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

Solution:

It is given that

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, 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

NCERT Solutions Class 11 Maths Chapter 2 Exercise ME Question 9

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

Summary:

Hence the following three given statements are not true