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

Answer

To determine the properties of the relation \( R = \{(1,1), (2,2), (3,1), (1,3)\} \) with respect to the set \( A = \{1, 2, 3\} \), we will check if the relation is reflexive, symmetric, transitive, or equivalent.

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

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

Since \( (3,3) \) is not present 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 \( (b,a) \) must also be in \( R \).

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

Since \( (3,1) \) does not have its symmetric counterpart \( (1,3) \) 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 \( R \).

- Check pairs:
- \( (1,1) \) and \( (1,3) \) implies \( (1,3) \) is in \( R \) (true).
- \( (1,3) \) and \( (3,1) \) implies \( (1,1) \) is in \( R \) (true).
- \( (3,1) \) and \( (1,1) \) implies \( (3,1) \) is in \( R \) (true).

However, we need to check all combinations:
- There are no pairs \( (a,b) \) and \( (b,c) \) that lead to a contradiction.

Since all conditions are satisfied, the relation is **transitive**.

### Step 4: Check for Equivalence
A relation is equivalent if it is reflexive, symmetric, and transitive. Since we found that the relation is not reflexive and not symmetric, it cannot be equivalent.

### Conclusion
The relation \( R \) is:
- Not Reflexive
- Not Symmetric
- Transitive
- Not Equivalent