QJEEM24Ma4eb7: JEE Main 2024 Mathematics - Class 11 JEEM24Ma4eb7 Solution: Let . Let and two relation on such that is divisible by is an…

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

$R_1$ includes pairs $(a, b)$ where $b$ is divisible by $a$ . This includes pairs like $(1,1), (1,2), \ldots, (1,20)$ 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.