Learning Objectives
- find the union of sets;
- find the intersection of sets;
- use ∪ and ∩ correctly;
- shade union and intersection regions;
- apply the two-set counting formula.
Introduction
This lesson develops Union and Intersection 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
Union and intersection allow us to combine groups and identify what they share. These operations appear frequently in surveys and probability questions.
Union
A ∪ B contains elements in A or B or both.
A = {1,2,3,4}, B = {3,4,5,6}
A ∪ B = {1,2,3,4,5,6}
Explore the shaded regions
Choose an operation above to highlight the correct region.
Intersection
A ∩ B contains elements common to both sets.
A = {1,2,3,4}, B = {3,4,5,6}
A ∩ B = {3,4}
Union and Intersection Compared
| Operation | Meaning | Key word |
|---|---|---|
| A ∪ B | In A or B or both | at least one |
| A ∩ B | In both A and B | common/shared |
The Counting Formula
The intersection is subtracted because it was counted once in n(A) and once again in n(B).
n(A)=18, n(B)=15 and n(A∩B)=6.
n(A∪B)=18+15−6=27.
🔍 Recognition Guide
- “A or B” → union.
- “A and B”, “both” or “common” → intersection.
- “at least one” → union.
- Never count an overlap twice.
💡 Exam Tip
Translate the symbols into words before calculating: union means “in A or B or both,” while intersection means “in both A and B.”
✨ Explore Union and Intersection
Use the buttons to reveal the region represented by each operation.
🧠 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- A={1,2,3,4}, B={3,4,5}. Find A∪B and A∩B.
- If A∩B=∅, what does this tell you?
- n(A)=12, n(B)=9 and n(A∩B)=4. Find n(A∪B).
- In a class, 20 learners study French, 17 study Shona and 8 study both. How many study at least one?
- U={1,2,...,12}, A=multiples of 2 and B=multiples of 3. Find A∪B and A∩B.
- n(A∪B)=40, n(A)=27 and n(B)=21. Find n(A∩B).
✅ Check Your Work
Show practice answers
- A∪B={1,2,3,4,5}; A∩B={3,4}.
- The sets are disjoint; they share no elements.
- 12+9−4=17.
- 20+17−8=29 learners.
- A={2,4,6,8,10,12}, B={3,6,9,12}; A∪B={2,3,4,6,8,9,10,12}; A∩B={6,12}.
- 40=27+21−n(A∩B), so n(A∩B)=8.
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
- Union means in at least one set.
- Intersection means in both sets.
- ∪ means union; ∩ means intersection.
- Subtract the overlap when counting a union.