MathematicsChapter 3: Sets and Relations
Word Problems With Two Sets
Word problems hide union and intersection in ordinary language.
Learning objectives
- Explain Word Problems With Two Sets in simple mathematical language.
- Read the comparison table and choose the correct rule or method.
- Use the symbols, conditions, and formulae accurately.
- Solve a short example and check the answer.
- Recognise and correct common mistakes.
What this grid covers
Word problems hide union and intersection in ordinary language. Houses with lawns or porches, students who play two games, and people who drink tea or coffee all use n(A union B) = n(A) + n(B) - n(A intersect B). Draw two overlapping regions inside a rectangle U. 'Only A' is n(A) - n(A intersect B). 'Neither' is n(U) - n(A union B). Read 'or' as union unless the paper says exclusive or.
Where / symbols
- Houses
- Houses with lawns or porches, students who play two games, and people who drink tea or coffee all use n(A union B) = n(A) + n(B) - n(A intersect B).
- lawns
- Houses with lawns or porches, students who play two games, and people who drink tea or coffee all use n(A union B) = n(A) + n(B) - n(A intersect B).
- porches
- Houses with lawns or porches, students who play two games, and people who drink tea or coffee all use n(A union B) = n(A) + n(B) - n(A intersect B).
- games
- Houses with lawns or porches, students who play two games, and people who drink tea or coffee all use n(A union B) = n(A) + n(B) - n(A intersect B).
- tea
- Houses with lawns or porches, students who play two games, and people who drink tea or coffee all use n(A union B) = n(A) + n(B) - n(A intersect B).
- coffee
- Houses with lawns or porches, students who play two games, and people who drink tea or coffee all use n(A union B) = n(A) + n(B) - n(A intersect B).
- n(A union B) = n(A) + n(B) - n(A intersect B)
- Houses with lawns or porches, students who play two games, and people who drink tea or coffee all use n(A union B) = n(A) + n(B) - n(A intersect B).
- union
- Word problems hide union and intersection in ordinary language.
- intersection
- Word problems hide union and intersection in ordinary language.
- both
- 'Both' is the intersection.
Relationship
- n(A union B) = n(A) + n(B) - n(A intersect B)
WHAT — Core idea
Word problems hide union and intersection in ordinary language. Houses with lawns or porches, students who play two games, and people who drink tea or coffee all use n(A union B) = n(A) + n(B) - n(A intersect B). Draw two overlapping regions inside a rectangle U. 'Only A' is n(A) - n(A intersect B). 'Neither' is n(U) - n(A union B). Read 'or' as union unless the paper says exclusive or.
WHY — Why it matters
This idea helps you recognise the right mathematical pattern, choose a safe method, and explain your working clearly.
HOW — Method and examples
Houses on a street: Lawns, porches — Both, only lawn, neither A class of students: Two games (e.g. cricket, hockey) — How many play at least one A canteen survey: Tea drinkers, coffee drinkers — Both drinks, only tea, neither Library readers: Maths books, science books — Union and overlap Club members: Music, drama — n(U) minus the union
HOW — Answer in an exam
Fill three numbers first: n(A), n(B), n(A intersect B) or n(A union B). Then the formula gives the missing one. 'At least one' is the union. 'Both' is the intersection. Do not add n(A) and n(B) if people can be in both.
Typical stories, and what to find:
| Story | Sets | Usual unknown |
|---|---|---|
| 1. Houses on a street | Lawns, porches | Both, only lawn, neither |
| 2. A class of students | Two games (e.g. cricket, hockey) | How many play at least one |
| 3. A canteen survey | Tea drinkers, coffee drinkers | Both drinks, only tea, neither |
| 4. Library readers | Maths books, science books | Union and overlap |
| 5. Club members | Music, drama | n(U) minus the union |
In simple words
Fill three numbers first: n(A), n(B), n(A intersect B) or n(A union B). Then the formula gives the missing one. 'At least one' is the union. 'Both' is the intersection. Do not add n(A) and n(B) if people can be in both.
Key points
- Houses
- lawns
- porches
- games
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Word Problems With Two Sets
Sticky Notes · Quick Revision
WHAT
- Word problems hide union and intersection in ordinary language.
Key Points
- Word problems hide union and intersection in ordinary language.
- Typical stories, and what to find: Review each row and apply its rule carefully.
- Fill three numbers first: n(A), n(B), n(A intersect B) or n(A union B).
HOW
- Houses on a street: Lawns, porches — Both, only lawn, neither
- A class of students: Two games (e.g. cricket, hockey) — How many play at least one
- A canteen survey: Tea drinkers, coffee drinkers — Both drinks, only tea, neither
- Library readers: Maths books, science books — Union and overlap
- Club members: Music, drama — n(U) minus the union
Examples
- Houses on a street: Lawns, porches
- A class of students: Two games (e.g. cricket, hockey)
- A canteen survey: Tea drinkers, coffee drinkers
- Library readers: Maths books, science books
- Club members: Music, drama
Common Mistake
- Fill three numbers first: n(A), n(B), n(A intersect B) or n(A union B).
- Then the formula gives the missing one.
Exam Tip
- Fill three numbers first: n(A), n(B), n(A intersect B) or n(A union B). Then the formula gives the missing one. 'At least one' is the union. 'Both' is the intersection. Do not add n(A) and n(B) if people can be in both.
60-second revision
- Word problems hide union and intersection in ordinary language.
- Typical stories, and what to find: Review each row and apply its rule carefully.
- Fill three numbers first: n(A), n(B), n(A intersect B) or n(A union B).
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