Question

Let $R$ be a relation on the set of all natural numbers given by $aRb\iff a$ divides $b^2.$
Which of the following properties does $R$ satisfy ?
I. Reflexivity
II. Symmetry
III. Transitivity

• A. I only
• B. III only
• C. I and III only
• D. I and II only

Answer 1

The correct option is A I only

$aRb\iff \frac{b^2}{a}$
(I) Reflexivity: This relation is reflexive relation because every natural number divides square of itself i.e.,
$a$ divides $a^2.$
$\Rightarrow aRa$

(II) Symmetry:
Let $a=3$ and $b=6$
So, $3R6\text{as}3$ divides $6^2.$
But $6$ does not divide $3^2.$
This relation is not symmetric.

(III) Transitivity:
This relation is not transitive.
For example: $27R9\text{and}9R3\nRightarrow 27R3$