JEE Main · 2026 · 06 Apr 2026 · Shift 2 · Sets Relations and Functions

The condition ${\log }_e(x+y)\le 2$ means $x+y\le e^2\approx 7.39$, and since $x,y\in \mathbb{N}$ this is $x+y\le 7$; so $R$ is all pairs with $1\le x,1\le y,x+y\le 7$. A chain $(a,b),(b,c)\in R$ exists exactly when some $b\ge 1$ has $b\le 7-a$ and $b\le 7-c$, that is when $\max (a,c)\le 6$. Transitivity then requires $(a,c)\in R$; this already holds when $a+c\le 7$ and must be supplied when $a+c\ge 8$. The required pairs are those with $a,c\in \{2,3,4,5,6\}$ and $a+c\ge 8$, which for $a=2,3,4,5,6$ number $1,2,3,4,5$ respectively. Adding these pairs introduces no further violations, so the minimum number of elements that must be added is $15$.