Check whether the relation R in R defined by $R = \{ (a,\;b):a \le {b^3}\}$ is reflexive, symmetric or transitive.

We have $R = \{ (a,\;b);\;\;a \le {b^3}\}$ where $a,\;b \in R.$

(i) Reflexive :
We observe that, $\cfrac{1}{2} \le {\left( {\cfrac{1}{2}} \right)^3}$ is not true.
Therefore, $\left( {\cfrac{1}{2},\;\cfrac{1}{2}} \right)\not \in R.$ So R is not reflexive.

(ii) Symmetric
We observe that $1 \le {(3)^2}\;\;but\;\;3\not \le {1^3}\;\;i.e.,\;\;(1,\;3) \in R\;\;but\;\;(3,\;1)\not \in R$
So, R is not symmetric.

(iii) Transitive
We observe that, $10 \le {3^3}\;\;and\;\;3 \le {2^3}\;\;but\;\;10\not \le {2^3}$
i.e., $(10,\;3) \in R\;\;and\;\;(3,\;2) \in R\;\;but\;\;(10,\;2)\not \in R$
So, R is not transitive.

R is neither reflexive nor symmetric nor transitive.