Counting with Venn diagrams: Class 11 practice
For two sets, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Set notation
The idea behind counting with venn diagrams
For two sets, n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Subtracting the overlap stops it being counted twice. 'Neither' = total − n(A ∪ B).
Step-by-step method
- Write n(A), n(B) and n(A ∩ B).
- Find n(A ∪ B) with the formula.
- Use the total for 'neither' or 'only A' questions.
Worked example 1 (Basic)
n(A) = 28, n(B) = 21, n(A ∩ B) = 11. Find n(A ∪ B).
- 28 + 21 − 11 = 38
- Answer: 38
Worked example 2 (Standard)
In a group of 51, 34 like tea, 17 like coffee and 14 like both. How many like neither?
- n(T ∪ C) = 37
- Neither = 51 − 37 = 14
- Answer: 14
Worked example 3 (Challenge)
Of 36 students, 23 take Maths, 21 take Physics and 3 take neither. How many take only Maths?
- n(M ∪ P) = 33
- n(M ∩ P) = 23 + 21 − 33 = 11
- Only Maths = 23 − 11 = 12
- Answer: 12
Common mistakes and misconceptions
- Adding without removing the overlap
- Confusing 'only A' with n(A)
Practice questions
- (Basic) n(A) = 20, n(B) = 23, n(A ∩ B) = 6. Find n(A ∪ B).
- (Basic) n(A) = 26, n(B) = 9, n(A ∩ B) = 6. Find n(A ∪ B).
- (Standard) In a group of 31, 18 like tea, 18 like coffee and 10 like both. How many like neither?
- (Standard) In a group of 43, 19 like tea, 15 like coffee and 5 like both. How many like neither?
- (Challenge) Of 30 students, 7 take Maths, 17 take Physics and 9 take neither. How many take only Maths?
- (Challenge) Of 42 students, 23 take Maths, 22 take Physics and 6 take neither. How many take only Maths?
Answer key with full working
- Q1: 20 + 23 − 6 = 37 → Answer: 37
- Q2: 26 + 9 − 6 = 29 → Answer: 29
- Q3: n(T ∪ C) = 26 → Neither = 31 − 26 = 5 → Answer: 5
- Q4: n(T ∪ C) = 29 → Neither = 43 − 29 = 14 → Answer: 14
- Q5: n(M ∪ P) = 21 → n(M ∩ P) = 7 + 17 − 21 = 3 → Only Maths = 7 − 3 = 4 → Answer: 4
- Q6: n(M ∪ P) = 36 → n(M ∩ P) = 23 + 22 − 36 = 9 → Only Maths = 23 − 9 = 14 → Answer: 14
Syllabus note
This skill is listed under "Sets" for Class 11 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is counting with venn diagrams in?
"Sets", Class 11 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.