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Let $R$ be a relation from $N$ to $N$ defined by $R=\left\{(a, b): a, b \in N\right.$ and $\left.a=b^2\right\}$.Are the following true ?
(i) $(a, a) \in R$ for all $a \in N$
(ii) $(a, b) \in \mathrm{R}$, implies $(b, a) \in R$
(iii) $(a, b) \in R,(b, c) \in R$ implies $(a, c) \in R$.
Justify your answer in each case.
(i) $(a, a) \in R$ for all $a \in N$
(ii) $(a, b) \in \mathrm{R}$, implies $(b, a) \in R$
(iii) $(a, b) \in R,(b, c) \in R$ implies $(a, c) \in R$.
Justify your answer in each case.
Solution:
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Verified Answer
(i) $a=a^2$ is true only, when $a=0$ or 1 . It is not a relation.
(ii) $a=b^2$ and $b=a^2$ is not true for all $a, b \in N$ It is not a relation.
(iii) $a=b^2, b=c^2, \therefore a=\left((c)^2\right)^2=c^4 \Rightarrow a \neq c^2$
$\therefore$ It is not a relation.
(ii) $a=b^2$ and $b=a^2$ is not true for all $a, b \in N$ It is not a relation.
(iii) $a=b^2, b=c^2, \therefore a=\left((c)^2\right)^2=c^4 \Rightarrow a \neq c^2$
$\therefore$ It is not a relation.
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