Let R1 and R2 be two relations defined as follows: R1= (a, b) ∈ R2: a2+b2 ∈ Q and R 2= ( a , b ) ∈ R 2: a 2+ b 2 ∉ Q where Q is the set of all rational numbers. Then:

Question

Let R1 and R2 be two relations defined as follows: R1= (a, b) ∈ R2: a2+b2 ∈ Q and R 2= ( a , b ) ∈ R 2: a 2+ b 2 ∉ Q 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

Tardigrade Admin · Accepted Answer

Let a2+b2 ∈ Q b2+c2 ∈ Q eg. a=2+√3 b=2-√3 a2+b2=14 ∈ Q Let c=(1+2 √3) b2+c2=20 ∈ Q But a2+c2=(2+√3)2+(1+2 √3)2 ∉ Q for R 2 Let a 2=1, b 2=√3 c 2=2 a2+b2 ∉ Q b2+c2 ∉ Q But a2+c2 ∈ Q

Q. Let $R_{1}$ and $R_{2}$ be two relations defined as follows:
$R_{1} = {(a, b) \in R^{2} : a^{2} + b^{2} \in Q}$ and
$R_{2} = {(a, b) \in R^{2} : a^{2} + b^{2} \in / Q}$
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

Q. Let R1​ and R2​ be two relations defined as follows: R1​={(a,b)∈R2:a2+b2∈Q} and R2​={(a,b)∈R2:a2+b2∈/Q} where Q is the set of all rational numbers. Then:

R2​ is transitive but R1​ is not transitive

R1​ is transitive but R2​ is not transitive

R1​ and R2​ are both transitive

Neither R1​ nor R2​ is transitive

Let a2+b2∈Q & b2+c2∈Q eg. a=2+3​ & b=2−3​ a2+b2=14∈Q Let c=(1+23​) b2+c2=20∈Q But a2+c2=(2+3​)2+(1+23​)2∈/Q for R2​ Let a2=1,b2=3​ & c2=2 a2+b2∈/Q & b2+c2∈/Q But a2+c2 & ∈Q

Q. Let $R_{1}$ and $R_{2}$ be two relations defined as follows:
$R_{1} = {(a, b) \in R^{2} : a^{2} + b^{2} \in Q}$ and
$R_{2} = {(a, b) \in R^{2} : a^{2} + b^{2} \in / Q}$
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