Set Theory Questions and Answers

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 4^th, 8^th, 12^th…… numbered students would be doing both English and Fashion design
∴ n(E∩F) = 40/4 =10
The 8, 16^th, 24^nd…… numbered students would be doing both English and Technical Eng
∴ n(E∩T) = 4 = 5
The 8^th, 16^th, …… 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

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

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

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.

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.

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.

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

 

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.

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

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