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NCERT Class XI Mathematics - Relations and Functions - Solutions

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Question : 33 of 36
Marks: +1, -0
Let R be a relation from N to N defined by R = {(a, b) : a, b ∈ N and a = b2b^2}
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:  
We have, R = {(a, b) : a, b ∈ N and a = b2} = {(b2, b) : b ∈ N}
(i) False
Only when a = 1, (a, a) = (1, 1) = (12, 1) ∈ R.
(ii) False
If (a, b) ∈ R ⇒ a = b2b^2 ⇏ b = a2a^2
∴ (a, b) ∈ R ⇒ (b, a) ∉ R.
(iii) False
If (a, b) ∈ R ⇒ a = b2b^2 ......(i)
and (b, c) ∈ R ⇒ b = c2c^2 ....(ii)
From (i) and (ii), a = (c2)2(c^2)^2 = c4c^4 ⇏ a = c2c^2
⇒ (a, b) ∈ R and (b, c) ∈ R but (a, c) ∉ R.
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