JEE Main 2024 — Sets And Relations Question with Solution

To determine the number of elements in $R_1-R_2$, let's first articulate the meaning of both relations on set $A=\{1,2,3,\dots ,20\}$:

$R_1$ includes pairs $(a,b)$ where $b$ is divisible by $a$. This includes pairs like $(1,1),(1,2),\dots ,(1,20)$ for $1$; similar series for $2$ up to $(2,20)$ (excluding odd numbers); for $3$ up to $(3,18)$; and so on, reflecting the divisibility condition.

$R_2$ consists of pairs $(a,b)$ where $a$ is an integral multiple of $b$. Essentially, this relationship is the reverse of $R_1$. However, for the essence of $R_1-R_2$, the key overlap comes with pairs where $a=b$, since those are the pairs that clearly reflect both conditions identically (i.e., a number is always divisible by itself, and it is always a multiple of itself). There are 20 such pairs corresponding to each integer in $A$ from 1 to 20, resulting in the common elements between $R_1$ and $R_2$ (the intersection $R_1\cap R_2$) being 20 pairs.

The difference $R_1-R_2$ seeks elements present in $R_1$ but not in $R_2$. Given that $R_1$ and $R_2$ share 20 elements that are identical, to find $R_1-R_2$, we subtract these 20 common elements from the total in $R_1$, resulting in $66-20=46$ pairs.