Question

Let $\rho$ be a relation defined on$N$, the set of natural numbers, as
$\rho =\{(x,y)\in N\times N:2x+y=41\}$ then

• A. $\rho$ is an equivalence relation
• B. $\rho$ is only reflexive relation
• C. $\rho$ is only symmetric relation
• D. $\rho$ is not transitive

Answer 1

The correct option is D $\rho$ is not transitive

$\rho =\{(x,y)\in N\times N:2x+y=41\}$
for reflexive relation $xRx\Rightarrow 2x+x=41\Rightarrow x=\frac{41}{3}\notin N$
for symmetric $\Rightarrow xRy\Rightarrow 2x+y=41\ne yRx$ (Not symmetric)
for transitive $xRy\Rightarrow 2x+y=41$ and $yRz\Rightarrow 2y+z=41,x$ (Not transitive)