Give an example of a relation. Which is Transitive but neither reflexive nor symmetric.
Give an example of a relation. Which is Transitive but neither reflexive nor symmetric.
Consider a relation $R$ in $R$ defined as:
$R =\{( a , b ): a < b \}$
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, $(1,2)\in R$ $($ as $1<2)$
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.
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