For the set `A={1,\ 2,\ 3}` , define a relation `R` on the set `A` as follows: `R={(1,\ 1),\ (2,\ 2),\ (3,\ 3),\ (1,\ 3)}` Write the ordered pairs to be added to `R` to make the smallest equivalence relation.

Explanation

To convert the relation \( R = \{(1, 1), (2, 2), (3, 3), (1, 3)\} \) into the smallest equivalence relation on the set \( A = \{1, 2, 3\} \), we need to ensure that the relation satisfies three properties: reflexivity, symmetry, and transitivity.

### Step 1: Check Reflexivity
A relation is reflexive if every element in the set is related to itself. For our set \( A \):

- We have \( (1, 1) \), \( (2, 2) \), and \( (3, 3) \) already included in \( R \).
- Therefore, \( R \) is reflexive.

**Hint:** Ensure that every element in the set has a pair that relates it to itself.

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

- We have \( (1, 3) \) in \( R \), but we do not have \( (3, 1) \).
- To make the relation symmetric, we need to add \( (3, 1) \).

**Hint:** For each ordered pair, check if the reverse pair is also present.

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

- After adding \( (3, 1) \), we need to check for transitivity:
- From \( (1, 3) \) and \( (3, 1) \), we can derive \( (1, 1) \) which is already in \( R \).
- From \( (3, 1) \) and \( (1, 3) \), we can derive \( (3, 3) \) which is also already in \( R \).
- We also need to check combinations with \( (2, 2) \) which does not affect transitivity with other pairs since it relates only to itself.

Thus, after adding \( (3, 1) \), the relation satisfies transitivity.

### Conclusion
The smallest equivalence relation that can be formed from \( R \) is:

\[ R = \{(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)\} \]

### Ordered Pair to be Added
The ordered pair to be added to \( R \) is \( (3, 1) \).