Question

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

Answer 1

$R=\left\{\right(a-b):a\le b^3\}$

(i) since $(a,a)\notin R$ as $a\le a^3$ is not always true

[Take a = 1/3. then $a\le a^3$ is not true]

$\therefore R$ is not reflexive

(ii) Also$(a,b)ϵR\Rightarrow (b,a)ϵR$

[Take a = 1, b=4, $\therefore 1\le 4^3$ but $4\left(1{)}^3\right]$

$\therefore$ R is not symmetric

(iii) Now (a,b)$ϵR(b,c)ϵR\Rightarrow (a,c)\therefore R$

[Take $a=100,b=5,c=3,\therefore 100\le 5^3,5\le 3^3$ but $100\ge 3^3]$

R is not symmetric.