Question

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

• A. Both $R_1$ and $R_2$ are transitive relations.
• B. Range of $R_2$ is $\{1,2,3,4\}$.
• C. Range of $R_1$ is $\{2,4,8\}$.
• D. Both $R_1$ and $R_2$ are symmetric relations.

Answer 1

The correct option is B Range of $R_2$ is $\{1,2,3,4\}$.

$R_1=\left\{\right(1,8),(2,6),(3,4),(4,2\left)\right\}{R}_2=\left\{\right(8,1),(6,2),(4,3),(2,4\left)\right\}\text{Range of}{R}_2=\{1,2,3,4\}$

$(1,8)\in R_1$ but $(8,1)\notin R_1$, so $R_1$ is not symmetric.
$(4,2)\in R_1\&(2,6)\in R_1$ but $(4,6)\notin R_1$, So $R_1$ is not transitive.

Similarly, $R_2$ is neither symmetric nor transitive.