Relations and Functions Test

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Relations and Functions Test - 11

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Weekly Quiz Competition

Question 1
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever Let I be the set of all citizens of India who were eligible to exercise their voting right in general election held in 2019. A relation ‘R’ is defined on I as follows:
ONE – NATION
ONE – ELECTION
FESTIVAL OF DEMOCRACY
GENERAL ELECTION - 2019
R = {(V₁ , V₂ ) : V₁ , V₂ ∈ I and both use their voting right in general election – 2019}
Two neighbours X and Y ∈ I. X exercised his voting right while Y did not cast her vote in general election - 2019. Which of the following is true?

A
A. (X, Y) ∈ R

B
B. (Y, X) ∈ R

C
C. (X, X) ∉ R

D
D. (X, Y) ∉ R

Solution

(X, Y) ∉ R
∵ X exercised his voting right while, Y did not cast her vote in general election-2019
And R = {(V₁ , V₂ ) : V₁ V₂ ∈ I and both use their voting right in general election-2019}
Question 2
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1, 2, 3, 4, 5, 6}. Let A be the set of players while B be the set of all possible outcomes.
A = {S, D}, B = {1, 2, 3, 4, 5, 6}
Let R : B → B be defined by R = {(1, 1), (1, 2), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}, then R is

A
A. Symmetric

B
B. Reflexive and Transitive

C
C. Transitive and symmetric

D
D. Equivalence

Solution

Check Reflexive
Since x is divisible by x
Therefore, (x,x) ∈ R
Therefore, R is reflexive.
Check Symmetric
To check whether symmetric or not,
If (x,y) ∈ R, then (y,x) ∈ R
Here, (2,4) ∈ R, as 4 is divisible by 2
But, (4,2) ∉ R as 2 is not divisible by 4
Therefore, R is not Symmetric.
Check Transitive
If y is divisible by x and z is divisible by y,
Then z is divisible by x
Therefore, if (x,y) ∈ R & (y,z) ∈ R, then (x,z) ∈ R
Therefore, R is transitive.
Question 3
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever Let I be the set of all citizens of India who were eligible to exercise their voting right in general election held in 2019. A relation ‘R’ is defined on I as follows:
ONE – NATION
ONE – ELECTION
FESTIVAL OF DEMOCRACY
GENERAL ELECTION - 2019
R = {(V₁ , V₂ ) : V₁ , V₂ ∈ I and both use their voting right in general election – 2019}
Mr. ’X’ and his wife ‘W’ both exercised their voting right in general election -2019, Which of the following is true?

A
A. both (X, W) and (W, X) ∈ R

B
B. (X, W) ∈ R but (W, X) ∉ R

C
C. both (X, W) and (W, X) ∉ R

D
D. (W, X) ∈ R but (X, W) ∉ R

Solution

Given R = {(V₁ , V₂ ∈ I and both use their voting rights}
Since X voted, and W also voted.
Therefore, both (X, W) ∈ R and (W, X) ∈ R
Question 4
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1, 2, 3, 4, 5, 6}. Let A be the set of players while B be the set of all possible outcomes.
A = {S, D}, B = {1, 2, 3, 4, 5, 6}
Let R be a relation on B defined by R = {(1, 2), (2, 2), (1, 3), (3, 4), (3, 1), (4, 3), (5, 5)}. Then R is

A
A. Symmetric

B
B. Reflexive

C
C. Transitive

D
D. None of these

Solution

R = {(1, 2), (2, 2), (1, 3), (3, 4), (3, 1), (4, 3), (5, 5)}
R is not reflexive.
Since, (1, 1), (3, 3), (4, 4), (6, 6) ∈ R
R is not symmetric.
Because, for (1, 2) ∈ R there does not exist (2, 1) ∈ R.
R is not transitive.
Because for all element of B there does not exist, (a, b) (b, c) ∈ R and (a, c) ∈ R.
Question 5
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever Let I be the set of all citizens of India who were eligible to exercise their voting right in general election held in 2019. A relation ‘R’ is defined on I as follows:
ONE – NATION
ONE – ELECTION
FESTIVAL OF DEMOCRACY
GENERAL ELECTION - 2019
R = {(V₁ , V₂ ) : V₁ , V₂ ∈ I and both use their voting right in general election – 2019}
The above defined relation R is __________

