Question

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

Answer 1

$n(R_1) = 20 + 10 + 6 + 5 + 4 + 3 + 3 + 2 + 2 + 2 + 1 + \cdots + 1 \quad \text{(10 times)}$

$n(R_1) = 66$

$R_1 \cap R_2 = \\{(1, 1), (2, 2), \ldots, (20, 20)\\}$

$n(R_1 \cap R_2) = 20$

$n(R_1 - R_2) = n(R_1) - n(R_1 \cap R_2)$

$= 66 - 20$

$R_1 - R_2 = 46 \text{ pairs}$