If R is a relation on the set `A={1,2,3}` given by `R={(1,1),(2,2),(3,3),(1,2)(2,3),(1,3)}`, then R is

Options

A. reflexive but not symmetric
B. reflexive but not transitive
C. symmetric and transitive
D. neither symmetric nor transitive

Answer

To determine the properties of the relation \( R \) on the set \( A = \{1, 2, 3\} \) given by \( R = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\} \), we will check if the relation is reflexive, symmetric, and transitive.

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

- For \( a = 1 \): \( (1, 1) \in R \)
- For \( a = 2 \): \( (2, 2) \in R \)
- For \( a = 3 \): \( (3, 3) \in R \)

Since all elements of \( A \) are related to themselves, \( R \) is **reflexive**.

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

- Check pairs in \( R \):
- For \( (1, 2) \): \( (2, 1) \notin R \)
- For \( (2, 3) \): \( (3, 2) \notin R \)
- For \( (1, 3) \): \( (3, 1) \notin R \)

Since there are pairs where the reverse is not present, \( R \) 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 \( R \).

- Check the pairs:
- From \( (1, 2) \) and \( (2, 3) \), we should have \( (1, 3) \in R \) (which is true).
- From \( (1, 3) \) and \( (3, 3) \), we should have \( (1, 3) \in R \) (which is true).
- From \( (2, 2) \) and \( (2, 3) \), we should have \( (2, 3) \in R \) (which is true).

Since all necessary conditions for transitivity are satisfied, \( R \) is **transitive**.

### Conclusion
The relation \( R \) is **reflexive** and **transitive**, but **not symmetric**.

### Final Answer
Thus, the relation \( R \) is reflexive but not symmetric.

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