$A = \{1, 2, 3, ..., 20\}$. Let $R_1$ and $R_2$ be two relations on $A$ such that $\begin{aligned} R_1 &= \{(a, b) : b ext{ is divisible by } a\} \ R_2 &= \{(a, b) : a ext{ is an integral multiple of } b\} \end{aligned}$ Then, the number of elements in $R_1 - R_2$ is equal to __________.

Question

$A = \{1, 2, 3, ..., 20\}$. Let $R_1$ and $R_2$ be two relations on $A$ such that $\begin{aligned} R_1 &= \{(a, b) : b 	ext{ is divisible by } a\} \ R_2 &= \{(a, b) : a 	ext{ is an integral multiple of } b\} \end{aligned}$ Then, the number of elements in $R_1 - R_2$ is equal to __________.

Aicharya · Accepted Answer

$A = \{1, 2, 3, ..., 20\}$, $R_1 = \{(a, b): b 	ext{ is divisible by } a\}$ and $R_2 = \{(a, b): a 	ext{ is an integral multiple of } b\}$. $\Rightarrow R_1 = \{(1, 1), (1, 2), ..., (1, 20), (2, 2), (2, 4), ..., (2, 20), (3, 3), (3, 6), ..., (3, 18), (4, 4), (4, 8), ..., (4, 20), (5, 5), (5, 10), ..., (5, 20), (6, 6), (6, 12), ..., (6, 18), (7, 7), (7, 14), (8, 8), (8, 16), (9, 9), (9, 18), (10, 10), (10, 20), ..., (20, 20)\}$. $\Rightarrow n(R_1) = 20 + 10 + 6 + 5 + 4 + 3 + 2 + 2 + 2 + 2 + 10 = 66$. $\Rightarrow R_1 \cap R_2 = \{(1, 1), (2, 2), (3, 3), (4, 4), ..., (20, 20)\}$. $\Rightarrow n(R_1 \cap R_2) = 20$. $\Rightarrow n(R_1 - R_2) = n(R_1) - n(R_1 \cap R_2)$. $\Rightarrow n(R_1 - R_2) = 66 - 20$. $\Rightarrow n(R_1 - R_2) = 46$.

$A = \{1, 2, 3, ..., 20\}$. Let $R_1$ and $R_2$ be two relations on $A$ such that $\begin{aligned} R_1 &=…

Solution