Question

The relation $R$ defined in $N$ as $aRb\iff b$ is divisible by $a$ is

• A. Reflexive but not symmetric
• B. Symmetric but not transitive
• C. Symmetric and transitive
• D. None of the above

Answer 1

The correct option is A

Reflexive but not symmetric

Explanation for the correct option:

Compute the required term:

$aRb\iff b$ is divisible by $a$

$1R1\iff 1$ is divisible by $1$

$(a,a)\in R\Rightarrow$ Reflexive

For symmetric, we know that $aRb\iff bRa$

Now, $2R4\Rightarrow 4$ is divisible by $2$ but $4R2\Rightarrow 2$ is not divisible by $4$.

Thus, it is not symmetric.

For transitive, we know that if $aRb\iff b$ is divisible by $a$, and $bRc\iff c$ is divisible by $b$, then $aRc\iff c$ is divisible by $a$.

Now $2R4\Rightarrow 4$ is divisible by $2$, and $4R8\iff 8$ is divisible by $4$, and $2R8\Rightarrow 8$ is also divisible by $2$.

Thus, it is transitive.

Hence, option $A$ is the correct answer.