Let N denotes the set of all natural numbers. Define two binary relations on N as:
`R_(1) ={(x,y) in N xx N : 2x + y = 10}`
`R_(2) = {(x,y) in N xx N: x + 2y = 10}`, Then

Options

A. range of `R_(1)` is {2,4,8}
B. range of `R_(2)` is {1,2,3,4}
C. both `R_(1)` and `R_(2)` are symmetric relations
D. both `R_(1)` and `R_(2)` are transitive relations

Answer

To solve the problem, we need to analyze the two binary relations \( R_1 \) and \( R_2 \) defined on the set of natural numbers \( N \).

### Step 1: Analyze the relation \( R_1 \)

The relation \( R_1 \) is defined as:
\[ R_1 = \{(x,y) \in N \times N : 2x + y = 10\} \]

To find the pairs \((x, y)\) that satisfy this equation, we can express \( y \) in terms of \( x \):
\[ y = 10 - 2x \]

### Step 2: Determine the valid values of \( x \)

Since \( x \) and \( y \) must be natural numbers, we need to find the values of \( x \) such that \( y \) remains a natural number:
- \( 10 - 2x > 0 \)
- This implies \( 2x < 10 \) or \( x < 5 \)

Thus, the possible values for \( x \) are \( 1, 2, 3, 4 \).

### Step 3: Calculate corresponding values of \( y \)

Now we can calculate \( y \) for each valid \( x \):
- For \( x = 1 \): \( y = 10 - 2(1) = 8 \) → Pair: \( (1, 8) \)
- For \( x = 2 \): \( y = 10 - 2(2) = 6 \) → Pair: \( (2, 6) \)
- For \( x = 3 \): \( y = 10 - 2(3) = 4 \) → Pair: \( (3, 4) \)
- For \( x = 4 \): \( y = 10 - 2(4) = 2 \) → Pair: \( (4, 2) \)

### Step 4: List the pairs in \( R_1 \)

Thus, the relation \( R_1 \) can be expressed as:
\[ R_1 = \{(1, 8), (2, 6), (3, 4), (4, 2)\} \]

### Step 5: Identify the range of \( R_1 \)

The range of \( R_1 \) is the set of all second elements in the pairs:
\[ \text{Range of } R_1 = \{2, 4, 6, 8\} \]

### Step 6: Analyze the relation \( R_2 \)

The relation \( R_2 \) is defined as:
\[ R_2 = \{(x,y) \in N \times N : x + 2y = 10\} \]

We can express \( y \) in terms of \( x \):
\[ y = \frac{10 - x}{2} \]

### Step 7: Determine the valid values of \( x \)

For \( y \) to be a natural number:
- \( 10 - x \) must be even, which implies \( x \) must be even.
- The possible even values for \( x \) are \( 0, 2, 4, 6, 8, 10 \), but since \( x \) must be a natural number, we consider \( 2, 4, 6, 8, 10 \).

### Step 8: Calculate corresponding values of \( y \)

Now we can calculate \( y \) for each valid \( x \):
- For \( x = 2 \): \( y = \frac{10 - 2}{2} = 4 \) → Pair: \( (2, 4) \)
- For \( x = 4 \): \( y = \frac{10 - 4}{2} = 3 \) → Pair: \( (4, 3) \)
- For \( x = 6 \): \( y = \frac{10 - 6}{2} = 2 \) → Pair: \( (6, 2) \)
- For \( x = 8 \): \( y = \frac{10 - 8}{2} = 1 \) → Pair: \( (8, 1) \)
- For \( x = 10 \): \( y = \frac{10 - 10}{2} = 0 \) → Not a natural number.

### Step 9: List the pairs in \( R_2 \)

Thus, the relation \( R_2 \) can be expressed as:
\[ R_2 = \{(2, 4), (4, 3), (6, 2), (8, 1)\} \]

### Step 10: Identify the range of \( R_2 \)

The range of \( R_2 \) is the set of all second elements in the pairs:
\[ \text{Range of } R_2 = \{1, 2, 3, 4\} \]

### Conclusion

- The range of \( R_1 \) is \( \{2, 4, 6, 8\} \).
- The range of \( R_2 \) is \( \{1, 2, 3, 4\} \).