# Set Theory Questions and Answers

## Questions on Set Theory

Definition – A Set is commonly defined as an assembly of items, recognized as foundations. These items could be any possible thing, including digits, alphabets, colors etc. However, none of the items of the set can be a set in itself. This page consists “Set Theory Questions and Answers” to enhance your concept and efficiency of this topic.

Presentation of sets: For example: P= {5, 6, 7, 8, 9, 10…….18} is the set of counting numbers beginning from 5. Q= {Black, white, Red, Yellow…} is a set of colors. R= {-3, -2, -1, ., 1, 2, 3….} is a set of all integers.

### Sets Theory Formula or Rule

• The set that consists of all the elements of a specified group is called the universal set and is denoted by the symbol ‘µ,’ also known as ‘mu.’
• For two sets P and Q,
• n(PᴜQ) is the number of items existent in either of the sets P or Q.
• n(P∩Q) is the number of items existent in both the sets P and Q.
• n(PᴜQ) = n(P) + (n(Q) – n(P∩Q)
For three sets P, Q and R,
• n(PᴜQᴜR) = n(P) + n(B) + n(R) – n(P∩Q) – n(Q∩R) – n(R∩P) + n(P∩Q∩R)

Now let us solve some Set Theory Questions and Answers.

## Also Check Out

### Sets Question with solution

1. A class had a strength of 40 students numbered from 1 to 40. Students with even roll numbers were doing English literature, while students with roll numbers in multiples of 4 were doing Fashion Design, and students with roll numbers in multiples of 8 were in the technical field.

Calculate the number of students who opted for none of the three streams?

20

20

67.71%

25

25

10.42%

30

30

5.21%

35

35

16.67%

Students doing Eng. Literature= n(E)

Students doing Fashion Design= n(F)

and Students doing Technical Eng.= n(T).
Now n(E) = 40/2= 20
n(F)= 40/4= 10

n(T) = 40/8 = 5
The 4th, 8th, 12th…… numbered students would be doing both English and Fashion design
∴ n(E∩F) = 40/4 =10
The 8, 16th, 24nd…… numbered students would be doing both English and Technical Eng
∴ n(E∩T) = 4 = 5
The 8th, 16th, …… numbered students would be doing both Fashion design and Technical Eng
∴ n(F∩T) = 40/8 =5
And a total of 5 students must be doing all 3 courses.
∴n(E∩F∩T) = 5
Now, n(E∪F∪T) = n(E)+n(F)+n(T) – n(E∩F) – n(E∩T) - n(F∩T) + n(E∩F∩T)
= 20+10+5-5-5-10+5

=20
∴ Number of students doing nothing = 40-20= 20

2. A group of tourists were traveling together on a train. 12 Tourists were able to speak Spanish, 30 could speak only French, and 12 could speak German. And no one from that group could speak any other language. If 4 persons in the group were able to speak two languages and 2 people were able to speak all the three languages, then calculate the total number of tourists in that group?

20

20

4.05%

42

42

83.78%

35

35

6.76%

10

10

5.41%

Let the 2 languages that these 4 people could speak easily be Spanish and French.

Mentioned that 2 people can speak all the three languages.

Tourists who could speak Spanish= 12.

Therefore tourists who could speak only Spanish= 12 – 4 – 2= 6

Tourists who could speak French= 30.

Therefore no. of tourists who could speak only French= 30 – 4 – 2= 24

Tourists who could speak German= 12.

Therefore no. of tourists who could speak only German= 12- 4 – 2= 6

Hence Tourist who could speak only 1 language= 6+24+6= 36

Tourists who could speak only 2 lang. = 4

Tourists who could speak all the lang = 2

Hence total number of tourists= 36+4+2= 42

3. A school had 240 students, from which 10% were able to play all the 3 games Basket Ball, Table Tennis and Hockey. 60 students were able to play any and only 2 games. Only 80 Students were able to play Basketball. Calculate the total number of students who were able to play only Table Tennis and only Hockey?

76

76

64.52%

100

100

12.9%

150

150

11.29%

60

60

11.29%

Given U= 240 Students

10% of 240= 24 Students

Hence the total number of students who could play only TT or Only Hockey

= 240- (60+80+24)= 240-164= 76

4. In a hostel, 20 residents want to watch football games, and 28 residents want to watch Cricket. However, 10 Residents were ready to watch both Football and Cricket. Calculate the number of residents who wanted to watch only football or Cricket?

