Learning Objectives
You should be able to:
- identify the seven regions inside a three-set Venn diagram;
- fill a three-set diagram from the centre outward;
- distinguish “exactly two” from “at least two”;
- solve and check three-set survey problems.
Introduction
This lesson develops Three-Set Problems from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.
Key Notes
Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.
🌍 Why This Matters
Three-set diagrams organise more complicated survey information into seven separate inside regions. Working from the centre outward keeps the data consistent.
Three-Set Venn Diagrams
Three-set problems contain seven regions inside the circles. Work from the centre outward.
Three-set Venn diagram: start at the centre
The Seven Inside Regions
- A only
- B only
- C only
- A and B only
- A and C only
- B and C only
- A and B and C
Worked Example
In a survey of 50 learners: n(A)=24, n(B)=22, n(C)=20, n(A∩B)=9, n(A∩C)=8, n(B∩C)=7 and n(A∩B∩C)=3.
Centre = 3.
A∩B only = 9−3=6.
A∩C only = 8−3=5.
B∩C only = 7−3=4.
A only = 24−6−5−3=10.
B only = 22−6−4−3=9.
C only = 20−5−4−3=8.
At least one = 45, so neither = 5.
Three-Set Counting Formula
The triple intersection is added back because subtracting the three pairwise intersections removes it too many times.
💡 Exam Tip
Fill the centre first, then the pairwise-only regions, then the single-only regions, and calculate neither last.
🧠 Let’s Think Together
Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.
Practice Questions
Foundation- n(A∩B)=10 and n(A∩B∩C)=4. Find the number in A and B only.
- n(A)=25. Inside A there are 7 in A∩B only, 5 in A∩C only and 3 in all three. Find A only.
- n(A)=30, n(B)=28, n(C)=24, pairwise intersections are 12, 10 and 9, and the triple intersection is 4. Find n(A∪B∪C).
- If n(U)=80 and the union in Question 3 is known, find neither.
- In a survey of 100 people, 56 chose A, 48 chose B, 40 chose C, 22 chose A and B, 18 chose A and C, 16 chose B and C, and 8 chose all three. How many chose none?
✅ Check Your Work
Show practice answers
- 10−4=6.
- 25−7−5−3=10.
- 30+28+24−12−10−9+4=55.
- 80−55=25.
-
Union = 56+48+40−22−18−16+8 = 96.
None = 100−96 = 4.
Common Mistakes
- Choosing a rule before identifying what the question describes.
- Skipping working or changing notation part-way through a solution.
- Accepting an answer without checking its size, sign or units.
Exam Focus
Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.
Summary
- Start with the triple intersection.
- Then fill pairwise-only regions.
- Then calculate single-set-only regions.
- Use inclusion–exclusion to check the total.