Question:
Let R be a relation from N to N defined by R = [(a, b) : a, b ∈ N and a = b2].
Are the following statements true?
(i) (a, a) ∈ R for all a ∈ N
(ii) (a, b) ∈ R ⇒ (b, a) ∈ R
(iii) (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R
Solution:
Given: R = [(a, b) : a, b ∈ N and a = b2]
(i) (a, a) ∈ R for all a ∈ N.
Here $2 \in \mathrm{N}$ hut? $\neq 2^{2}$
$\therefore(2,2) \notin \mathrm{R}$
False
(ii) $(a, b) \in \mathrm{R} \Rightarrow(b, a) \in \mathrm{R}$
$\because 4=2^{2}$
$(4,2) \in \mathrm{R}$, but $(2,4) \notin \mathrm{R}$
False
(iii) $(a, b) \in \mathbf{R}$ and $(b, c) \in \mathbf{R} \Rightarrow(a, c) \in \mathbf{R}$
$\because 16=4^{2}$ and $4=2^{2}$
$\therefore(16,4) \in \mathrm{R}$ and $(4,2) \in \mathrm{R}$
Here,
$(16,2) \notin \mathrm{R}$
False