Question

Determine whether the Relation R in the set $A=1,2,3,4,5,6$ as

$R=\left\{\right(x,y):yisdivisiblebyx\}$ is reflexive, symmetric and transitive

Answer 1

$A=\{1,2,3,4,5,6\}$

$R=\{(x,y):yisdivisiblebyx\}$

We know that any number $\left(x\right)$ is divisible by itself.

$(x,x)\in R$

$\therefore R$ is reflexive.

Now,$(2,4)\in R$ [as $4$ is divisible by $2$]

But,$(4,2)\notin R$. [as $2$ is not divisible by $4$]

$\therefore R$ is not symmetric.

Let $(x,y),(y,z)\in R$. Then, $y$ is divisible by $x$ and $z$ is divisible by $y$.

$\therefore z$ is divisible by $x$.

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

$\therefore R$ is transitive.

Hence, $R$ is reflexive and transitive but not symmetric.