The relation R is defined on the set of natural numbers as {(a, b) : a = 2b}. Then, `R^(-1)` is given by

Options

A. {(2,1), (4, 2), (6, 3),…}
B. {(1,2),(2,4),(3,6),….}
C. `R^(-1)` is not defined
D. None of these

Answer

To find the inverse of the relation \( R \) defined on the set of natural numbers as \( R = \{(a, b) : a = 2b\} \), we will follow these steps:

### Step 1: Understand the relation \( R \)
The relation \( R \) consists of pairs \( (a, b) \) such that \( a \) is equal to \( 2b \). This means for every natural number \( b \), there exists a corresponding \( a \).

### Step 2: Generate pairs from the relation
Let's generate some pairs from the relation:
- For \( b = 1 \): \( a = 2 \times 1 = 2 \) → Pair: \( (2, 1) \)
- For \( b = 2 \): \( a = 2 \times 2 = 4 \) → Pair: \( (4, 2) \)
- For \( b = 3 \): \( a = 2 \times 3 = 6 \) → Pair: \( (6, 3) \)
- For \( b = 4 \): \( a = 2 \times 4 = 8 \) → Pair: \( (8, 4) \)
- Continuing this, we can see that the relation \( R \) can be expressed as:
\[ R = \{(2, 1), (4, 2), (6, 3), (8, 4), \ldots\} \]

### Step 3: Define the inverse relation \( R^{-1} \)
The inverse relation \( R^{-1} \) consists of pairs \( (b, a) \) such that \( (a, b) \in R \). Thus, we will swap the elements in each pair from \( R \):
- From \( (2, 1) \) we get \( (1, 2) \)
- From \( (4, 2) \) we get \( (2, 4) \)
- From \( (6, 3) \) we get \( (3, 6) \)
- From \( (8, 4) \) we get \( (4, 8) \)

### Step 4: Write the inverse relation
The inverse relation \( R^{-1} \) can be expressed as:
\[ R^{-1} = \{(1, 2), (2, 4), (3, 6), (4, 8), \ldots\} \]

### Step 5: Determine if \( R^{-1} \) is defined
The pairs in \( R^{-1} \) indicate that for every natural number \( a \), there is a corresponding \( b \) such that \( a = 2b \). However, since \( b \) must be a natural number, and \( a \) can be any even natural number, \( R^{-1} \) is not defined for odd natural numbers.

### Conclusion
Thus, the final answer is that \( R^{-1} \) consists of pairs where the first element is any natural number \( b \) and the second element is \( 2b \), but it is not defined for all natural numbers.