Let R be a relation from N to N defined by R = {(a, b): a, b, εN and a = b2}. Are the following true?
(i) (a,a) ε R, for all a ε N
(ii) (a,b) ε R, implies (b,a) ε R
(iii) (a,b) ε R, (b,c) ε R implies (a,c) ε R.
Justify your answer in each case.
Given: R = {(a,b): a, b Є N and a = b2}
(i) (a,a) Є R, for all a Є N
As we can see that 3 Є N but 3 ≠ 32 = 9.
Hence, the statement is not true.
(ii) (a,b) Є R, implies (b,a) Є R
As we can see that (4,2) Є N and 4 = 22 = 4 but 2 ≠ 42 = 16 hence, (2,4) does not belong to N.
Hence, the statement is not true.
(iii) (a,b) Є R, (b,c) 
R implies (a,c) Є R.
As we see, (9, 3) Є R, (16, 4) Є R because 3,4,9,16 Є N and 9 = 32 and 16 = 42.
Now, 9 ≠ 42 = 16; therefore, (9, 4) does not belong to N.
Hence, the statement is not true.