Set Theory / Venn Diagrams Questions and Answers

Q: What is the ratio of the number of cinema artists who are not proficient in any of these three skills (X0) to those who are proficient in all three (X3) ?

The ratio X0 : X3 is 3 : 5.Add every region of the Venn diagram to find how many artists have at least one skill:\[ \begin{aligned} 70 + 80 + 125 &= 275 \\ 25 + 35 + 45 &= 105 \\ 275 + 105 + 75 &= 455 \end{aligned} \]So X0 = 500 - 455 = 45 artists have none of the skills, and X3, the centre region, is 75.\[ 45 : 75 = 3 : 5 \]The other ratios come from misreading a region: 1 : 9, 5 : 7 and 6 : 7 cannot be formed from 45 and 75.

Q: If all other data remains same, how many cinema artists who are not proficient in any of these three skills should become proficient in all three, so that the above said ratio between number of cine-artists who are not proficient in any of these three skills (X0) to those who are proficient in all three (X3) should now become X0 : X3 = 1 : 4 ?

21 artists must change groups.From the Venn diagram, 45 artists have none of the skills and 75 have all three (see the previous question). If x artists move from the first group to the second:\[ \begin{aligned} \frac{45 - x}{75 + x} &= \frac{1}{4} \\ 180 - 4x &= 75 + x \\ x &= 21 \end{aligned} \]Check: 45 - 21 = 24 and 75 + 21 = 96, and 24 : 96 = 1 : 4.12, 15 and 10 all fail the check: moving 15 artists, for example, gives 30 : 90 = 1 : 3, and moving 12 gives 33 : 87.

Q: If A = {x : x is a prime number less than 10}, find the cardinality of A.

We need to list the prime numbers less than 10.Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves.The prime numbers less than 10 are 2, 3, 5, and 7.So, A = {2, 3, 5, 7}.The cardinality of A, denoted n(A), is the number of elements in the set, which is 4.

Important Instructions

Direction for [ Question No: 1 To 2 ] :

Directions : These questions are based on the information and diagram given below :

A survey was conducted on a sample of 500 cinema artists with reference to their proficiency in the skill of acting, singing and dancing. The results of the survey are depicted in the given Venn diagram.

What is the ratio of the number of cinema artists who are not proficient in any of these three skills (X₀) to those who are proficient in all three (X₃) ?

Venn diagram of three circles, Singing, Acting and Dancing. Singing only 70, Acting only 80, Dancing only 125, Singing and Acting only 25, Singing and Dancing only 35, Acting and Dancing only 45, all three 75.

A.5 : 7
B.6 : 7
C.3 : 5
D.1 : 9

Exam Year(s)

Punjab Civil Service - 2021,

Answer: C

The ratio X₀ : X₃ is 3 : 5.

Add every region of the Venn diagram to find how many artists have at least one skill:

\[ \begin{aligned} 70 + 80 + 125 &= 275 \\ 25 + 35 + 45 &= 105 \\ 275 + 105 + 75 &= 455 \end{aligned} \]

So X₀ = 500 - 455 = 45 artists have none of the skills, and X₃, the centre region, is 75.

\[ 45 : 75 = 3 : 5 \]

The other ratios come from misreading a region: 1 : 9, 5 : 7 and 6 : 7 cannot be formed from 45 and 75.

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If all other data remains same, how many cinema artists who are not proficient in any of these three skills should become proficient in all three, so that the above said ratio between number of cine-artists who are not proficient in any of these three skills (X₀) to those who are proficient in all three (X₃) should now become X₀ : X₃ = 1 : 4 ?

The same Venn diagram of singing, acting and dancing among 500 cinema artists, with 75 artists in the region common to all three circles.

A.12
B.21
C.15
D.10

Exam Year(s)

Punjab Civil Service - 2021,

Answer: B

21 artists must change groups.

