Question

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

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

Answer 1

Answer: (D)

We have $R=\left\{\left(x,y\right):\left|x^2-y^2\right|<16\right\}$
Let $x=1,\left|x^2-y^2\right|<16$
$\Rightarrow \left|1-y^2\right|<16$
$\Rightarrow \left|y^2-1\right|<16$
$\Rightarrow y=1,2,3,4$.
Let $x=2,\left|x^2-y^2\right|<16$
$\Rightarrow \left|4-y^2\right|<16$
$\Rightarrow \left|y^2-4\right|<16$
$\Rightarrow y=1,2,3,4$.
Let $x=3,\left|x^2-y^2\right|<16$
$\Rightarrow \left|9-y^2\right|<16$
$\Rightarrow \left|y^2-9\right|<16$
$\Rightarrow y=1,2,3,4$.
Let $x=4,\left|x^2-y^2\right|<16$
$\Rightarrow \left|16-y^2\right|<16$
$\Rightarrow \left|y^2-16\right|<16$
$\Rightarrow y=1,2,3,4,5$.
Let $x=5,\left|x^2-y^2\right|<16$
$\Rightarrow \left|25-y^2\right|<16$
$\Rightarrow \left|y^2-25\right|<16$
$\Rightarrow y=4,5$.
$\therefore B=\{(1,1),(1,2),(1,3),(1,4),(2,1),(2,2),(2,3)$,
$(2,4),(3,1),(3,2),(3,3),(3,4),(4,1),(4,2),(4,3),(4,4)$,
$(4,5),(5,4),(5,5)\}$.