JEE Main 2024 — Sets And Relations Question with Solution

$$\begin{array}{rl}& A=\{1,2,3,4,5\}\\ & xRy\iff 4x\le 5y\\ & 4x\le 5y\Rightarrow \frac{x}{y}\le \frac{5}{4}\Rightarrow \frac{x}{y}\le 1.25\end{array}$$

$$\begin{array}{rl}& R=\{(1,2),(1,3),(1,4),(1,5),(1,1),(2,2),(2,3),(2,4),\\ & (2,5),(3,3),(3,4),(3,5),(4,4),(4,5),(5,4),(5,5)\}\\ & \therefore n(R)=m=16\end{array}$$

Elements to be added to $$R$$ to make it symmetric

$$(1,2)\in R\Rightarrow (2,1)$$ should be added

Similarly, $$(3,1),(4,1),(5,1),(3,2),(4,2),(5,2),(4,3),(5,3)$$

$$\therefore 9$$ elements should be added

$$\begin{array}{ll}\therefore & n=9\\ \therefore & m+n=25\end{array}$$