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.

Question

Philoid · Accepted Answer

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.

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.