Let `A= {1, 2, 3, . . 20}`. Let `R_1` and `R_2` two relation on A such that
`R_1` = {(a, b) : b is divisible by a}
`R_2` = {(a, b) : a is an integral multiple of b}.
Then, number of elements in `R_1 – R_2` is equal to ________.

Answer

To solve the problem, we need to find the number of elements in the relation \( R_1 - R_2 \) where:

- \( R_1 = \{(a, b) : b \text{ is divisible by } a\} \)
- \( R_2 = \{(a, b) : a \text{ is an integral multiple of } b\} \)

### Step 1: Determine the elements of \( R_1 \)

We will find all pairs \( (a, b) \) such that \( b \) is divisible by \( a \) for \( a, b \in A = \{1, 2, 3, \ldots, 20\} \).

- For \( a = 1 \): \( b \) can be \( 1, 2, 3, \ldots, 20 \) (20 pairs)
- For \( a = 2 \): \( b \) can be \( 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \) (10 pairs)
- For \( a = 3 \): \( b \) can be \( 3, 6, 9, 12, 15, 18 \) (6 pairs)
- For \( a = 4 \): \( b \) can be \( 4, 8, 12, 16, 20 \) (5 pairs)
- For \( a = 5 \): \( b \) can be \( 5, 10, 15, 20 \) (4 pairs)
- For \( a = 6 \): \( b \) can be \( 6, 12, 18 \) (3 pairs)
- For \( a = 7 \): \( b \) can be \( 7, 14 \) (2 pairs)
- For \( a = 8 \): \( b \) can be \( 8, 16 \) (2 pairs)
- For \( a = 9 \): \( b \) can be \( 9 \) (1 pair)
- For \( a = 10 \): \( b \) can be \( 10 \) (1 pair)
- For \( a = 11 \): \( b \) can be \( 11 \) (1 pair)
- For \( a = 12 \): \( b \) can be \( 12 \) (1 pair)
- For \( a = 13 \): \( b \) can be \( 13 \) (1 pair)
- For \( a = 14 \): \( b \) can be \( 14 \) (1 pair)
- For \( a = 15 \): \( b \) can be \( 15 \) (1 pair)
- For \( a = 16 \): \( b \) can be \( 16 \) (1 pair)
- For \( a = 17 \): \( b \) can be \( 17 \) (1 pair)
- For \( a = 18 \): \( b \) can be \( 18 \) (1 pair)
- For \( a = 19 \): \( b \) can be \( 19 \) (1 pair)
- For \( a = 20 \): \( b \) can be \( 20 \) (1 pair)

Now, we sum these pairs:

\[ 20 + 10 + 6 + 5 + 4 + 3 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 66 \]

Thus, the number of elements in \( R_1 \) is \( |R_1| = 66 \).

### Step 2: Determine the elements of \( R_2 \)

Next, we find all pairs \( (a, b) \) such that \( a \) is an integral multiple of \( b \).

- For \( b = 1 \): \( a \) can be \( 1, 2, 3, \ldots, 20 \) (20 pairs)
- For \( b = 2 \): \( a \) can be \( 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \) (10 pairs)
- For \( b = 3 \): \( a \) can be \( 3, 6, 9, 12, 15, 18 \) (6 pairs)
- For \( b = 4 \): \( a \) can be \( 4, 8, 12, 16, 20 \) (5 pairs)
- For \( b = 5 \): \( a \) can be \( 5, 10, 15, 20 \) (4 pairs)
- For \( b = 6 \): \( a \) can be \( 6, 12, 18 \) (3 pairs)
- For \( b = 7 \): \( a \) can be \( 7, 14 \) (2 pairs)
- For \( b = 8 \): \( a \) can be \( 8, 16 \) (2 pairs)
- For \( b = 9 \): \( a \) can be \( 9 \) (1 pair)
- For \( b = 10 \): \( a \) can be \( 10 \) (1 pair)
- For \( b = 11 \): \( a \) can be \( 11 \) (1 pair)
- For \( b = 12 \): \( a \) can be \( 12 \) (1 pair)
- For \( b = 13 \): \( a \) can be \( 13 \) (1 pair)
- For \( b = 14 \): \( a \) can be \( 14 \) (1 pair)
- For \( b = 15 \): \( a \) can be \( 15 \) (1 pair)
- For \( b = 16 \): \( a \) can be \( 16 \) (1 pair)
- For \( b = 17 \): \( a \) can be \( 17 \) (1 pair)
- For \( b = 18 \): \( a \) can be \( 18 \) (1 pair)
- For \( b = 19 \): \( a \) can be \( 19 \) (1 pair)
- For \( b = 20 \): \( a \) can be \( 20 \) (1 pair)

Now, we sum these pairs:

\[ 20 + 10 + 6 + 5 + 4 + 3 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 66 \]

Thus, the number of elements in \( R_2 \) is \( |R_2| = 66 \).

### Step 3: Determine the intersection \( R_1 \cap R_2 \)

The intersection \( R_1 \cap R_2 \) consists of pairs \( (a, b) \) such that both conditions hold. This occurs when \( a = b \).

Thus, the pairs in \( R_1 \cap R_2 \) are:

\[ (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (7, 7), (8, 8), (9, 9), (10, 10), (11, 11), (12, 12), (13, 13), (14, 14), (15, 15), (16, 16), (17, 17), (18, 18), (19, 19), (20, 20) \]

This gives us \( |R_1 \cap R_2| = 20 \).

### Step 4: Calculate \( |R_1 - R_2| \)

Using the formula:

\[ |R_1 - R_2| = |R_1| - |R_1 \cap R_2| \]

Substituting the values we found:

\[ |R_1 - R_2| = 66 - 20 = 46 \]

Thus, the number of elements in \( R_1 - R_2 \) is \( \boxed{46} \).