Question

Let R be a relation on the set of all natural numbers given by $aRb\iff$ a divided $b^2$. 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

1. For reflexivity: $aRa=aRa=\frac{a^2}{a}=a\Rightarrow R$ is reflexive

2. For Symmetry: $aRb=\frac{b^2}{a}$ and $bRa=\frac{a^2}{b}\Rightarrow aRb\ne bRa\Rightarrow R$ is not symmetric.

3. For transitivity: $aRb=\frac{b^2}{a}=bRc=\frac{c^2}{b}$ but $aRc=\frac{c^2}{a}=\frac{c^4}{b^3}\ne \frac{c^2}{b}\Rightarrow R$ is not transitive