Venn Diagrams
CSAT · Logical Reasoning
Introduction
Venn-diagram questions count overlapping groups – people who like tea or coffee, students in several clubs. Inclusion–exclusion gives the answer, and filling the diagram from the centre outwards prevents double counting.
Key concepts
Two sets
n(A ∪ B) = n(A) + n(B) − n(A ∩ B). Only A = n(A) − n(A ∩ B). Neither = total − n(A ∪ B).
Three sets
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C).
Fill from the centre
- Write the 'all three' number in the centre.
- Subtract it from each pairwise overlap to get 'exactly two'.
- Subtract overlaps from each set total to get 'only one'.
- Add all seven regions, then compare with the total for 'none'.
Exactly and at least
- Exactly two = (sum of pairwise) − 3 × (all three).
- At least two = (sum of pairwise) − 2 × (all three).
Classification diagrams
Some questions ask which diagram represents, say, 'India, Asia, Earth' (nested circles) or 'doctors, men, women' (two separate circles overlapping a third). Decide whether each pair is subset, overlapping or separate.
- n(A ∪ B) = A + B − AB
- n(A ∪ B ∪ C) = A + B + C − AB − BC − CA + ABC
- Exactly two = ΣAB − 3ABC
- At least two = ΣAB − 2ABC
Solved examples
Example 1
Of 63 students, 37 play cricket, 21 play football and 21 play neither. How many play only cricket?
(a) 16 (b) 21 (c) 5 (d) 37
- At least one = 63 − 21 = 42.
- Both = 37 + 21 − 42 = 16.
- Only cricket = 37 − 16 = 21.
Answer: (b) 21
Example 2
In a survey of 84 people, 32 read newspaper X, 74 read newspaper Y and 26 read both. How many read neither?
(a) -22 (b) 5 (c) 30 (d) 4
- Read at least one = 32 + 74 − 26 = 80.
- Neither = 84 − 80 = 4.
Answer: (d) 4
Example 3
In a survey of 135 people, 66 read newspaper X, 52 read newspaper Y and 12 read both. How many read neither?
(a) 17 (b) 29 (c) 32 (d) 41
- Read at least one = 66 + 52 − 12 = 106.
- Neither = 135 − 106 = 29.
Answer: (b) 29
Example 4
In a class, 49 students like Hindi, 36 like English and 24 like Maths. 19 like Hindi and English, 5 like English and Maths, 13 like Hindi and Maths, and 2 like all three. Every student likes at least one. How many students are there?
(a) 76 (b) 72 (c) 74 (d) 109
- n(H ∪ E ∪ M) = H + E + M − (HE + EM + HM) + HEM.
- = 49 + 36 + 24 − (19 + 5 + 13) + 2 = 74.
Answer: (c) 74
Example 5
In a survey of 147 people, 82 read newspaper X, 76 read newspaper Y and 33 read both. How many read neither?
(a) 22 (b) 24 (c) 55 (d) -11
- Read at least one = 82 + 76 − 33 = 125.
- Neither = 147 − 125 = 22.
Answer: (a) 22
Example 6
In a survey of 80 people, 54 read newspaper X, 60 read newspaper Y and 39 read both. How many read neither?
(a) 44 (b) -34 (c) 5 (d) 6
- Read at least one = 54 + 60 − 39 = 75.
- Neither = 80 − 75 = 5.
Answer: (c) 5
Example 7
In a class, 49 students like Hindi, 34 like English and 49 like Maths. 21 like Hindi and English, 20 like English and Maths, 24 like Hindi and Maths, and 8 like all three. Every student likes at least one. How many students are there?
(a) 75 (b) 67 (c) 132 (d) 83
- n(H ∪ E ∪ M) = H + E + M − (HE + EM + HM) + HEM.
- = 49 + 34 + 49 − (21 + 20 + 24) + 8 = 75.
Answer: (a) 75
Example 8
In a survey of 83 people, 37 read newspaper X, 26 read newspaper Y and 5 read both. How many read neither?
(a) 27 (b) 20 (c) 30 (d) 25
- Read at least one = 37 + 26 − 5 = 58.
- Neither = 83 − 58 = 25.
Answer: (d) 25
- Always start from the innermost region.
- Check that all regions add to the total.
- For classification, test each pair: subset, overlap or disjoint.
- Forgetting to add back the triple overlap.
- Treating 'likes A and B' as 'likes only A and B'.
- Ignoring the 'none' region.
Practice set
Attempt all 12 questions (about 90 seconds each), then check the worked solutions below.
