Question

Let R be a relation from N to N defined by R = {(a, b): a, b ∈ N and a = b2}. 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 and a = b²} 

(i) It can be seen that 2 ∈ N; however, 2 ≠ 2² = 4. 

Therefore, the statement “(a, a) ∈ R, for all a ∈ N” is not true. 

(ii) It can be seen that (9, 3) ∈ N because 9, 3 ∈ N and 9 = 3². 

Now, 3 ≠ 9² = 81; 

therefore, (3, 9) ∉ N Therefore, the statement “(a, b) ∈ R, implies (b, a) ∈ R” is not true.

 (iii) It can be seen that (9, 3) ∈ R, (16, 4) ∈ R because 9, 3, 16, 4 ∈ N and 9 = 3² and 16 = 4². 

Now, 9 ≠ 4² = 16; therefore, (9, 4) ∉ N Therefore, the statement “(a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R” is not true.