Question

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

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

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

$(x,x)\notin R$

As $x$ cannot be the father of himself.

$\therefore R$ is not reflexive.

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

$\Rightarrow x$ 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 $(y,z)\in R$.

$\Rightarrow x$ 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.