Question

Relation $R$ in the set $A$ of human beings in a town at a particular time given by $R=\left\{\right(x,y):xiswifeofy\}$

enter 1-reflexive and transitive but not symmetric

2-reflexive only

3-Transitive only

4-Equivalence

5-Neither reflexive, nor symmetric, nor transitive

Answer 1

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

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 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.