The minimum number of elements that must be added to the relation `R={(a,b),(b,c)}` on the set `{a,b,c}` so that it becomes symmetric and transitive is :

Options

A. `3`
B. `7`
C. `4`
D. `5`

Answer

To solve the problem, we need to determine the minimum number of elements that must be added to the relation \( R = \{(a,b), (b,c)\} \) on the set \( \{a,b,c\} \) so that it becomes both symmetric and transitive.

### Step-by-Step Solution:

1. **Understanding the Relation**:
- The given relation \( R \) consists of two pairs: \( (a,b) \) and \( (b,c) \).
- The set of elements involved is \( \{a, b, c\} \).

2. **Making the Relation Symmetric**:
- A relation is symmetric if for every pair \( (x,y) \) in the relation, the pair \( (y,x) \) is also in the relation.
- From \( R \):
- For \( (a,b) \), we need to add \( (b,a) \).
- For \( (b,c) \), we need to add \( (c,b) \).
- After adding these pairs, the relation becomes:
\[ R = \{(a,b), (b,c), (b,a), (c,b)\} \]

3. **Making the Relation Transitive**:
- A relation is transitive if whenever \( (x,y) \) and \( (y,z) \) are in the relation, then \( (x,z) \) must also be in the relation.
- We currently have \( (a,b) \) and \( (b,c) \). By transitivity, we need to add \( (a,c) \).
- We also have \( (b,a) \) and \( (a,b) \) which requires \( (b,b) \) to maintain transitivity.
- Lastly, we have \( (c,b) \) and \( (b,c) \) which requires \( (c,c) \) for transitivity.

4. **Final Relation**:
- After adding all necessary pairs for symmetry and transitivity, the complete relation is:
\[ R = \{(a,b), (b,c), (b,a), (c,b), (a,c), (b,b), (c,c)\} \]
- The total number of pairs in this relation is 7.

5. **Calculating the Minimum Number of Elements to Add**:
- Initially, we had 2 pairs in \( R \).
- To achieve 7 pairs, we need to add:
\[ 7 - 2 = 5 \text{ pairs} \]

### Conclusion:
The minimum number of elements that must be added to the relation \( R \) so that it becomes symmetric and transitive is **5**.