Let R be a relation defined by `R={(1,3),(2,4),(5,1)}` on the set of natural number N. Then `R^(-1)` is equal to

Options

A. `{(3,1),(4,2),(1,5)}`
B. `{(5,1),(4,2),(1,3)}`
C. `{(5,1),(2,4),(1,3)}`
D. None of these

Answer

To find the inverse of the relation \( R \), we will follow these steps:

### Step 1: Understand the Relation
The relation \( R \) is given as:
\[ R = \{(1, 3), (2, 4), (5, 1)\} \]
This means that:
- 1 is related to 3
- 2 is related to 4
- 5 is related to 1

### Step 2: Define the Inverse Relation
The inverse of a relation \( R \), denoted as \( R^{-1} \), is obtained by swapping the elements in each ordered pair of \( R \). That is, if \( (a, b) \) is in \( R \), then \( (b, a) \) will be in \( R^{-1} \).

### Step 3: Apply the Definition of Inverse
Now, we will apply the definition of the inverse relation to each ordered pair in \( R \):
- From \( (1, 3) \), we get \( (3, 1) \)
- From \( (2, 4) \), we get \( (4, 2) \)
- From \( (5, 1) \), we get \( (1, 5) \)

### Step 4: Write the Inverse Relation
Combining all the pairs we obtained, we have:
\[ R^{-1} = \{(3, 1), (4, 2), (1, 5)\} \]

### Final Answer
Thus, the inverse relation \( R^{-1} \) is:
\[ R^{-1} = \{(3, 1), (4, 2), (1, 5)\} \]

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