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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
$R _{2}$ is transitive but $R _{1}$ is not transitive
$R _{1}$ is transitive but $R _{2}$ is not transitive
$R _{1}$ and $R _{2}$ are both transitive
Neither $R _{1}$ nor $R _{2}$ is transitive
Solution
Let $a^{2}+b^{2} \in Q \& b^{2}+c^{2} \in Q$
eg. $\quad a =2+\sqrt{3} \& b =2-\sqrt{3}$
$a^{2}+b^{2}=14 \in Q$
Let $\quad c =(1+2 \sqrt{3})$
$b ^{2}+ c ^{2}=20 \in Q$
But $\quad a^{2}+c^{2}=(2+\sqrt{3})^{2}+(1+2 \sqrt{3})^{2} \notin Q$
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 $a^{2}+c^{2} \in Q$
