Question

Reflexive and transitive but not symmetric.

Answer 1

Define a relation $R$ in $R$ as:
$R=\left\{\right(a,b):a^3\ge b^3\}$
Clearly $(a,a)\in R$ as $a^3=a^3$.
$\therefore R$ is reflexive.
Now, $(2,1)\in R$ $(\text{as}{2}^3\ge 1^3)$
But, $(1,2)\notin R(\text{as}{1}^3<2^3)$
$\therefore R$ is not symmetric.
Let $(a,b),(b,c)\in R$
$\Rightarrow a^3\ge b^3$ and $b^3\ge c^3$
$\Rightarrow a^3\ge c^3$
$\Rightarrow (a,c)\in R$
$\therefore R$ is transitive.
Hence, relation $R$ is reflexive and transitive but not symmetric.