Question

The relation $R$ defined on the set $A=\{1,2,3,4,5\}$ by $R=\left\{\right(x,y):|x^2-y^2|<16\}$ is given by

• A. $\left\{\right(1,1),(2,1),(3,1),(4,1),(2,3\left)\right\}$
• B. $\left\{\right(2,2),(3,2),(4,2),(2,4\left)\right\}$
• C. $\left\{\right(3,3),(4,3),(5,4),(3,4\left)\right\}$
• D. None of these

Answer 1

Answer: (D)

None of these

We have $R=\left\{\right(x,y):|x^2-y^2|<16\}$.

Take $x=1,$ we have

$|x^2-y^2|<16\Rightarrow |1-y^2|<16$
$\Rightarrow |y^2-1|<16\Rightarrow y=1,2,3,4$.

Take $x=2,$ we have

$|x^2-y^2|<16\Rightarrow |4-y^2|<16$
$\Rightarrow |y^2-4|<16\Rightarrow y=1,2,3,4$.

Take $x=3,$ we have

$|x^2-y^2|<16\Rightarrow |9-y^2|<16$
$\Rightarrow |y^2-9|<16\Rightarrow y=1,2,3,4$.

Take $x=4,$ we have

$|x^2-y^2|<16\Rightarrow |16-y^2|<16$
$\Rightarrow |y^2-16|<16\Rightarrow y=1,2,3,4,5$.

Take $x=5,$

$|x^2-y^2|<16\Rightarrow |25-y^2|<16$
$\Rightarrow |y^2-25|<16\Rightarrow y=4,5$

$\therefore R=\left\{\right(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3),(2,4),(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4),(4,5),(5,4),(5,5)\}$
$\therefore$ The correct answer is $D$.