Mathematics - Sets and Relations Question with Solution | TestHub

Given: $A=\left\{1,2,3,...20\right\}$, $R_1=\{\left(a,b\right):b$ is divisible by $a$} and $R_2=\{\left(a,b\right):a$ is an integral multiple of $b$}.

$\Rightarrow R_1=\left\{\left(1,1\right),\left(1,2\right),...,\left(1,20\right),\left(2,2\right),\left(2,4\right),...,\left(2,20\right),\left(3,3\right),\left(3,6\right),...,\left(3,18\right),\left(4,4\right),\left(4,8\right),...,\left(4,20\right),\left(5,5\right),\left(5,10\right),...,\left(5,20\right),\left(6,6\right),\left(6,18\right),...,\left(6,18\right),\left(7,7\right),\left(7,14\right),\left(8,8\right),\left(8,16\right),\left(9,9\right),\left(9,18\right),\left(10,10\right),\left(10,20\right),\left(11,11\right),...,\left(20,20\right)\right\}$

$\Rightarrow n\left(R_1\right)=20+10+6+5+4+3+2+2+2+2+10=66$

$\Rightarrow R_1\cap R_2=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(4,4\right),...,\left(20,20\right)\right\}$

$\Rightarrow n\left(R_1\cap R_2\right)=20$

$\Rightarrow n\left(R_1-R_2\right)=n\left(R_1\right)-n\left(R_1\cap R_2\right)$

$\Rightarrow n\left(R_1-R_2\right)=66-20$

$\Rightarrow n\left(R_1-R_2\right)=46$