Quantitative Aptitude

Set Theory: Practice Questions with Answers

14 practice questions on Set Theory, with answers; most also have an explanation or a worked solution. Try each one before you open its answer.

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1. Q.1

   In a survey of 100 people, 60 like tea and 50 like coffee. If 20 like both, how many like neither?

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   Answer: 10

   Solution:

   1. |Tea ∪ Coffee| = 60 + 50 − 20 = 90.
   2. Neither = Total − union = 100 − 90 = 10.

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2. Q.2

   Using the same data as above, how many students play exactly two of the three games?

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   Answer: 22

   Solution:

   1. Exactly two = (sum of pairwise intersections) − 3 × (all three)
   2. = (15+12+10) − 3(5) = 37 − 15
   3. = 22.

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3. Q.3

   In a class of 60 students, 35 play cricket and 25 play football. If 10 students play both, how many play at least one of the two games?

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   Answer: 50

   Solution:

   1. |Cricket ∪ Football| = 35 + 25 − 10 = 50.

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4. Q.4

   In a group of 50 students, 30 study Math, 25 study Physics, and 15 study both. How many study only Math (not Physics)?

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   Answer: 15

   Solution:

   1. Only Math = |Math| − |Math ∩ Physics| = 30 − 15 = 15.

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5. Q.5

   In a class of 100 students: 50 play cricket, 40 play football, 30 play hockey. 15 play both cricket and football, 12 play both football and hockey, 10 play both cricket and hockey, and 5 play all three. How many play at least one of the three games?

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   Answer: 88

   Solution:

   1. |C∪F∪H| = |C|+|F|+|H| − |C∩F| − |F∩H| − |C∩H| + |C∩F∩H|
   2. = 50+40+30 − 15−12−10 + 5
   3. = 120 − 37 + 5 = 88.

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6. Q.6

   Using the same data as above, how many students play none of the three games?

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   Answer: 12

   Solution:

   1. None = Total − (at least one) = 100 − 88 = 12.

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7. Q.7

   In a class of 80 students, 50 play cricket, 40 play football, and 20 play both. How many play exactly one of the two games?

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   Answer: 50

   Solution:

   1. |Cricket ∪ Football| = 50 + 40 − 20 = 70.
   2. Exactly one = union − both = 70 − 20 = 50.

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8. Q.8

   In a survey of 120 people, 70 like movies and 60 like music. If 25 like both, how many like neither?

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   Answer: 15

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9. Q.9

   Using the same data, how many read none of the three?

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   Answer: 2. 80 − 78.

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10. Q.10

    Using the same data, how many read exactly two of the three newspapers?

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    Answer: 22. (15+12+10) − 3(5).

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11. Q.11

    In a class of 70 students, 40 play chess and 30 play carrom. If 15 play both, how many play at least one game?

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    Answer: 55

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12. Q.12

    In a group of 90 students, 55 study English, 45 study Hindi, and 20 study both. How many study only English?

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    Answer: 35

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13. Q.13

    In a class of 55 students, 32 play basketball and 28 play volleyball. If 10 play both, how many play neither?

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    Answer: 5. Union = 32+28−10 = 50; neither = 55−50.

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14. Q.14

    In a survey of 80 people: 45 read newspaper A, 35 read B, 30 read C. 15 read both A and B, 12 read both B and C, 10 read both A and C, and 5 read all three. How many read at least one?

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    Answer: 78. 45+35+30 − 15−12−10 + 5.

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