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?

1. (a, a) ∈ R, for all a ∈ N
2. (a, b) ∈ R, implies (b, a) ∈ R
3. (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.

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(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.

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(iii) It can be seen that (16, 4) ∈ R, (4, 2) ∈ R because 16, 4, 2 ∈ N and 16 = 4²
and 4 = 2².
Now, 16 ≠ 2² = 4; therefore, (16, 2) ∈ N

Therefore, the statement “(a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R” is not true.