Mathematics - Sets and Relations Question with Solution | TestHub

Here, $p,p^n\in \left\{1,2,\dots 50\right\}$

Now $p$ can take values

$2,3,5,7,11,13,17,23,29,31,37,41,43$ and $47$.

$\therefore$ $R_1$$=\{\left(2,2^0\right),\left(2,2^1\right)\dots \left(2,2^5\right),$$\left(3,3^0\right),\dots \left(3,3^3\right),$$\left(5,5^0\right),\dots \left(5,5^2\right),$$\left(7,7^0\right),\dots \left(7,7^2\right),$$\left(11,{11}^0\right),\left(11,{11}^1\right),...\left(47,{47}^0\right),\left(47,{47}^1\right)\}$

$R_2=\left\{\left(2,2^0\right),\left(2,2^1\right),\left(3,3^0\right),\left(3,3^1\right)....\left(47,{47}^0\right),\left(47,{47}^1\right)\right\}$

Now $R_1-R_2\equiv R_1\cap {R_2}^{\text{'}}$

So $R_1-R_2=\left\{\left(2,2^2\right),\left(2,2^3\right),\left(2,2^4\right),\left(2,2^5\right),\left(3,3^2\right),\left(3,3^3\right),\left(5,5^2\right),\left(7,7^2\right)\right\}$

$\therefore n\left(R_1-R_2\right)=8$