Let a = {a, b, c} and R = {(a, a), (b, b), (c, c), (b, c), (a, b)} be a relation on A, then the

Options

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

Answer

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

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if every element in the set \( A \) is related to itself. This means that for every \( x \in A \), the pair \( (x, x) \) must be in \( R \).

- Elements in \( A \): \( a, b, c \)
- Required pairs for reflexivity: \( (a, a), (b, b), (c, c) \)
- Given pairs in \( R \): \( (a, a), (b, b), (c, c), (b, c), (a, b) \)

Since all required pairs \( (a, a), (b, b), (c, c) \) are present in \( R \), the relation is **reflexive**.

### Step 2: Check for Symmetry
A relation \( R \) is symmetric if for every pair \( (x, y) \in R \), the pair \( (y, x) \) is also in \( R \).

- Given pairs in \( R \): \( (a, a), (b, b), (c, c), (b, c), (a, b) \)
- Check each pair:
- \( (a, a) \) → \( (a, a) \) (symmetric)
- \( (b, b) \) → \( (b, b) \) (symmetric)
- \( (c, c) \) → \( (c, c) \) (symmetric)
- \( (b, c) \) → \( (c, b) \) (not in \( R \))
- \( (a, b) \) → \( (b, a) \) (not in \( R \))

Since \( (b, c) \) does not have \( (c, b) \) and \( (a, b) \) does not have \( (b, a) \) in \( R \), the relation is **not symmetric**.

### Step 3: Check for Transitivity
A relation \( R \) is transitive if whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \) must also be in \( R \).

- Given pairs in \( R \): \( (a, a), (b, b), (c, c), (b, c), (a, b) \)
- Check transitivity:
- From \( (a, b) \) and \( (b, c) \), we need \( (a, c) \) → **not in \( R \)**
- From \( (b, b) \) and \( (b, c) \), we need \( (b, c) \) → **in \( R \)**
- From \( (a, a) \) and \( (a, b) \), we need \( (a, b) \) → **in \( R \)**

Since \( (a, c) \) is not present, the relation is **not transitive**.

### Conclusion
The relation \( R \) is:
- Reflexive: Yes
- Symmetric: No
- Transitive: No

### Summary of Properties
- **Reflexive**: Yes
- **Symmetric**: No
- **Transitive**: No