Question

Let R be a relation from N to N defined by R = {( a , b ): a , b ∈ N and a = b 2 }. Are the following true? (i) ( a , a ) ∈ R, for all a ∈ N (ii) ( a , b ) ∈ R, implies ( b , a ) ∈ R (iii) ( a , b ) ∈ R, ( b , c ) ∈ R implies ( a , c ) ∈ R. Justify your answer in each case.

Answer 1

R={ ( a,​​b ):a,b∈N, a= b 2 }

(i)

Consider ( a,a )∈R for all a∈N

Condition 1- ( a,a )∈N that is a natural number for relation of R ,

Condition2-For a= a 2 ,

a= a 2 1= 1 1 2≠ 2 2 3≠ 3 3

The above equations show that condition 2isnot satisfied, that is a= a 2 .

Therefore, the first statement is false.

(ii)

Consider ( a,b )∈R , implies ( b,a )∈R

Check the conditions with the help of example of ( a,b )∈R and ( b,a )∈R .

Let b = 3.

a= b 2 b=3 a= 3 2 a=9

Using a = 9,

b= a 2 3≠ 9 2 3≠81

The above equation clearly shows that if a= b 2 , then values of b= a 2 are not true for the function.

Thus, the second statement is false, that is ( b,a )∉R .

(iii)

Let ( a,b )∈R,( b,c )∈R implies( a,c )∈R

Check the condition with the help of an example,

Let b = 9

a= b 2 b=9 a= 9 2 a=81

Now, using b = 9,

b= c 2 9= c 2 9 =c 3=c

Checking whether ( a,c ) form the relation R,

a= c 2 81≠ ( 3 ) 2 81≠9

The above equation shows that if a= b 2 and b= c 2 , then the condition a= c 2 is not true, that is ( a,c )∉R .

Thus, the third statement is false.