Question

Given an example of a relation. Which is transitive but neither reflexive nor symmetric.

Answer 1

Consider a relation $R$ in $R$ defined as
$R=\left\{\right(a,b):a<b\}$
For any $a\in R$, we have $(a,a)\notin R$ since $a$ cannot be strictly less than a itself. In fact, $a=a$.
$\therefore R$ is not reflexive.
Now, $(1,2)\in R(as1<2)$
But, $2$ is not less than $1$.
$\therefore (2,1)\notin R$
$\therefore R$ is not symmetric.
Now, let $(a,b),(b,c)\in R$.
$\Rightarrow a<b$ and $b<c$
$\Rightarrow a<c$
$\Rightarrow (a,c)\in R$
$\therefore R$ is transitive.
Hence, relation $R$ is transitive but not reflexive and symmetric.