The relation R is defines 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 the above

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
The relation \( R \) consists of ordered pairs \( (a, b) \) where \( a \) is twice \( b \). This means for every natural number \( b \), we can find \( a \) by the formula \( a = 2b \).

### Step 2: Generate Ordered Pairs
Let's generate some ordered pairs for the relation \( R \):
- If \( b = 1 \), then \( a = 2 \times 1 = 2 \) → The pair is \( (2, 1) \).
- If \( b = 2 \), then \( a = 2 \times 2 = 4 \) → The pair is \( (4, 2) \).
- If \( b = 3 \), then \( a = 2 \times 3 = 6 \) → The pair is \( (6, 3) \).

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

### Step 3: Find the Inverse Relation
The inverse relation \( R^{-1} \) is obtained by swapping the elements in each ordered pair of \( R \). Therefore, we will have:
- From \( (2, 1) \) we get \( (1, 2) \).
- From \( (4, 2) \) we get \( (2, 4) \).
- From \( (6, 3) \) we get \( (3, 6) \).

Thus, the inverse relation \( R^{-1} \) is:
\[ R^{-1} = \{(1, 2), (2, 4), (3, 6)\} \]

### Step 4: Conclusion
The final answer for the inverse relation \( R^{-1} \) is:
\[ R^{-1} = \{(1, 2), (2, 4), (3, 6)\} \]

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