A
A. Symmetric and transitive but not reflexive

B
B. Universal relation

C
C. Equivalence relation

D
D. Reflexive but not symmetric and transitive

Solution

R is reflexive, since every person is friend or itself.
i.e., (F₁ , F₂ ) ∈ R
Further, (F₁ , F₂ ) ∈ R
⇒ F₁ is friend of F₂
⇒ F₂ is friend of F₁
⇒ (F₂ , F₁ ) ∈ R
⇒ R is symmetric
Moreover, (F₁ , F₂ ), (F₂ , F₃ ) ∈ R
⇒ F₁ is friend of F₂ and F₂ is friend of F₃ .
⇒ F₁ is a friend of F₃ .
⇒ (F₁ , F₃ ) ∈ R
Therefore, R is an equivalence relation.
Question 6
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1, 2, 3, 4, 5, 6}. Let A be the set of players while B be the set of all possible outcomes.
A = {S, D}, B = {1, 2, 3, 4, 5, 6}
Let R : B → B be defined by R = {(x, y) : y is divisible by x} is

A
A. Reflexive and transitive but not symmetric

B
B. Reflexive and symmetric but not transitive

C
C. Not reflexive but symmetric and transitive

D
D. Equivalence

Solution

R is reflexive, since every element of B i.e.,
B = {1, 2, 3, 4, 5, 6} is divisible by itself.
i.e., (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) ∈ R
further, (1, 2) ∈ R
but (2, 1) ∉ R
Moreover,
(1, 2), (2, 4) ∈ R
⇒ (1, 4) ∈ R
⇒ R is transitive.
Therefore, R is reflexive and transitive but not symmetric.
Question 7
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever Let I be the set of all citizens of India who were eligible to exercise their voting right in general election held in 2019. A relation ‘R’ is defined on I as follows:
ONE – NATION
ONE – ELECTION
FESTIVAL OF DEMOCRACY
GENERAL ELECTION - 2019
R = {(V₁ , V₂ ) : V₁ , V₂ ∈ I and both use their voting right in general election – 2019}
Mr. Shyam exercised his voting right in General Election - 2019, then Mr. Shyam is related to which of the following?

A
A. All those eligible voters who cast their votes

B
B. Family members of Mr. Shyam

C
C. All citizens of India

D
D. Eligible voters of India

Solution

Given R = {((V₁ , V₂ ) : V₁ , V₂ ∈ I and both use their voting rights}
So, Mr. Shyam will be related to all eligible voters who casted their votes.
Question 8
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1, 2, 3, 4, 5, 6}. Let A be the set of players while B be the set of all possible outcomes.
A = {S, D}, B = {1, 2, 3, 4, 5, 6}
Raji wants to know the number of relations possible from A to B. How many numbers of relations are possible?

A
A. 6²

B
B. 2⁶

C
C. 6!

D
D. 2¹²

Solution

Given A = {S,D}, B = {1, 2, 3, 4, 5, 6}
Number of relation from A to B
= 2 Numbers of elements of A x Number of elements of B
= 2^{2 x 6}
= 2¹²
Question 9
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever Let I be the set of all citizens of India who were eligible to exercise their voting right in general election held in 2019. A relation ‘R’ is defined on I as follows:
ONE – NATION
ONE – ELECTION
FESTIVAL OF DEMOCRACY
GENERAL ELECTION - 2019
R = {(V₁ , V₂ ) : V₁ , V₂ ∈ I and both use their voting right in general election – 2019}
Three friends F₁ , F₂ and F₃ exercised their voting right in general election-2019, then which of the following is true?

