JEE Main · 2023 · 08 Apr 2023 · Shift 1 · Sets Relations and Functions

The set $A$ has five even elements $\{0,4,6,8,10\}$ and three odd elements $\{3,7,9\}$. A relation is symmetric when $(y,x)\in R$ for every $(x,y)\in R$, so the number of elements to add equals the number of pairs in $R$ whose reverse is not already present.

If $x-y$ is a positive odd integer, then $x$ and $y$ have opposite parity with $x>y$; each of the $5\times 3=15$ even-odd pairs yields exactly one such ordering, giving $15$ pairs. For each, the reverse has $y-x$ negative, which is neither a positive odd integer nor $2$, so all $15$ reverses are missing.

If $x-y=2$, the two elements share parity: among the evens these are $(6,4),(8,6),(10,8)$ and among the odds $(9,7)$, giving $4$ pairs, and again each reverse (with $y-x=-2$) is absent.

The two cases are disjoint, so the minimum number of elements to add is $15+4=19.$