Question

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

enter 1-reflexive and transitive but not symmetric

2-reflexive only

3-Transitive only

4-Equivalence

5-None

Answer 1

$R=\left\{\right(x,y):\text{x and y live in the same locality}\}$

Clearly $(x,y)\in R$ as $x$ and $x$ is the same human being.

$\therefore R$ is reflexive.

If $(x,y)\in R$, then $x$ and $y$ live in the same locality.

$\Rightarrow y$ and $x$ live in the same locality.

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

$\therefore R$ is symmetric.

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

$\Rightarrow$ $x$ and $y$ live in the same locality and $y$ and $z$ live in the same locality.

$\Rightarrow$ $x$ and $z$ live in the same locality.

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

$\therefore R$ is transitive.

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