Question

Let $R_{1}$ and $R_{2}$ be two relations defined as follows:
$R_{1}=\left\{(a, b) \in R^{2}: a^{2}+b^{2} \in Q\right\}$ and
$R _{2}=\left\{( a , b ) \in R ^{2}: a ^{2}+ b ^{2} \notin Q \right\}$
where $Q$ is the set of all rational numbers. Then:

• A. $R _{2}$ is transitive but $R _{1}$ is not transitive
• B. $R _{1}$ is transitive but $R _{2}$ is not transitive
• C. $R _{1}$ and $R _{2}$ are both transitive
• D. Neither $R _{1}$ nor $R _{2}$ is transitive

Answer 1

Answer: (D)

Let $a^{2}+b^{2} \in Q$ & $b^{2}+c^{2} \in Q$
eg. $a=2+\sqrt{3}$ & $b=2-\sqrt{3}$
$a^{2}+b^{2}=14 \in Q$
Let $c=(1+2 \sqrt{3})$
$b^{2}+c^{2}=20 \in Q$
But $a^{2}+c^{2}=(2+\sqrt{3})^{2}+(1+2 \sqrt{3})^{2} \notin Q$
for $R _{2}$ Let $a ^{2}=1, b ^{2}=\sqrt{3}$ & $c ^{2}=2$
$a^{2}+b^{2} \notin Q$ & $b^{2}+c^{2} \notin Q$
But $a^{2}+c^{2}$ & $\in Q$