Application of Sets
These QuickNotes give you the most important points to remember for the ESSLCE. They are based on the MoE Grade 9 Mathematics Textbook, Unit 1, Chapter 1.5.
~3 min read- The main tool is the formula n(A ∪ B) = n(A) + n(B) – n(A ∩ B).
- n(A) means the number of elements in set A.
- You subtract n(A ∩ B) so you do not count the shared elements twice.
- If the two sets share nothing (disjoint), then n(A ∪ B) = n(A) + n(B).
Key Words
- n(A) is the number of elements in set A.
- Disjoint sets share no elements, so n(A ∩ B) = 0.
- The union formula counts the elements that are in A or B, without counting the shared ones twice.
Why we subtract the overlap
n(A) and n(B) both count the “both” part, so adding them counts it twice. We subtract n(A ∩ B) once to fix this.
Counting the Elements of a Union
- The number of elements in A ∪ B is n(A ∪ B) = n(A) + n(B) – n(A ∩ B).
- You subtract n(A ∩ B) because the shared elements were counted twice, once in n(A) and once in n(B).
- If the two sets are disjoint, then n(A ∩ B) = 0, so n(A ∪ B) = n(A) + n(B).
- You can rearrange the formula to find a missing number, for example n(A ∩ B) = n(A) + n(B) – n(A ∪ B).
- For example, if n(A) = 20, n(B) = 28, and n(A ∪ B) = 36, then n(A ∩ B) = 20 + 28 – 36 = 12.
Common Mistakes to Avoid
- Do not just add n(A) and n(B). If the sets share elements, you must subtract n(A ∩ B), or you count the shared ones twice.
- Only when the sets are disjoint is n(A ∪ B) = n(A) + n(B).
- n(A ∩ B) is the number of shared elements, not the shared elements themselves.
Easy Ways to Remember
- Add the two sets, then take away the overlap once.
- For a word problem, draw a Venn diagram and write the “both” number in the middle first, then fill in the rest.
Quiz
Tap an answer to check it.
1. If n(A) = 20, n(B) = 28, and n(A ∪ B) = 36, what is n(A ∩ B)?
Rearrange the formula: n(A ∩ B) = n(A) + n(B) – n(A ∪ B) = 20 + 28 – 36 = 12.
2. If n(A) = 34, n(B) = 46, and n(A ∪ B) = 70, what is n(A ∩ B)?
n(A ∩ B) = 34 + 46 – 70 = 10. The trap is 80, which is just n(A) + n(B) without subtracting the overlap.
3. At a meeting of 60 people, 42 drink tea and 27 drink coffee. Everyone drinks at least one. How many drink both?
Here n(A ∪ B) = 60 (everyone drinks at least one). So both = 42 + 27 – 60 = 9.
Remember: To count a union, use n(A ∪ B) = n(A) + n(B) - n(A ∩ B). Add the two sets, then subtract the overlap so you do not count it twice.