Question

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

enter 1-reflexive and transitive but not symmetric

2-reflexive only

3-Transitive only

4-Equivalence

5-None

Answer 1

$R=\left\{\right(x,y):xandyworkatthesameplace\}$

$\Rightarrow (x,x)\in R$

$\therefore R$ is reflexive.

If $(x,y)\in R$, then $x$ and $y$ work at the same place.

$\Rightarrow y$ and $x$ work at the same place.

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

$\therefore R$ is symmetric.

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

$\Rightarrow x$ and $y$ work at the same place and $y$ and $z$ work at the same place.

$\Rightarrow x$ and $z$ work at the same place.

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

$\therefore R$ is transitive.

Hence, $R$ is reflexive, symmetric and transitive.