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 wife of  $y\}$

$R =\{( x , y ): x$ is the wife of $y \}$

Now,

$(x, x) \notin R$

since $x$ cannot be the wife of herself.

$\therefore R$ is not reflexive.

Now, let $(x, y) \in R$

$\Rightarrow x$ is the wife of $y$ Clearly y is not the wife of $x$. $\therefore $ $( y , x ) \notin R$

Indeed, if $x$ is the wife of $y$, then $y$ is the husband of $x$.

$\therefore $ $R$ is not transitive.

Let $( x , \,y ),\,( y , \,z ) \in R$

$\Rightarrow $ $x$ is the wife of $y$ and $y$ is the wife of $z$.

This case is not possible. Also, this does not imply that $x$ is the wife of $z$.

$\therefore $ $( x ,\, z ) \notin R$

$\therefore $ $R$ is not transitive.

Hence, $R$ is neither reflexive, nor symmetric, nor transitive.