JEE Main · 2025 · 08 Apr 2025 · Shift 2 · Sets Relations and Functions

Split the pairs $(x,y)$ with $\max \{x,y\}\in \{3,4\}$ by their maximum. With maximum $3$, both coordinates lie in $\{0,1,2,3\}$ and at least one equals $3$: $4^2-3^2=7$ pairs. With maximum $4$, both lie in $\{0,1,2,3,4\}$ with at least one equal to $4$: $5^2-4^2=9$ pairs. Hence $|R|=16$, so $S_1$ (claiming $18$) is false. Because $\max$ is symmetric, $(x,y)\in R\Rightarrow (y,x)\in R$, so $R$ is symmetric; it is not reflexive since $(0,0)\notin R$, and not transitive since $(0,3)\in R$ and $(3,1)\in R$ while $(0,1)\notin R$. Only $S_2$ holds.