Set Theory / Venn Diagrams Questions and Answers

Q: Forty students watched films A, B and C over a week. Each student watched either only one film or all three. Thirteen students watched film A, sixteen students watched film B and nineteen students watched film C. How many students watched all three films?

4 students watched all three films.Let x students watch all three; the other \(40 - x\) watched exactly one film each. Counting viewings film by film, each single-film student is counted once and each all-three student three times:\[ \begin{aligned} 13 + 16 + 19 &= (40 - x) + 3x \\ 48 &= 40 + 2x \Rightarrow x = 4 \end{aligned} \]Check: 36 students saw one film and 4 saw all three, giving \(36 + 12 = 48\) viewings. With 0 or 2 students watching all three, the viewings would total only 40 or 44; with 8 they would total 56.

Q: In a group of students, 10 students like Mathematics, 12 students like English, 4 students like both Mathematics and English, and 6 students like neither Mathematics nor English. The number of students in the group is ____

The group has 24 students.Those who like at least one subject number, by inclusion and exclusion,\[ n(M \cup E) = 10 + 12 - 4 = 18 \]Adding the 6 who like neither gives \(18 + 6 = 24\).18 forgets the 6 students who like neither subject. 20 adds 10 and 12 but forgets both corrections, and 32 adds every number in the question, counting the 4 students who like both twice and treating them as a separate group.

Q: To pass a test, a candidate needs to answer at least 2 out of 3 questions correctly. A total of 6,30,000 candidates appeared for the test. Question A was correctly answered by 3,30,000 candidates. Question B was answered correctly by 2,50,000 candidates. Question C was answered correctly by 2,60,000 candidates. Both questions A and B were answered correctly by 1,00,000 candidates. Both questions B and C were answered correctly by 90,000 candidates. Both questions A and C were answered correctly by 80,000 candidates. If the number of students answering all questions correctly is the same as the number answering none, how many candidates failed to clear the test?

The number who failed is 4,20,000.Work in thousands. Let \(x\) be the number answering all three correctly, which equals the number answering none. By inclusion-exclusion, the number answering at least one is\[ \begin{aligned} &330 + 250 + 260 - 100 - 90 - 80 + x \\ &= 570 + x \end{aligned} \]so the number answering none is \(630 - 570 - x = 60 - x\). Setting this equal to \(x\) gives \(x = 30\). Exactly two correct: \((100 + 90 + 80) - 3x = 180\). Those who pass have two or three correct: \(180 + 30 = 210\). So \(630 - 210 = 420\) thousand fail.30,000 is the number answering all (or none), and 2,70,000 is the sum of the pairwise counts, which double-counts those answering all three.

Q: In a population, patients who have high cholesterol also have high blood-pressure (BP). Some patients with high BP also have diabetes. There are no patients who have both high cholesterol and diabetes. Furthermore,1. the total number of patients with at least one of these conditions is 75,2. the number of patients with high cholesterol is 10,3. the number of patients with high BP is 45, and4. the number of patients with only high BP and no other conditions is 20.Then the number of patients who have both diabetes and high BP is _______

The number with both diabetes and high BP is 15.Everyone with high cholesterol also has high BP, and nobody has both high cholesterol and diabetes, so the 10 cholesterol patients sit inside the high-BP group and outside the diabetes group. The 45 high-BP patients then split into three parts:\[ \begin{aligned} 45 &= 10 + 20 + x \\ x &= 15 \end{aligned} \]where 10 have cholesterol and BP, 20 have only BP, and \(x\) have BP and diabetes. The remaining 75 − 45 = 30 patients have diabetes alone, which is consistent with the total of 75.0 and 10 ignore part of the high-BP count, and 20 confuses this group with the 'only high BP' patients.

Q: Among 150 faculty members in an institute, 55 are connected with each other through Facebook® and 85 are connected through WhatsApp®. 30 faculty members do not have Facebook® or WhatsApp® accounts. The number of faculty members connected only through Facebook® accounts is ______________.

35 faculty members are connected only through Facebook.Of the 150, 30 use neither, so 120 use at least one. By inclusion-exclusion, those on both are\[ \begin{aligned} |F \cap W| &= 55 + 85 - 120 = 20 \\ \text{only } F &= 55 - 20 = 35 \end{aligned} \]65 is the number connected only through WhatsApp (85 - 20). 45 and 90 do not fit the counts: with 45 on Facebook only, the 55 Facebook users would include just 10 on both, and the union would be 130, not 120.

