Question

Let $R_1$ and $R_2$ be relations on the set $\{1,2,\dots ,50\}$ such that
$R_1=\{\left(p,p^n\right)$ : $p$ is a prime and $n\ge 0$ is an integer $\}$ and $R_2=\{\left(p,p^n\right):p$ is a prime and $n=0$ or 1$\}$.
Then, the number of elements in $R_1-R_2$ is ______

Answer 1

Answer: 8

Here, $p,p^n\in \{1,2,\dots 50\}$
Now $p$ can take values
$2,3,5,7,11,13,17,23,29,31,37,41,43$ and 47 .
we can calculate no. of elements in R, as
$\left(2,2^{\circ }\right),\left(2,2^1\right)..\left(2,2^5\right)$
$\left(3,3^{\circ }\right),\dots \left(3,3^3\right)$
$\left(5,5^{\circ }\right),\dots \left(5,5^2\right)$
$\left(7,7^{\circ }\right),\dots \left(7,7^2\right)$
$\left(11,11^{\circ }\right),\dots \left(11,11^1\right)$
And rest for all other two elements each
$\therefore n\left(R_1\right)=6+4+3+3+(2\times 10)=36$
Similarly for $R_2$
$\left(2,2^{\circ }\right),\left(2,2^1\right)$
$\left(47,47^{\circ }\right),\left(47,47^1\right)$
$\therefore n\left(R_2\right)=2\times 14=28$
$\therefore n\left(R_1\right)-n\left(R_2\right)=36-28=8$