Question

Let R be a relation from N to N defined by R = {(a, b): a, b $\in$ N and a = $b^2$ }. Which of the following is/are true?

(i) (a, a) $\in$ R, for all a $\in$ N

(ii) (a, b) $\in$ R, implies (b, a) $\in$ R

(iii) (a, b) $\in$ R, (b, c) $\in$ R implies (a, c) $\in$ R

• A. Only (i)
• B. Only (i) and (ii)
• C. Only (ii) and (iii)
• D. None of these

Answer 1

The correct option is D

None of these

R = {(a, b): a, b ∈ N and a = $b^2$}

(i) It can be seen that 2 $\in$ N and $2^2\ne 2$ . Therefore (a, a) $\in$ R for all a $\in$ N is NOT true.

(ii) It can be seen that (9, 3) $\in$ R because 9, 3 $\in$ M and 9 = $3^2$

But now $3\ne 9^2$ (3, 9) ∉ R

Therefore the statement (a, b) $\in$ R implies (b, a) $\in$ R is NOT true.

(iii) it can be seen that (a, b) $\in$ R a = $b^2$ - - - - - (1)

(b, c) $\in$ R b = $c^2$ - - - - - (2)

From these two we get,

A = $c^2\ne c^4$ (except for a = c = 1)

Hence (a, c) ∉ R

Therefore, the statement (a, b) $\in$ R, (b, c) $\in$ R implies (c, a) $\in$ R is NOT true.