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

Answer

To find the composition of the relation \( R \) with itself, denoted as \( R \circ R \), we need to follow these steps:

### Step 1: Understand the Relation
The relation \( R \) is given as:
\[ R = \{(4, 5), (1, 4), (4, 6), (7, 6), (3, 7)\} \]

### Step 2: Identify the Elements
We need to find pairs \( (a, c) \) such that there exists an element \( b \) where \( (a, b) \in R \) and \( (b, c) \in R \).

### Step 3: List the Pairs
We will check each pair in \( R \) to see if we can form new pairs by finding a common element \( b \).

1. **For \( (4, 5) \)**:
- \( b = 5 \) does not appear as the first element in any pair in \( R \).

2. **For \( (1, 4) \)**:
- \( b = 4 \) appears in \( (4, 5) \) and \( (4, 6) \).
- From \( (1, 4) \) and \( (4, 5) \), we get \( (1, 5) \).
- From \( (1, 4) \) and \( (4, 6) \), we get \( (1, 6) \).

3. **For \( (4, 6) \)**:
- \( b = 6 \) does not appear as the first element in any pair in \( R \).

4. **For \( (7, 6) \)**:
- \( b = 6 \) does not appear as the first element in any pair in \( R \).

5. **For \( (3, 7) \)**:
- \( b = 7 \) appears in \( (7, 6) \).
- From \( (3, 7) \) and \( (7, 6) \), we get \( (3, 6) \).

### Step 4: Compile the Results
From the above checks, we have found the following pairs:
- From \( (1, 4) \) and \( (4, 5) \): \( (1, 5) \)
- From \( (1, 4) \) and \( (4, 6) \): \( (1, 6) \)
- From \( (3, 7) \) and \( (7, 6) \): \( (3, 6) \)

Thus, the composition \( R \circ R \) is:
\[ R \circ R = \{(1, 5), (1, 6), (3, 6)\} \]

### Final Answer
The relation \( R \circ R \) is:
\[ R \circ R = \{(1, 5), (1, 6), (3, 6)\} \]