Question

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:

R = {(v₁, v₂) ∶ v₁, v₂ ∈ \(I\)and both use their voting right in general election – 2019}

1. Two neighbors 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. (X,Y) ∈R

b. (Y,X) ∈R

c. (X,X) ∉R

d. (X,Y) ∉R

2. Mr.’’ and his wife ‘’both exercised their voting right in general election -2019, Which of the following is true?

a. both (X,W) and (W,X) ∈ R

b. (X,W) ∈ R but (W,X) ∉ 

 c. both (X,W) and (W,X) ∉ R

d. (W,X) ∈ R but (X,W) ∉ R

3. Three friends F₁, F₂ and F₃ exercised their voting right in general election-2019, then which of the following is true?

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 

4. The above defined relation R is __________

a. Symmetric and transitive but not reflexive

b. Universal relation

c. Equivalence relation

d. Reflexive but not symmetric and transitive

5. Mr. Shyam exercised his voting right in General Election – 2019, then Mr. Shyam is related to which of the following

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

Answer 1

1. (d) (X,Y) ∉ R

Given R = {(V₁, V₂): V₁, V₂ ∉ I and both use their voting rights}

Since X voted, and Y did not vote

So, we can write (X, Y) ∉ R

2. (a) both (X,W) and (W,X) ∈ R

Given R = {(V₁, V₂): 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

3. (a) (F₁,F₂) ∈ R, (F₂,F₃) ∈ R and (F₁,F₃) ∈ R

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

4. (c) Equivalence relation

Given

R = {(V₁, V₂): V₁, V₂ ∉ I and both use their voting rights}

Check reflexive

Here, (V, V) ∉ R

So, R is reflexive,

Check symmetric

If V₁ and V₂ both use their voting rights

Then, if (V₁, V₂) ∉ R, then (V₂, V₁) ∉ R.

Check symmetric

If V₁ and V₂ both use their voting rights 

Then, if (V₁, V₂) ∉ R, then (V₂, V₁) ∉ R. 

Hence, R is symmetric.

Check transitive

If (V₁, V₂) and (V₂, V₃) ∉ R, then (V₁, V₃) ∉ R.

So, R is transitive

Since R is reflexive, symmetric & transitive.

Therefore, R is an equivalence relation.

Since R is reflexive, symmetric & transitive. 

Therefore, R is an equivalence relation.

5. (a) All those eligible voters who cast their votes

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.

Answer 2

1. (d) (X,Y) ∉R

2. (a) both (X,W) and (W,X) ∈ R

3. (a) (F1,F2 ) ∈R, (F2,F3) ∈ R and (F1,F3) ∈ R

4. (c) Equivalence relation

5. (a) All those eligible voters who cast their votes