From the Venn diagram, 45 artists have none of the skills and 75 have all three (see the previous question). If x artists move from the first group to the second:

\[ \begin{aligned} \frac{45 - x}{75 + x} &= \frac{1}{4} \\ 180 - 4x &= 75 + x \\ x &= 21 \end{aligned} \]

Check: 45 - 21 = 24 and 75 + 21 = 96, and 24 : 96 = 1 : 4.

12, 15 and 10 all fail the check: moving 15 artists, for example, gives 30 : 90 = 1 : 3, and moving 12 gives 33 : 87.

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If A = {1, 2, 3, 4} and B = {3, 4, 5, 6}, what is A ∪ B?

A.{3, 4}
B.{1, 2, 3, 4, 5, 6}
C.{1, 2, 5, 6}
D.{1, 2}

Answer: B

The union collects every element of either set, listed once.

\[ A \cup B = \{1, 2, 3, 4, 5, 6\} \]

3 and 4 belong to both but are not repeated.

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If A = {a, b, c, d} and B = {c, d, e, f}, what is A ∩ B?

A.{a, b, c, d, e, f}
B.{a, b}
C.{c, d}
D.{e, f}

Answer: C

The intersection keeps only the elements common to both sets.

\[ A \cap B = \{c, d\} \]

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Let U = {1, 2, 3, 4, 5, 6, 7, 8} be the universal set and A = {1, 3, 5, 7}. Find A'.

A.{1, 3, 5, 7}
B.{}
C.{2, 4, 6, 8}
D.{1, 2, 3, 4, 5, 6, 7, 8}

Answer: C

The complement contains everything in the universal set that is not in \( A \).

\[ A' = \{2, 4, 6, 8\} \]

Together \( A \) and \( A' \) make up the whole of \( U \) and share nothing.

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If a set A has 5 elements, what is the number of elements in its power set, n(P(A))?

A.5
B.10
C.25
D.32

Answer: D

The power set contains every subset, and each element is either in or out.

\[ n(P(A)) = 2^{n(A)} = 2^{5} = 32 \]

That count includes the empty set and \( A \) itself.

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If n(A) = 20, n(B) = 30 and n(A ∪ B) = 40, find n(A ∩ B).

A.5
B.10
C.15
D.20

Answer: B

We use the formula for the cardinality of the union of two sets:

n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

Substituting the given values:

40 = 20 + 30 - n(A ∩ B)

40 = 50 - n(A ∩ B)

n(A ∩ B) = 50 - 40 = 10

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In a class of 50 students, 30 like Math, 25 like Science, and 10 like both. How many students like neither Math nor Science?

A.5
B.10
C.15
D.20

Answer: A

Let M be the set of students who like Math and S be the set of students who like Science.

n(M) = 30, n(S) = 25, n(M ∩ S) = 10.

Number of students who like at least one subject is n(M ∪ S) = n(M) + n(S) - n(M ∩ S) = 30 + 25 - 10 = 45.

Total students = 50.

Number of students who like neither = Total students - n(M ∪ S) = 50 - 45 = 5.

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Which of the following represents the shaded region in the Venn diagram?

A.\( A \cup B \)
B.\( A \cap B \)
C.\( A - B \)
D.\( B - A \)

Answer: C

The shaded region lies inside circle A but outside circle B - the part of \( A \) that \( B \) does not overlap.

\[ A - B = \{\, x \mid x \in A \text{ and } x \notin B \,\} \]

\( A \cap B \) would be the lens where they overlap, and \( B - A \) the mirror image on the other side.

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10.

If A = {x : x is a prime number less than 10}, find the cardinality of A.

Answer: B

We need to list the prime numbers less than 10.

Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves.

The prime numbers less than 10 are 2, 3, 5, and 7.

So, A = {2, 3, 5, 7}.

The cardinality of A, denoted n(A), is the number of elements in the set, which is 4.

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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

Important Instructions

Direction for [ Question No: 1 To 2 ] :

Topics

Other Subjects

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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