JEE Main 2026 Apr 6 Shift 2, Mathematics Q21: Relations

${\log }_e(x+y)\le 2$ $\Rightarrow$ $x+y\le e^2\approx 7.389$. With $x,y\in \mathbb{N}$, this means $x+y\le 7$.

So $R$ consists of all pairs $(x,y)$ with $x,y\ge 1$ and $x+y\le 7$: 21 pairs in total.

For transitivity, whenever $(a,b)$ and $(b,c)$ both lie in $R$, the pair $(a,c)$ must also lie in $R$. The set of pairs $(a,c)$ forced by chaining through some valid $b$ (with $1\le b\le 6$) covers every $(a,c)$ with $a,c\in \{1,2,3,4,5,6\}$ — that is, all 36 such pairs.

Of those 36 pairs, the ones already in $R$ are exactly those with $a+c\le 7$, namely $6+5+4+3+2+1=21$. The remaining $36-21=15$ pairs must be added:

$(6,2),(6,3),(6,4),(6,5),(6,6);$ $(5,3),(5,4),(5,5),(5,6);$ $(4,4),(4,5),(4,6);$ $(3,5),(3,6);$ $(2,6)$.

Minimum number of elements to add $=15$.