Question

If a relation $R$ defined on $A=\{1,3,5,7\}$, then which of the following is/are void relation?

• A. $R=\left\{\right(a,b):a+b=8:a,b\in A\}$
• B. $R=\left\{\right(a,b):2a+b=10:a,b\in A\}$
• C. $R=\left\{\right(a,b):2ab=30:a,b\in A\}$
• D. $R=\left\{\right(a,b):a+b=11:a,b\in A\}$

Answer 1

Answer: (B), (D)

$R=\left\{\right(a,b):a+b=11:a,b\in A\}$

$R=\left\{\right(a,b):a+b=8:a,b\in A\}$
$R=\left\{\right(1,7),(3,5),(5,3),(7,1\left)\right\}$
It is not a void relation.

$R=\left\{\right(a,b):2a+b=10:a,b\in A\}$
Here $2a$ is even and $b$ is odd
$\therefore 2a+b$ can never be even.
It is a void relation.

$R=\left\{\right(a,b):2ab=30:a,b\in A\}$
$R=\left\{\right(3,5),(5,3\left)\right\}$
It is not a void relation.

$R=\left\{\right(a,b):a+b=11:a,b\in A\}$
Here $a$ and $b$ is odd
$\therefore a+b$ can never be odd.
It is a void relation.