Let `S = {1, 2, 3,……..20}` be a given set. Relation `R_1` is defined on S as `R_1 = {(x, y) : 2x – 3y = 2}` and relation `R_2` is defined on S as `R_2 = {(x, y): 4x = 5y}`. If m denotes the number of elements required to add to make `R_1` symmetric and n denotes the number of elements required to add to make `R_2` symmetric, then value of `m + n` is

Options

A. 12
B. 4
C. 10
D. 6

Answer

To solve the problem, we need to analyze the relations \( R_1 \) and \( R_2 \) defined on the set \( S = \{1, 2, 3, \ldots, 20\} \) and determine how many elements need to be added to make each relation symmetric.

### Step 1: Analyze Relation \( R_1 \)

The relation \( R_1 \) is defined as:
\[ R_1 = \{(x, y) : 2x - 3y = 2\} \]
Rearranging this gives:
\[ 2x = 3y + 2 \implies x = \frac{3y + 2}{2} \]
For \( x \) to be an integer, \( 3y + 2 \) must be even. This implies that \( y \) must be even (since \( 3y \) is odd when \( y \) is odd, making \( 3y + 2 \) odd). The even values of \( y \) in set \( S \) are \( 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 \).

Now we will calculate the corresponding \( x \) values for these \( y \) values:
- For \( y = 2 \): \( x = \frac{3(2) + 2}{2} = 5 \)
- For \( y = 4 \): \( x = \frac{3(4) + 2}{2} = 8 \)
- For \( y = 6 \): \( x = \frac{3(6) + 2}{2} = 11 \)
- For \( y = 8 \): \( x = \frac{3(8) + 2}{2} = 14 \)
- For \( y = 10 \): \( x = \frac{3(10) + 2}{2} = 17 \)
- For \( y = 12 \): \( x = \frac{3(12) + 2}{2} = 20 \)

Thus, the pairs in \( R_1 \) are:
\[ R_1 = \{(5, 2), (8, 4), (11, 6), (14, 8), (17, 10), (20, 12)\} \]

### Step 2: Check for Symmetry in \( R_1 \)

For \( R_1 \) to be symmetric, if \( (x, y) \) is in \( R_1 \), then \( (y, x) \) must also be in \( R_1 \). We need to check which pairs are missing their symmetric counterparts:
- \( (5, 2) \) is missing \( (2, 5) \)
- \( (8, 4) \) is missing \( (4, 8) \)
- \( (11, 6) \) is missing \( (6, 11) \)
- \( (14, 8) \) is missing \( (8, 14) \)
- \( (17, 10) \) is missing \( (10, 17) \)
- \( (20, 12) \) is missing \( (12, 20) \)

Thus, we need to add 6 pairs to make \( R_1 \) symmetric.

### Step 3: Analyze Relation \( R_2 \)

The relation \( R_2 \) is defined as:
\[ R_2 = \{(x, y) : 4x = 5y\} \]
Rearranging gives:
\[ y = \frac{4}{5}x \]
For \( y \) to be an integer, \( x \) must be a multiple of 5. The multiples of 5 in set \( S \) are \( 5, 10, 15, 20 \).

Now we will calculate the corresponding \( y \) values for these \( x \) values:
- For \( x = 5 \): \( y = \frac{4}{5}(5) = 4 \)
- For \( x = 10 \): \( y = \frac{4}{5}(10) = 8 \)
- For \( x = 15 \): \( y = \frac{4}{5}(15) = 12 \)
- For \( x = 20 \): \( y = \frac{4}{5}(20) = 16 \)

Thus, the pairs in \( R_2 \) are:
\[ R_2 = \{(5, 4), (10, 8), (15, 12), (20, 16)\} \]

### Step 4: Check for Symmetry in \( R_2 \)

Similarly, we check for missing symmetric pairs:
- \( (5, 4) \) is missing \( (4, 5) \)
- \( (10, 8) \) is missing \( (8, 10) \)
- \( (15, 12) \) is missing \( (12, 15) \)
- \( (20, 16) \) is missing \( (16, 20) \)

Thus, we need to add 4 pairs to make \( R_2 \) symmetric.

### Step 5: Calculate \( m + n \)

Now, we can calculate \( m + n \):
\[ m = 6 \quad \text{(for } R_1\text{)} \]
\[ n = 4 \quad \text{(for } R_2\text{)} \]
\[ m + n = 6 + 4 = 10 \]

### Final Answer
The value of \( m + n \) is \( \boxed{10} \).