Question

Determine whether each of the following relations are reflexive, symmetric and transitive

Relation $R$ in the set $A=\{1,2,3,4,5,6\}$ as $R=\left\{\right(x,y):y\text{is divisible by}x\}$

enter 1-reflexive and transitive but not symmetric

2-reflexive only

3-Transitive only

4-Equivalence

5-None

Answer 1

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

$R=\left\{\right(x,y):\text{y is divisible by x}\}$

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

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

$\therefore R$ 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.