Q1. In a class, 30 students like Hindi, 36 like English and 22 like Maths. 9 like Hindi and English, 17 like English and Maths, 5 like Hindi and Maths, and 2 like all three. Every student likes at least one. How many students are there?
(a) 61 (b) 88 (c) 59 (d) 57
Q2. In a class, 50 students like Hindi, 57 like English and 44 like Maths. 28 like Hindi and English, 20 like English and Maths, 26 like Hindi and Maths, and 11 like all three. Every student likes at least one. How many students are there?
(a) 151 (b) 99 (c) 77 (d) 88
Q3. In a survey of 130 people, 89 read newspaper X, 48 read newspaper Y and 35 read both. How many read neither?
(a) 63 (b) 28 (c) -7 (d) 31
Q4. In a class, 33 students like Hindi, 37 like English and 50 like Maths. 12 like Hindi and English, 21 like English and Maths, 21 like Hindi and Maths, and 7 like all three. Every student likes at least one. How many students are there?
(a) 80 (b) 66 (c) 120 (d) 73
Q5. Of 100 students, 32 play cricket, 70 play football and 18 play neither. How many play only cricket?
(a) 20 (b) 50 (c) 12 (d) 32
Q6. In a class, 53 students like Hindi, 59 like English and 54 like Maths. 25 like Hindi and English, 39 like English and Maths, 32 like Hindi and Maths, and 19 like all three. Every student likes at least one. How many students are there?
(a) 108 (b) 166 (c) 89 (d) 70
Q7. Of 93 students, 29 play cricket, 43 play football and 27 play neither. How many play only cricket?
(a) 6 (b) 37 (c) 29 (d) 23
Q8. In a survey of 137 people, 91 read newspaper X, 65 read newspaper Y and 35 read both. How many read neither?
(a) 51 (b) 16 (c) -19 (d) 18
Q9. Of 109 students, 56 play cricket, 60 play football and 19 play neither. How many play only cricket?
(a) 30 (b) 26 (c) 34 (d) 56
Q10. In a survey of 97 people, 30 read newspaper X, 68 read newspaper Y and 10 read both. How many read neither?
(a) 9 (b) -1 (c) 19 (d) 10
Q11. Of 98 students, 55 play cricket, 53 play football and 1 play neither. How many play only cricket?
(a) 42 (b) 44 (c) 11 (d) 55
Q12. In a class, 55 students like Hindi, 46 like English and 51 like Maths. 29 like Hindi and English, 19 like English and Maths, 28 like Hindi and Maths, and 11 like all three. Every student likes at least one. How many students are there?
(a) 87 (b) 98 (c) 76 (d) 152
Q1 – (c) 59. n(H ∪ E ∪ M) = H + E + M − (HE + EM + HM) + HEM. = 30 + 36 + 22 − (9 + 17 + 5) + 2 = 59.
Q2 – (d) 88. n(H ∪ E ∪ M) = H + E + M − (HE + EM + HM) + HEM. = 50 + 57 + 44 − (28 + 20 + 26) + 11 = 88.
Q3 – (b) 28. Read at least one = 89 + 48 − 35 = 102. Neither = 130 − 102 = 28.
Q4 – (d) 73. n(H ∪ E ∪ M) = H + E + M − (HE + EM + HM) + HEM. = 33 + 37 + 50 − (12 + 21 + 21) + 7 = 73.
Q5 – (c) 12. At least one = 100 − 18 = 82. Both = 32 + 70 − 82 = 20. Only cricket = 32 − 20 = 12.
Q6 – (c) 89. n(H ∪ E ∪ M) = H + E + M − (HE + EM + HM) + HEM. = 53 + 59 + 54 − (25 + 39 + 32) + 19 = 89.
Q7 – (d) 23. At least one = 93 − 27 = 66. Both = 29 + 43 − 66 = 6. Only cricket = 29 − 6 = 23.
Q8 – (b) 16. Read at least one = 91 + 65 − 35 = 121. Neither = 137 − 121 = 16.
Q9 – (a) 30. At least one = 109 − 19 = 90. Both = 56 + 60 − 90 = 26. Only cricket = 56 − 26 = 30.
Q10 – (a) 9. Read at least one = 30 + 68 − 10 = 88. Neither = 97 − 88 = 9.
Q11 – (b) 44. At least one = 98 − 1 = 97. Both = 55 + 53 − 97 = 11. Only cricket = 55 − 11 = 44.
Q12 – (a) 87. n(H ∪ E ∪ M) = H + E + M − (HE + EM + HM) + HEM. = 55 + 46 + 51 − (29 + 19 + 28) + 11 = 87.
- Inclusion–exclusion
- Fill from the centre
- Exactly two vs at least two
- Regions must add to the total
