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Determine whether each of the following relations are reflexive, symmetric and transitive:
Relation $R$ in the set $A$ of human beings in a town at a particular time given by
$R =\{(x, y): x $ is father of $y\}$
Solution
$R =\{( x , y ): x$ is the father of $y \}$
$( x , x ) \notin R$
As $x$ cannot be the father of himself.
$\therefore R$ is not reflexive.
Now, let $( x , y ) \notin R$
$\Rightarrow x$ is the father of $y$
$\Rightarrow y$ cannot be the father of $y$
Indeed, $y$ is the son or the daughter of $y$.
$\therefore(y, x) \notin R$
$\therefore R$ is not symmetric.
Now, let $(x, y) \in R$ and $(y, z) \notin R$
$\Rightarrow x$ is the father of $y$ and $y$ is the father of $z$.
$\Rightarrow x$ is not the father of $z$
Indeed, $x$ is the grandfather of $z$
$\therefore $ $( x , z ) \notin R$
$\therefore R$ is not transitive.
Hence, $R$ is neither reflexive, nor symmetric, nor transitive.
