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Question

Let R be a relation from N to N defined by R = {(a, b): a, b  N and a = b2}. Which of the following is/are 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


  1. Only (i)

  2. Only (ii) and (iii)

  3.  Only (i) and (ii)

  4. None of these


Solution

The correct option is D

None of these


 R = {(a, b): a, b ∈ N and a = b2}

(i) It can be seen that 2  N and 222 . Therefore (a, a)  R for all a  N is NOT true.

(ii) It can be seen that (9, 3)  R because 9, 3  M and 9 = 32 

But now 392  (3, 9) R

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

(iii) it can be seen that (a, b)  R a = b2    - - - - - (1)

(b, c)  R b = c2                                    - - - - - (2)

From these two we get,

A = c2c4  (except for a = c = 1)

Hence (a, c) R

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

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