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 $y\}$

$R =\{( x , y ): x$  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.