Let N denote the set of all natural numbers, define two binary relations on N as $R_1=\left\{(x,y)\in N\times N:2x+y=10\right\}$ and $R_2=\left\{(x,y)\in N\times N:x+2y=10\right\}$, then

 

	List – I		List – II
I)	$R_2$ is	P)	$\left\{1,2,3,4\right\}$
II)	$R_1$ is	Q)	not symmetric, not transitive
III)	Range of $R_1$	R)	Symmetric, transitive
IV)	Range of $R_2$	S)	$\left\{2,4,6,8\right\}$

 

i) $R_1=\left\{(x,y)\in N\times N:2x+y=10\right\}$

So, possible values for x and y are

$x=1,y=8\Rightarrow (1,8)$

$x=2,y=6\Rightarrow (2,6)$

$x=3,y=4\Rightarrow (3,4)$

$x=4,y=2\Rightarrow (4,2)$

Range of $R_1=\left\{2,4,6,8\right\}$$\Rightarrow R_1$ is  not symmetric, not transitive

ii) $R_2=\left\{(x,y)\in N\times N:x+2y=10\right\}$

So, possible values for x and y are

$x=8,y=1\Rightarrow (8,1)$

$x=6,y=2\Rightarrow (6,2)$

$x=4,y=3\Rightarrow (4,3)$

$x=2,y=4\Rightarrow (2,4)$

Range of $R_2=\left\{1,2,3,4\right\}$$\Rightarrow R_2$  is  not symmetric, not transitive