Set Theory / Venn Diagrams Questions and Answers

Q: In a society with 100 members, each member plays at least one sport amongst cricket, hockey and volleyball. If 20 members play cricket only, 30 play hockey only, 25 play volleyball only, 12 play both cricket and hockey, 7 play both hockey and volleyball and 3 members play all the sports, then how many members play at least two sports ?

The number who play at least two sports is 25.Every one of the 100 members plays at least one sport, so each member is in exactly one of two groups: those who play exactly one sport and those who play two or more.\[ \begin{aligned} \text{exactly one} &= 20 + 30 + 25 = 75 \\ \text{at least two} &= 100 - 75 = 25 \end{aligned} \]The pairwise figures are not needed: everyone who does not play exactly one sport plays at least two. Option 22 is the number who play exactly two sports, \(25 - 3\), which leaves out the 3 who play all three.

Q: In a group of 18 people, 10 read French, 11 read English, while 6 read none of these two. How many of them read both French and English ?

9 people read both French and English.Of the 18 people, 6 read neither language, so 12 read at least one. By the inclusion-exclusion rule:\[ \begin{aligned} n(F \cup E) &= n(F) + n(E) - n(F \cap E) \\ 12 &= 10 + 11 - n(F \cap E) \\ n(F \cap E) &= 9 \end{aligned} \]Check: 1 reads French only, 2 read English only, 9 read both and 6 neither, making 18. 3 is the sum of the 'only' groups, and 6 and 4 do not satisfy the totals.

Q: Set M has 42 students and set N has 60 students. At least 12 of the students in set M are not in set N. Which of the following cannot be the possible number of students in set N that are not in set M ?

The number that is not possible is 28.At least 12 of set M's 42 students are outside N, so at most 42 - 12 = 30 students are in both sets. Set N has 60 students, so the number in N but not in M is\[ \begin{aligned} 60 - (\text{common}) &\ge 60 - 30 \\ &= 30 \end{aligned} \]Any value from 30 to 60 is possible, depending on how many students the two sets share. 32, 42 and 45 all lie in that range, but 28 is below the minimum of 30, so it cannot occur.

Q: In a school, a total of 180 students are enrolled in the foreign languages program. The foreign languages offered are : French, German and Chinese and the number of people studying each of these subjects are 80, 90 and 100 respectively. The number of students who study more than one of the three subjects is 50% more than the number of students who study all the three subjects. How many students study all three foreign languages ?

36 students study all three languages.Let x study all three and y exactly two. Those studying more than one number 1.5x, so y + x = 1.5x and y = 0.5x. Let a study exactly one language.\[ \begin{aligned} a + y + x &= 180 \\ a + 2y + 3x &= 80 + 90 + 100 = 270 \\ y + 2x &= 90 \\ 0.5x + 2x &= 90 \Rightarrow x = 36 \end{aligned} \]Check: 18 study exactly two, 126 exactly one, and 126 + 2(18) + 3(36) = 270.18 is the number studying exactly two, a likely confusion; 24 and 42 fail the 270 total.

Important Instructions

81.

In a society with 100 members, each member plays at least one sport amongst cricket, hockey and volleyball. If 20 members play cricket only, 30 play hockey only, 25 play volleyball only, 12 play both cricket and hockey, 7 play both hockey and volleyball and 3 members play all the sports, then how many members play at least two sports ?

A.25
B.22
C.32
D.35

Exam Year(s)

Kerala PSC KAS - 2025,

Answer: A

The number who play at least two sports is 25.

Every one of the 100 members plays at least one sport, so each member is in exactly one of two groups: those who play exactly one sport and those who play two or more.

\[ \begin{aligned} \text{exactly one} &= 20 + 30 + 25 = 75 \\ \text{at least two} &= 100 - 75 = 25 \end{aligned} \]

The pairwise figures are not needed: everyone who does not play exactly one sport plays at least two. Option 22 is the number who play exactly two sports, \(25 - 3\), which leaves out the 3 who play all three.

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82.

In a group of 18 people, 10 read French, 11 read English, while 6 read none of these two. How many of them read both French and English ?

Exam Year(s)

MPPSC State Service - 2024, MPPSC Forest Service - 2024,

Answer: B

9 people read both French and English.

Of the 18 people, 6 read neither language, so 12 read at least one. By the inclusion-exclusion rule:

\[ \begin{aligned} n(F \cup E) &= n(F) + n(E) - n(F \cap E) \\ 12 &= 10 + 11 - n(F \cap E) \\ n(F \cap E) &= 9 \end{aligned} \]

Check: 1 reads French only, 2 read English only, 9 read both and 6 neither, making 18. 3 is the sum of the 'only' groups, and 6 and 4 do not satisfy the totals.

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83.

Set M has 42 students and set N has 60 students. At least 12 of the students in set M are not in set N. Which of the following cannot be the possible number of students in set N that are not in set M ?

A.32
B.45
C.28
D.42

Exam Year(s)

Punjab Civil Service - 2021,

Answer: C

The number that is not possible is 28.

At least 12 of set M's 42 students are outside N, so at most 42 - 12 = 30 students are in both sets. Set N has 60 students, so the number in N but not in M is

\[ \begin{aligned} 60 - (\text{common}) &\ge 60 - 30 \\ &= 30 \end{aligned} \]

Any value from 30 to 60 is possible, depending on how many students the two sets share. 32, 42 and 45 all lie in that range, but 28 is below the minimum of 30, so it cannot occur.

Discuss in Forum

84.

In a school, a total of 180 students are enrolled in the foreign languages program. The foreign languages offered are : French, German and Chinese and the number of people studying each of these subjects are 80, 90 and 100 respectively. The number of students who study more than one of the three subjects is 50% more than the number of students who study all the three subjects. How many students study all three foreign languages ?

A.18
B.24
C.36
D.42

Exam Year(s)

Punjab Civil Service - 2021,

Answer: C

36 students study all three languages.

Let x study all three and y exactly two. Those studying more than one number 1.5x, so y + x = 1.5x and y = 0.5x. Let a study exactly one language.

\[ \begin{aligned} a + y + x &= 180 \\ a + 2y + 3x &= 80 + 90 + 100 = 270 \\ y + 2x &= 90 \\ 0.5x + 2x &= 90 \Rightarrow x = 36 \end{aligned} \]

Check: 18 study exactly two, 126 exactly one, and 126 + 2(18) + 3(36) = 270.

18 is the number studying exactly two, a likely confusion; 24 and 42 fail the 270 total.

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Set Theory / Venn Diagrams Questions and Answers

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Tags: Set Theory Questions and Answers || Venn Diagrams MCQ Questions || Set Operations GK || Venn Diagram Quantitative Aptitude GK || Logical Reasoning Venn Diagrams

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