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