Q: In an engineering college of 10,000 students, 1,500 like neither their core branches nor other branches. The number of students who like their core branches is 1/4th of the number of students who like other branches. The number of students who like both their core and other branches is 500.The number of students who like their core branches is

The number who like their core branches is 1,800.Of the 10,000 students, \(10000 - 1500 = 8500\) like at least one kind of branch. Let \(C\) like their core branches and \(O\) like other branches, with \(C = O/4\) and 500 liking both. By inclusion and exclusion,\[ \begin{aligned} C + O - 500 &= 8500 \\ C + 4C &= 9000 \\ C &= 1800 \end{aligned} \]So 1,800 like their core branches and 7,200 like other branches. 1,600 comes from subtracting the 500 who like both instead of adding it back (\(5C = 8000\)), and 3,500 and 1,500 are numbers from the question rather than results.

Q: In a company, 35% of the employees drink coffee, 40% of the employees drink tea and 10% of the employees drink both tea and coffee. What % of employees drink neither tea nor coffee?

The answer is 35 per cent.By inclusion and exclusion, the share drinking at least one of the two is\[ \begin{aligned} 35 + 40 - 10 &= 65\% \\ 100 - 65 &= 35\% \end{aligned} \]so 35% drink neither. 25% comes from adding the overlap instead of subtracting it (35 + 40 + 10 = 85), and 15 or 40 do not follow from any consistent count. The 10% who drink both are inside both the 35% and the 40%, which is why they are subtracted once.

Q: In a class of 100 students,(i) there are 30 students who neither like romantic movies nor comedy movies,(ii) the number of students who like romantic movies is twice the number of students who like comedy movies, and(iii) the number of students who like both romantic movies and comedy movies is 20.How many students in the class like romantic movies?

60 students like romantic movies.Of the 100 students, 30 like neither kind, so 70 like at least one. Let C be the number who like comedy; then \(2C\) like romance and 20 like both. By inclusion and exclusion:\[ \begin{aligned} 2C + C - 20 &= 70 \\ 3C &= 90 \Rightarrow C = 30 \end{aligned} \]So \(2C = 60\) students like romantic movies. 30 is the number who like comedy, 40 is the number who like romance only (60 - 20), and 20 is the number who like both.

Q: A survey of 450 students about their subjects of interest resulted in the following outcome.150 students are interested in Mathematics.200 students are interested in Physics.175 students are interested in Chemistry.50 students are interested in Mathematics and Physics.60 students are interested in Physics and Chemistry.40 students are interested in Mathematics and Chemistry.30 students are interested in Mathematics, Physics and Chemistry.Remaining students are interested in Humanities.Based on the above information, the number of students interested in Humanities is

The number interested in Humanities is 45.By inclusion and exclusion, the students interested in at least one of the three sciences are\[ \begin{aligned} &150 + 200 + 175 - 50 - 60 - 40 + 30 \\ &= 525 - 150 + 30 = 405 \end{aligned} \]The remaining \(450 - 405 = 45\) are interested in Humanities. The triple overlap must be added back because subtracting the three pairwise overlaps removes the 30 students interested in all three subjects once too often. 30 and 40 are figures taken from the list, and 10 does not come from any consistent count.

Q: In a college 200 students took admission in different subjects. 56 students took Biology, 62 took Maths, 84 took Psychology. 18 students took Biology and Maths, 20 took Biology and Psychology and 12 took Maths and Psychology. If 8 students took all three subjects how many students took none of these subjects?

40 students took none of the three subjects.Use the inclusion-exclusion rule for three sets to count students taking at least one subject.\[ \begin{aligned} n &= 56 + 62 + 84 \\ &\quad - 18 - 20 - 12 + 8 \\ &= 202 - 50 + 8 = 160 \end{aligned} \]So 160 students took at least one subject, and \(200 - 160 = 40\) took none.Forgetting to add back the 8 who took all three gives 152 and 48 with none, which is not offered. Simply adding the three subject totals (202) exceeds the 200 students, which shows why the overlaps must be subtracted.

Important Instructions

Forty students watched films A, B and C over a week. Each student watched either only one film or all three. Thirteen students watched film A, sixteen students watched film B and nineteen students watched film C. How many students watched all three films?

Exam Year(s)

GATE Mathematics - 2018, GATE ME - 2018, GATE AE - 2018,

Answer: C

4 students watched all three films.

Let x students watch all three; the other \(40 - x\) watched exactly one film each. Counting viewings film by film, each single-film student is counted once and each all-three student three times:

\[ \begin{aligned} 13 + 16 + 19 &= (40 - x) + 3x \\ 48 &= 40 + 2x \Rightarrow x = 4 \end{aligned} \]

Check: 36 students saw one film and 4 saw all three, giving \(36 + 12 = 48\) viewings. With 0 or 2 students watching all three, the viewings would total only 40 or 44; with 8 they would total 56.

