Question

Relation $R$ in the set $Z$ of all integers defined as $R=\left\{\right(x,y):(x-y\left)isaninteger\right\}$

enter 1-reflexive and transitive but not symmetric

2-reflexive only

3-Transitive only

4-Equivalence

5-None

Answer 1

$R=\left\{\right(x,y):x-y\text{is an integer}\}$

Now, for every $x\in Z,(x,x)\in R$ as $x-x=0$ is an integer.

$\therefore R$ is reflexive.

Now, for every $x,y\in Z$ if $(x,y)\in R$, then $x-y$ is an integer.

$\Rightarrow -(x-y)$ is also an integer.

$\Rightarrow (y-x)$ is an integer.

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

$\Rightarrow R$ is symmetric.

Now,

Let $(x,y)$ and $(y,z)\in R$, where $x,y,z\in Z$.

$\Rightarrow (x-y)$ and $(y-z)$ are integers.

$\Rightarrow x-z=(x-y)+(y-z)$ is an integer.

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

$\therefore R$ is transitive.

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