 − 8 − 7 + 4 = 52. It matches the diagram.
The mistake that costs marks
The common slip is subtracting only the pair total and forgetting that the pair total contains the centre. A student writes S and R only = 10, which counts the 4 students in all three clubs twice.
| Region | Wrong | Right |
|---|---|---|
| S and R only | 10 | 10 − 4 = 6 |
| Scouts only | 28 − 10 − 8 = 10 | 28 − (6 + 4 + 4) = 14 |
| Total in circles | Too large or too small | 52 |
The wrong scouts-only value looks plausible, which is why the final total check matters. Always add every region and compare with the given total.
A reliable checklist
- Write the centre value in the middle.
- Subtract the centre from each pair total.
- Subtract all regions already inside that circle from each set total.
- Add all seven regions, then add the outside count.
If any region comes out negative, a number has been copied wrongly or the question gives inconsistent data.
Check yourself
In a class, n(A) = 15, n(B) = 14, n(C) = 12, n(A ∩ B) = 6, n(A ∩ C) = 5, n(B ∩ C) = 4, n(A ∩ B ∩ C) = 2, and 3 students are in none of the sets. Find the total number of students.
Answer
Pairs only: A and B only = 6 − 2 = 4, A and C only = 5 − 2 = 3, B and C only = 4 − 2 = 2.
Singles only: A only = 15 − (4 + 3 + 2) = 6, B only = 14 − (4 + 2 + 2) = 6, C only = 12 − (3 + 2 + 2) = 5.
Inside the circles: 6 + 6 + 5 + 4 + 3 + 2 + 2 = 28. Adding 3 outside gives 31 students.
What to study next
Practise the skill in the set operations practice set. For turning sentences into the set statements these diagrams need, see translating word statements into set notation.
If you want a teacher to check your diagrams step by step, see online one-to-one Mathematics tuition.