Question

State the reason why the relation $S=(a,b)\in R\times R:a\le b^3$ on the set $R$ of real numbers is not transitive.

Answer 1

$S=(a,b)\in R\times R;a\le b^3$

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

it $a<b^3\&b<c^3$ then $a<c^3$

$\begin{array}{cccccccc}a& b& c& b^3& c^3& a<b^3& b<c^3& a<c^3\\ 1& 2& 3& 4& 27& true& True& True\\ 3& 3/2& 4/3& 3.375& 2.37& True& True& False\end{array}$

$a<b^3,b<c^3$ then $a<c^3$ is not

true for all real numbers.

Hence, relatipn is not transitive.