Question

Let $R$ be a relation from $N$ to $N$ defined by $R=\left\{\right(a,b):a,b\in N\text{and}a=b^2\}$. Is the statement true?

(iii) $(a,b)\in R$ and $(b,c)\in R\Rightarrow (a,c)\in R$.

Answer 1

Given: $R=\left\{\right(a,b):a,b\in N\text{and}a=b^2\}$

If $(a,b)\in R,$ then

$a=b^2$ ...$\left(1\right)$

And $(b,c)\in R$

$b=c^2$ ...$\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$, we get

$a=c^4$ ...$\left(3\right)$

If $(a,c)\in R,$ then

$a=c^2$

From equation $\left(3\right)$, we get

$c^2=c^4$

$\Rightarrow c^2(c^2-1)=0$

$\Rightarrow c=0,\pm 1$

As relation is defined from $N$ to $N$, so $c=1$

So, $(a,b)\in R$ and $(b,c)\in R$ doesn't imply that $(a,c)\in R$

Hence, the given statement is False.