A
A. (F₁ , F₂ ) ∈ R, (F₂ , F₃ ) ∈ R and (F₁ , F₃ ) ∈ R

B
B. (F₁ , F₂ ) ∈ R, (F₂ , F₃ ) ∈ R and (F₁ , F₃ ) ∉ R

C
C. (F₁ , F₂ ) ∈ R, (F₂ , F₂ ) ∈ R but (F₃ , F₃ ) ∉ R

D
D. (F₁ , F₂ ) ∉ R, (F₂ , F₃ ) ∉ R and (F₁ , F₃ ) ∉ R

Solution

Given R = {((V₁ , V₂ ) : V₁ , V₂ ∈ I and both use their voting rights}
Since all 3 friends F₁ , F₂ and F₃ voted
Therefore, (F₁ , F₂ ) ∈ R, (F₂ , F₃ ) ∈ R and (F₁ , F₃ ) ∈ R
Question 10
1 / -0.25

Direction: Read the following text and answer the following questions on the basis of the same:
Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1, 2, 3, 4, 5, 6}. Let A be the set of players while B be the set of all possible outcomes.
A = {S, D}, B = {1, 2, 3, 4, 5, 6}
Raji wants to know the number of functions from A to B. How many number of functions are possible?

A
A. 6²

B
B. 2⁶

C
C. 6!

D
D. 2¹²

Solution

Given A = {S,D}, B = {1, 2, 3, 4, 5, 6}
So, A has 2 elements, B has 6 elements
Number of functions from A to B = 6*6

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Relations and Functions Test - 11

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Relations and Functions Test - 11

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Question 1

A. (X, Y) ∈ R

B. (Y, X) ∈ R

C. (X, X) ∉ R

D. (X, Y) ∉ R

Solution

Question 2

A. Symmetric

B. Reflexive and Transitive

C. Transitive and symmetric

D. Equivalence

Solution

Question 3

A. both (X, W) and (W, X) ∈ R

B. (X, W) ∈ R but (W, X) ∉ R

C. both (X, W) and (W, X) ∉ R

D. (W, X) ∈ R but (X, W) ∉ R

Solution

Question 4

A. Symmetric

B. Reflexive

C. Transitive

D. None of these

Solution

Question 5

A. Symmetric and transitive but not reflexive

B. Universal relation

C. Equivalence relation

D. Reflexive but not symmetric and transitive

Solution

Question 6

A. Reflexive and transitive but not symmetric

B. Reflexive and symmetric but not transitive

C. Not reflexive but symmetric and transitive

D. Equivalence

Solution

Question 7

A. All those eligible voters who cast their votes

B. Family members of Mr. Shyam

C. All citizens of India

D. Eligible voters of India

Solution

Question 8

A. 62

B. 26

C. 6!

D. 212

Solution

Question 9

A. (F1 , F2 ) ∈ R, (F2 , F3 ) ∈ R and (F1 , F3 ) ∈ R

B. (F1 , F2 ) ∈ R, (F2 , F3 ) ∈ R and (F1 , F3 ) ∉ R

C. (F1 , F2 ) ∈ R, (F2 , F2 ) ∈ R but (F3 , F3 ) ∉ R

D. (F1 , F2 ) ∉ R, (F2 , F3 ) ∉ R and (F1 , F3 ) ∉ R

Solution

Question 10

A. 62

B. 26

C. 6!

D. 212

Solution

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A. 6²

B. 2⁶

D. 2¹²

A. (F₁ , F₂ ) ∈ R, (F₂ , F₃ ) ∈ R and (F₁ , F₃ ) ∈ R

B. (F₁ , F₂ ) ∈ R, (F₂ , F₃ ) ∈ R and (F₁ , F₃ ) ∉ R

C. (F₁ , F₂ ) ∈ R, (F₂ , F₂ ) ∈ R but (F₃ , F₃ ) ∉ R

D. (F₁ , F₂ ) ∉ R, (F₂ , F₃ ) ∉ R and (F₁ , F₃ ) ∉ R

A. 6²

B. 2⁶

D. 2¹²