Question

Check whether the relation R on R defined by R = {(a, b) : a ≤ b³} is reflexive, symmetric or transitive.

Answer 1

Reflexivity:

$$\begin{gathered} \mathrm{Since}\frac{1}{2}>{\left(\frac{1}{2}\right)}^3, \\ \left(\frac{1}{2},\frac{1}{2}\right)\notin R \\ \mathrm{So},R\mathrm{is}\mathrm{not}\mathrm{reflexive}. \end{gathered}$$

Symmetry:

$$\begin{gathered} \mathrm{Since}\left(\frac{1}{2},2\right)\in R, \\ \frac{1}{2}<2^3 \\ \mathrm{But}2>{\left(\frac{1}{2}\right)}^3 \\ \Rightarrow \left(2,\frac{1}{2}\right)\in R \\ \mathrm{So},R\mathrm{is}\mathrm{not}\mathrm{symmetric}. \end{gathered}$$

Transitivity:
$$\begin{gathered} \mathrm{Since}\left(7,3\right)\in R\mathrm{and}\left(3,3^{\frac{1}{3}}\right)\in R\mathit{,} \\ 7<3^3\mathrm{and}3={\left(3^{\frac{1}{3}}\right)}^3 \\ \mathrm{But}7>{\left(3^{\frac{1}{3}}\right)}^3 \\ \Rightarrow \left(7,3^{\frac{1}{3}}\right)\notin R \\ \mathrm{So},R\mathrm{is}\mathrm{not}\mathrm{transitive}. \end{gathered}$$