Question

Let $A=\{x:x=2n+1,n\in W\}$ and $B=\{y:y=2n,n\in N\}.$ If a relation $R$ is defined from $A$ to $B,$ then which of the following is void relation?

• A. $R=\left\{\right(x,y):x+y=11\}$
• B. $R=\left\{\right(x,y):x+2y=21\}$
• C. $R=\left\{\right(x,y):2x+y=14\}$
• D. $R=\left\{\right(x,y):x+y=8\}$

Answer 1

The correct option is D $R=\left\{\right(x,y):x+y=8\}$

$A=\{x:x=2n+1,n\in W\}$
$\Rightarrow A=\{1,3,5,7,9,...\}$
$B=\{y:y=2n,n\in N\}$
$\Rightarrow B=\{2,4,6,8,10,...\}$

$R=\left\{\right(x,y):x+y=11\}$
Here, $R=\left\{\right(1,10),(3,8),(5,6),(7,4),(9,2\left)\right\}$
It is not a void relation.

$R=\left\{\right(x,y):x+2y=21\}$
Here, $R=\left\{\right(1,10),(5,8),(9,6),(13,4),(17,2\left)\right\}$
It is not a void relation.

$R=\left\{\right(x,y):2x+y=14\}$
Here, $R=\left\{\right(1,12),(3,8),(5,4\left)\right\}$
It is not a void relation.

$R=\left\{\right(x,y):x+y=8\}$
Since, $x$ is odd number and $y$ is even number, so sum of $x+y$ can never be even number.
Hence, it is a void relation.