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

Correct Answer: 46

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

\[ n(R_1) = 66 \]

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

\[ n(R_1 \cap R_2) = 20 \]

\[ n(R_1 - R_2) = n(R_1) - n(R_1 \cap R_2) \]

\[ = 66 - 20 \]

\[ R_1 - R_2 = 46 \text{ pairs} \]