Let $R$ be relation defined on the set of natural number $N$ as follows,
$R = \{ (x,y):x \in N,y \in N,2x + y = 41\}$.
Find the domain and range of the relation $R$.
Also verify whether $R$ is reflexive, symmetric and transitive.

It is given that,, $R = \{ (x,y):x \in N,y \in N,2x + y = 41\}$.

Domain $= \{ 1,2,3, \ldots ,20\}$

Range $= \{ 1,3,5,7, \ldots ,39\}$

$R = \{ (1,39),(2,37),(3,35), \ldots ,(19,3),(20,1)\}$

$R$ is not reflexive as (2,2)$\notin R$

$2 \times 2 + 2 \ne 41$

So, $R$ is not symmetric.

As (1,39)$\in R$ but (39,1)$\notin R$

So, $R$ is not transitive.

As $(11,19) \in R,(19,3) \in R$

But (11,3)$\notin R$

Hence we can say that, $R$ is neither reflexive, nor symmetric and nor transitive.