If A=(1,2,3), then the relation R={(1,1)(2,2),(3,1),(1,3)}` is

Options

A. reflexive.
B. symmetric
C. transitive.
D. equivalence.

Explanation

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

Given:
- Set \( A = \{1, 2, 3\} \)
- Relation \( R = \{(1, 1), (2, 2), (3, 1), (1, 3)\} \)

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

- For \( 1 \): \( (1, 1) \in R \) ✔️
- For \( 2 \): \( (2, 2) \in R \) ✔️
- For \( 3 \): \( (3, 3) \notin R \) ❌

Since \( (3, 3) \) is not in \( R \), the relation is **not reflexive**.

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

- For \( (1, 1) \): \( (1, 1) \in R \) ✔️
- For \( (2, 2) \): \( (2, 2) \in R \) ✔️
- For \( (3, 1) \): \( (1, 3) \in R \) ✔️
- For \( (1, 3) \): \( (3, 1) \notin R \) ❌

Since \( (3, 1) \) does not have its symmetric pair \( (1, 3) \) in \( R \), the relation is **not symmetric**.

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

- Consider \( (3, 1) \) and \( (1, 3) \):
- Here, \( (3, 1) \in R \) and \( (1, 3) \in R \) implies \( (3, 3) \) should be in \( R \) for transitivity.
- But \( (3, 3) \notin R \) ❌

Since \( (3, 3) \) is not in \( R \), the relation is **not transitive**.

### Conclusion
Since the relation \( R \) is neither reflexive, symmetric, nor transitive, it does not satisfy the properties of any standard relation type.

### Final Answer
The relation \( R \) is **neither reflexive, symmetric, nor transitive**.

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