10

10

21.67%

18

18

61.67%

15

15

5%

20

20

11.67%

Given:

Residents who wanted to watch Football= n(F) = 20

Residents who wanted to see Cricket = n(C) = 28

Residents who wanted to watch both football and Cricket = n(F∩C)= 10

Therefore Residents who watch only Football = n(F) = 20 - 10 = 10

Residents who watch only Cricket = n(C) = 28 - 10 = 18

Hence, Residents who watch only Football or Cricket= n(FUC) = n(F) + n(C) - n(f∩C)

= 18+10 - 10= 18

Therefore Option A is the correct one.

5. A class had 120 students, out of which 54 students liked to have Pizza, and 84 preferred to have a Salad. However, each student preferred to have at least one of the two dishes. Calculate the number of students who liked both Pizza and Salad?

50

50

3.08%

20

20

7.69%

18

18

86.15%

70

70

3.08%

Let number of students who like Pizza be P

And number of students who like Salad be Q

Given,

(P ∪ Q) = 120     n(P) = 54   and  n(Q) = 84

Then;

n(P ∩ Q) = n(P) + n(Q) - n(P ∪ Q)

= 54 + 84 - 120

= 18

Therefore, 18 Students liked both Pizza and Salad.

6.  In the below-mentioned diagram Triangle represents Girls Square represents Singers, and Circle represents Musicians. Identify the Girls who are both Singers and Musicians?

Eva

Eva

20.59%

Diya

Diya

75%

Cia

Cia

2.94%

Bina

Bina

1.47%

It is evident by looking at the figure that Diya is the Girl who’s both Singer and a Musician.

Therefore Option B is the correct one.

7. In a class of 40 students, 12 students selected Spanish and Chinese, 22 students opted for Chinese. However, each student selected at least one of the two languages. Calculate the total number of students who chose only Spanish as their primary learning language and not Chinese?

10

10

9.23%

20

20

6.15%

18

18

58.46%

None of above

None of above

26.15%

Let the students who selected Spanish be ‘S’ and students who chose Chinese be ‘C.’

Therefore number of students who selected at least 1 subject= n(S∪C)

Given

Each student selected at least 1 subject

Therefore n(S∪C)= 40

n(S∪C)= n(S) + n(C) – n(S∩C)

= 40= n(S)+ 22- 12

n(S)= 30 (Students who selected Spanish )

But since we’ve to find students who selected only Spanish

Therefore n(Only Spanish)= n only Spanish - n(S∩C)= 30-12= 18

8. If 70% of the audience prefers to read comic Novels, 90% loves to read Romantic Novels, and 98% like to read Fiction Novels. Calculate the proportion of readers who like to read all three languages?

60%

60%

3.64%

58%

58%

74.55%

90%

90%

12.73%

50%

50%

9.09%

Percentage of readers who like to read both Comic and Romantic Novels

= 70+90- 100= 60%

The proportion of people who like to have Comic, romantic, and Fiction Novels

= 60+98-100= 58%

Hence Option B is the correct one.

9. In a recent survey conducted in a school, it was discovered that 80 students preferred an I-phone, whereas 60 students went for an Android phone. However, 20 Students were not that particular and accordingly could go with any of them. If each student preferred to have at least one of the items, find out how many students the survey was conducted?

120

120

70.97%

60

60

9.68%

100

100

11.29%

50

50

8.06%

It is evident from the diagram that the total number of students on whom the survey was conducted= 60+20+40= 120

Students who preferred I- Phone= n(P)

Students who preferred Android= n(A)

Therefore n(P∩A) Students who could prefer both.
Thus using the formula n(PUA) = n(P) + n(A) – n(P∩A)
n(PUQ) = 80 + 60 – 20= 120

10. An office consisted of 200 Employees, 144 people came by Car, and 86 employees came by bus. Calculate the number of employees who came exclusively via car? How many employees came by Bus only and how many employees traveled both by car and by bus?

114, 56

114, 56

76.27%

100, 120

100, 120

6.78%

114, 24

114, 24

10.17%

104, 35

104, 35

6.78%

Let employees who came via car be C.

Employees who came via Bus= B.

Employees who came by car and not by bus= C - B

Employees who came by bus and not Car= B - C

C ∩ B Employees who came both by bus and car.

Given,

n(C) = 144

n(B) = 86

n(C ∪ B) = 200

Now, n(C ∩ B) = n(C) + n(B) - n(C ∪ B)

= 144 + 86 - 200

= 30

Therefore, Employees who came both by car and bus= 30

n(C) = n(C - B) + n(C ∩ B) ⇒

n(C - B) = n(C) - n(C ∩ B)

= 144 - 30

= 114

and n(B - C) = n(B) - n(C ∩ B)

= 86 - 30

= 56

Therefore, Number of employees who came by car only= 114

Number of employees who came by bus only= 56

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