Let a relation R be defined by
`R={(4,5),(1,4),(4,6),(7,6),(3,7)}`. Find
`R^(-1)` o R

Answer

To solve the problem of finding \( R^{-1} \circ R \) for the relation \( R = \{(4,5), (1,4), (4,6), (7,6), (3,7)\} \), we will follow these steps:

### Step 1: Find the Inverse of Relation \( R \)

The inverse of a relation \( R \), denoted as \( R^{-1} \), is obtained by swapping the elements of each ordered pair in \( R \).

Given:
\[ R = \{(4,5), (1,4), (4,6), (7,6), (3,7)\} \]

The inverse \( R^{-1} \) will be:
- For \( (4,5) \), we get \( (5,4) \)
- For \( (1,4) \), we get \( (4,1) \)
- For \( (4,6) \), we get \( (6,4) \)
- For \( (7,6) \), we get \( (6,7) \)
- For \( (3,7) \), we get \( (7,3) \)

Thus,
\[ R^{-1} = \{(5,4), (4,1), (6,4), (6,7), (7,3)\} \]

### Step 2: Find the Composition \( R^{-1} \circ R \)

The composition of two relations \( R^{-1} \) and \( R \), denoted as \( R^{-1} \circ R \), is defined as follows:
For every pair \( (a,b) \in R^{-1} \) and \( (b,c) \in R \), we include \( (a,c) \) in the composition.

Now, we will check each pair in \( R^{-1} \) and see if we can find a corresponding pair in \( R \):

1. **For \( (5,4) \)** in \( R^{-1} \):
- \( 4 \) is in \( R \) with pairs \( (4,5) \) and \( (4,6) \).
- Thus, we can form \( (5,5) \) and \( (5,6) \).

2. **For \( (4,1) \)** in \( R^{-1} \):
- \( 1 \) is in \( R \) with pair \( (1,4) \).
- Thus, we can form \( (4,4) \).

3. **For \( (6,4) \)** in \( R^{-1} \):
- \( 4 \) is in \( R \) with pairs \( (4,5) \) and \( (4,6) \).
- Thus, we can form \( (6,5) \) and \( (6,6) \).

4. **For \( (6,7) \)** in \( R^{-1} \):
- \( 7 \) is in \( R \) with pair \( (7,6) \).
- Thus, we can form \( (6,6) \).

5. **For \( (7,3) \)** in \( R^{-1} \):
- \( 3 \) is in \( R \) with pair \( (3,7) \).
- Thus, we can form \( (7,7) \).

### Step 3: Compile the Results

Now, we compile all the pairs we found:
- From \( (5,4) \): \( (5,5), (5,6) \)
- From \( (4,1) \): \( (4,4) \)
- From \( (6,4) \): \( (6,5), (6,6) \)
- From \( (6,7) \): \( (6,6) \)
- From \( (7,3) \): \( (7,7) \)

Thus, the final relation \( R^{-1} \circ R \) is:
\[ R^{-1} \circ R = \{(5,5), (5,6), (4,4), (6,5), (6,6), (7,7)\} \]

### Final Answer
The relation \( R^{-1} \circ R \) is:
\[ \{(5,5), (5,6), (4,4), (6,5), (6,6), (7,7)\} \]

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