Let `A=(1,2,3,4}`, and let `R={(2,2),(3,3),(4,4),(1,2)}` be a relation on A. Then R is

Options

A. reflexive
B. symmetric
C. transitive
D. none of these

Answer

To determine the properties of the relation \( R \) on the set \( A = \{1, 2, 3, 4\} \), we need to check whether \( R \) is reflexive, symmetric, and transitive.

The relation \( R \) is given as:
\[ R = \{(2, 2), (3, 3), (4, 4), (1, 2)\} \]

### Step 1: Check for Reflexivity
A relation \( R \) is reflexive if for every element \( a \in A \), the pair \( (a, a) \) is in \( R \).

- For \( 1 \): \( (1, 1) \) is not in \( R \)
- For \( 2 \): \( (2, 2) \) is in \( R \)
- For \( 3 \): \( (3, 3) \) is in \( R \)
- For \( 4 \): \( (4, 4) \) is in \( R \)

Since \( (1, 1) \) is not in \( R \), the relation is **not reflexive**.

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

- For \( (2, 2) \): \( (2, 2) \) is in \( R \) (symmetric)
- For \( (3, 3) \): \( (3, 3) \) is in \( R \) (symmetric)
- For \( (4, 4) \): \( (4, 4) \) is in \( R \) (symmetric)
- For \( (1, 2) \): \( (2, 1) \) is not in \( R \)

Since \( (1, 2) \) is in \( R \) but \( (2, 1) \) is not, the relation is **not symmetric**.

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

- We have \( (1, 2) \) and \( (2, 2) \) in \( R \). Since \( (1, 2) \) and \( (2, 2) \) are in \( R \), we check if \( (1, 2) \) is in \( R \) (it is).
- For \( (2, 2) \) and \( (2, 2) \): \( (2, 2) \) is in \( R \).
- For \( (3, 3) \) and \( (3, 3) \): \( (3, 3) \) is in \( R \).
- For \( (4, 4) \) and \( (4, 4) \): \( (4, 4) \) is in \( R \).

Since all necessary pairs are satisfied, the relation is **transitive**.

### Conclusion
The relation \( R \) is:
- Not reflexive
- Not symmetric
- Transitive

Thus, the answer is that \( R \) is transitive but not reflexive or symmetric.