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In a group of students, 10 students like Mathematics, 12 students like English, 4 students like both Mathematics and English, and 6 students like neither Mathematics nor English. The number of students in the group is ____

A.18
B.20
C.24
D.32

Exam Year(s)

GATE Chemistry - 2026, GATE Physics - 2026, GATE AE - 2026,

Answer: C

The group has 24 students.

Those who like at least one subject number, by inclusion and exclusion,

\[ n(M \cup E) = 10 + 12 - 4 = 18 \]

Adding the 6 who like neither gives \(18 + 6 = 24\).

18 forgets the 6 students who like neither subject. 20 adds 10 and 12 but forgets both corrections, and 32 adds every number in the question, counting the 4 students who like both twice and treating them as a separate group.

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To pass a test, a candidate needs to answer at least 2 out of 3 questions correctly. A total of 6,30,000 candidates appeared for the test. Question A was correctly answered by 3,30,000 candidates. Question B was answered correctly by 2,50,000 candidates. Question C was answered correctly by 2,60,000 candidates. Both questions A and B were answered correctly by 1,00,000 candidates. Both questions B and C were answered correctly by 90,000 candidates. Both questions A and C were answered correctly by 80,000 candidates. If the number of students answering all questions correctly is the same as the number answering none, how many candidates failed to clear the test?

A.30,000
B.2,70,000
C.3,90,000
D.4,20,000

Exam Year(s)

GATE Physics - 2018, GATE AG - 2018,

Answer: D

The number who failed is 4,20,000.

Work in thousands. Let \(x\) be the number answering all three correctly, which equals the number answering none. By inclusion-exclusion, the number answering at least one is

\[ \begin{aligned} &330 + 250 + 260 - 100 - 90 - 80 + x \\ &= 570 + x \end{aligned} \]

so the number answering none is \(630 - 570 - x = 60 - x\). Setting this equal to \(x\) gives \(x = 30\). Exactly two correct: \((100 + 90 + 80) - 3x = 180\). Those who pass have two or three correct: \(180 + 30 = 210\). So \(630 - 210 = 420\) thousand fail.

30,000 is the number answering all (or none), and 2,70,000 is the sum of the pairwise counts, which double-counts those answering all three.

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In a population, patients who have high cholesterol also have high blood-pressure (BP). Some patients with high BP also have diabetes. There are no patients who have both high cholesterol and diabetes. Furthermore,
1. the total number of patients with at least one of these conditions is 75,
2. the number of patients with high cholesterol is 10,
3. the number of patients with high BP is 45, and
4. the number of patients with only high BP and no other conditions is 20.

Then the number of patients who have both diabetes and high BP is _______

A.0
B.15
C.20
D.10

Exam Year(s)

GATE Mathematics - 2026, GATE Life Sciences - 2026, GATE AG - 2026,

Answer: B

The number with both diabetes and high BP is 15.

Everyone with high cholesterol also has high BP, and nobody has both high cholesterol and diabetes, so the 10 cholesterol patients sit inside the high-BP group and outside the diabetes group. The 45 high-BP patients then split into three parts:

\[ \begin{aligned} 45 &= 10 + 20 + x \\ x &= 15 \end{aligned} \]

where 10 have cholesterol and BP, 20 have only BP, and \(x\) have BP and diabetes. The remaining 75 − 45 = 30 patients have diabetes alone, which is consistent with the total of 75.

0 and 10 ignore part of the high-BP count, and 20 confuses this group with the 'only high BP' patients.

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Among 150 faculty members in an institute, 55 are connected with each other through Facebook® and 85 are connected through WhatsApp®. 30 faculty members do not have Facebook® or WhatsApp® accounts. The number of faculty members connected only through Facebook® accounts is ______________.

A.35
B.45
C.65
D.90

Exam Year(s)

GATE CS - 2016,

Answer: A

35 faculty members are connected only through Facebook.

Of the 150, 30 use neither, so 120 use at least one. By inclusion-exclusion, those on both are

\[ \begin{aligned} |F \cap W| &= 55 + 85 - 120 = 20 \\ \text{only } F &= 55 - 20 = 35 \end{aligned} \]

65 is the number connected only through WhatsApp (85 - 20). 45 and 90 do not fit the counts: with 45 on Facebook only, the 55 Facebook users would include just 10 on both, and the union would be 130, not 120.

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In an engineering college of 10,000 students, 1,500 like neither their core branches nor other branches. The number of students who like their core branches is 1/4^th of the number of students who like other branches. The number of students who like both their core and other branches is 500.

