Let A = {7, 8, 9, 10} and R = {(8, 8), (9, 9), (10, 10), (7, 8)} be a relation on A, then R is

Options

A. Transitive only
B. Reflexive only
C. Symmetric only
D. None of these

Answer

To determine the properties of the relation \( R \) on set \( A \), we need to check if \( R \) is reflexive, symmetric, and transitive.

### Step 1: Check if \( R \) is Reflexive
A relation \( R \) is reflexive if every element \( a \) in set \( A \) is related to itself, meaning \( (a, a) \) must be in \( R \) for all \( a \in A \).

- Set \( A = \{7, 8, 9, 10\} \)
- Relation \( R = \{(8, 8), (9, 9), (10, 10), (7, 8)\} \)

Check for each element in \( A \):
- For \( 7 \): \( (7, 7) \) is not in \( R \)
- For \( 8 \): \( (8, 8) \) is in \( R \)
- For \( 9 \): \( (9, 9) \) is in \( R \)
- For \( 10 \): \( (10, 10) \) is in \( R \)

Since \( (7, 7) \) is missing, \( R \) is **not reflexive**.

### Step 2: Check if \( R \) is Symmetric
A relation \( R \) is symmetric if for every \( (a, b) \in R \), \( (b, a) \) must also be in \( R \).

- Check the pairs in \( R \):
- For \( (8, 8) \): \( (8, 8) \) is symmetric.
- For \( (9, 9) \): \( (9, 9) \) is symmetric.
- For \( (10, 10) \): \( (10, 10) \) is symmetric.
- For \( (7, 8) \): Check if \( (8, 7) \) is in \( R \). It is not.

Since \( (7, 8) \) does not have \( (8, 7) \) in \( R \), \( R \) is **not symmetric**.

### Step 3: Check if \( R \) is Transitive
A relation \( R \) is transitive if whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \) must also be in \( R \).

- Check the pairs:
- We have \( (7, 8) \) in \( R \). We need to check if there exists \( (8, c) \) in \( R \) for any \( c \). The only pair involving \( 8 \) is \( (8, 8) \), which gives us \( (7, 8) \) leading to \( (7, 8) \) (which is already in \( R \)).
- For \( (8, 8) \), it leads to itself, which is fine.
- For \( (9, 9) \) and \( (10, 10) \), they also lead to themselves.

Since there are no violations of transitivity in the existing pairs, \( R \) is **transitive**.

### Conclusion
Based on the checks:
- \( R \) is **not reflexive**.
- \( R \) is **not symmetric**.
- \( R \) is **transitive**.

Thus, the final answer is that \( R \) is **transitive**.

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