Let R = {(1, 2), (2, 3)} be a relation defined on set {1, 2, 3}. The minimum number of ordered pairs required to be added in R, such that enlarged relation becomes an equivalence relation is

Options

A. 3
B. 5
C. 7
D. 9

Answer

To determine the minimum number of ordered pairs required to be added to the relation \( R = \{(1, 2), (2, 3)\} \) so that it becomes an equivalence relation on the set \( \{1, 2, 3\} \), we need to ensure that the enlarged relation satisfies the properties of reflexivity, symmetry, and transitivity.

### Step-by-Step Solution:

1. **Identify the Current Relation**:
The current relation \( R \) is given as:
\[ R = \{(1, 2), (2, 3)\} \]

2. **Check for Reflexivity**:
An equivalence relation must be reflexive, meaning every element must relate to itself. For the set \( \{1, 2, 3\} \), we need to add:
\[ (1, 1), (2, 2), (3, 3) \]
So, we add 3 pairs for reflexivity.

3. **Check for Symmetry**:
The relation must also be symmetric. This means that if \( (a, b) \) is in \( R \), then \( (b, a) \) must also be in \( R \).
- From \( (1, 2) \), we need to add \( (2, 1) \).
- From \( (2, 3) \), we need to add \( (3, 2) \).

Thus, we need to add 2 more pairs for symmetry.

4. **Check for Transitivity**:
The relation must be transitive. This means if \( (a, b) \) and \( (b, c) \) are in \( R \), then \( (a, c) \) must also be in \( R \).
- We have \( (1, 2) \) and \( (2, 3) \), so we need to add \( (1, 3) \).
- We also have \( (2, 1) \) and \( (1, 2) \), which requires adding \( (2, 2) \) (already added).
- We have \( (3, 2) \) and \( (2, 1) \), which requires adding \( (3, 1) \).

5. **Count the Total Ordered Pairs Added**:
Now, let's summarize the pairs added:
- Reflexive pairs: \( (1, 1), (2, 2), (3, 3) \) → 3 pairs
- Symmetric pairs: \( (2, 1), (3, 2) \) → 2 pairs
- Transitive pairs: \( (1, 3), (3, 1) \) → 2 pairs

Total pairs added:
\[ 3 + 2 + 2 = 7 \]

6. **Final Count**:
The minimum number of ordered pairs required to be added to \( R \) to make it an equivalence relation is **7**.

### Answer:
The minimum number of ordered pairs required to be added in \( R \) is **7**.