The number of students who like their core branches is

A.1,800
B.3,500
C.1,600
D.1,500

Exam Year(s)

GATE Life Sciences - 2024, GATE CS - 2024, GATE XE - 2024,

Answer: A

The number who like their core branches is 1,800.

Of the 10,000 students, \(10000 - 1500 = 8500\) like at least one kind of branch. Let \(C\) like their core branches and \(O\) like other branches, with \(C = O/4\) and 500 liking both. By inclusion and exclusion,

\[ \begin{aligned} C + O - 500 &= 8500 \\ C + 4C &= 9000 \\ C &= 1800 \end{aligned} \]

So 1,800 like their core branches and 7,200 like other branches. 1,600 comes from subtracting the 500 who like both instead of adding it back (\(5C = 8000\)), and 3,500 and 1,500 are numbers from the question rather than results.

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In a company, 35% of the employees drink coffee, 40% of the employees drink tea and 10% of the employees drink both tea and coffee. What % of employees drink neither tea nor coffee?

A.15
B.25
C.35
D.40

Exam Year(s)

GATE Chemistry - 2021,

Answer: C

The answer is 35 per cent.

By inclusion and exclusion, the share drinking at least one of the two is

\[ \begin{aligned} 35 + 40 - 10 &= 65\% \\ 100 - 65 &= 35\% \end{aligned} \]

so 35% drink neither. 25% comes from adding the overlap instead of subtracting it (35 + 40 + 10 = 85), and 15 or 40 do not follow from any consistent count. The 10% who drink both are inside both the 35% and the 40%, which is why they are subtracted once.

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In a class of 100 students,

(i) there are 30 students who neither like romantic movies nor comedy movies,
(ii) the number of students who like romantic movies is twice the number of students who like comedy movies, and
(iii) the number of students who like both romantic movies and comedy movies is 20.

How many students in the class like romantic movies?

A.40
B.20
C.60
D.30

Exam Year(s)

GATE Chemistry - 2023,

Answer: C

60 students like romantic movies.

Of the 100 students, 30 like neither kind, so 70 like at least one. Let C be the number who like comedy; then \(2C\) like romance and 20 like both. By inclusion and exclusion:

\[ \begin{aligned} 2C + C - 20 &= 70 \\ 3C &= 90 \Rightarrow C = 30 \end{aligned} \]

So \(2C = 60\) students like romantic movies. 30 is the number who like comedy, 40 is the number who like romance only (60 - 20), and 20 is the number who like both.

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A survey of 450 students about their subjects of interest resulted in the following outcome.

150 students are interested in Mathematics.
200 students are interested in Physics.
175 students are interested in Chemistry.
50 students are interested in Mathematics and Physics.
60 students are interested in Physics and Chemistry.
40 students are interested in Mathematics and Chemistry.
30 students are interested in Mathematics, Physics and Chemistry.
Remaining students are interested in Humanities.

Based on the above information, the number of students interested in Humanities is

A.10
B.30
C.40
D.45

Exam Year(s)

GATE Life Sciences - 2022, GATE XE - 2022,

Answer: D

The number interested in Humanities is 45.

By inclusion and exclusion, the students interested in at least one of the three sciences are

\[ \begin{aligned} &150 + 200 + 175 - 50 - 60 - 40 + 30 \\ &= 525 - 150 + 30 = 405 \end{aligned} \]

The remaining \(450 - 405 = 45\) are interested in Humanities. The triple overlap must be added back because subtracting the three pairwise overlaps removes the 30 students interested in all three subjects once too often. 30 and 40 are figures taken from the list, and 10 does not come from any consistent count.

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10.

In a college 200 students took admission in different subjects. 56 students took Biology, 62 took Maths, 84 took Psychology. 18 students took Biology and Maths, 20 took Biology and Psychology and 12 took Maths and Psychology. If 8 students took all three subjects how many students took none of these subjects?

A.10
B.20
C.30
D.40

Exam Year(s)

Kerala PSC KAS - 2020,

Answer: D

40 students took none of the three subjects.

Use the inclusion-exclusion rule for three sets to count students taking at least one subject.

\[ \begin{aligned} n &= 56 + 62 + 84 \\ &\quad - 18 - 20 - 12 + 8 \\ &= 202 - 50 + 8 = 160 \end{aligned} \]

So 160 students took at least one subject, and \(200 - 160 = 40\) took none.

Forgetting to add back the 8 who took all three gives 152 and 48 with none, which is not offered. Simply adding the three subject totals (202) exceeds the 200 students, which shows why the overlaps must be subtracted.

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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

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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