Question

Show that the relation $R$ in the set $R$ of real numbers, defined as $R=\left\{\right(a,b):a<b^2\}$, is neither reflexive nor symmetric nor transitive.

Answer 1

$\frac{1}{2},\frac{1}{4}$ Not reflexive

$a=2,b=5\Rightarrow (a,b):2\le 25$, is true

but $(b,a):5<4$ is not true

hence $(a,b)\in R(b,a)$ : Not symmetric

$a=3,b=-2,c=-1$

$(a,b)\in R,(b,c)\in R$

$\Rightarrow 3\le 4,-2\le 1$

$(a,c)\in R\Rightarrow 3\le 1$ which is not true

hence $(a,b)\in R,(b,c)\in R$

$(a,c)\in R$: not transitive.