If A = {1,2,3,4,} then minimum number of ordered pair added to make it equivalence relation on set A containing (1,3) and (1,2) is

Options

A. 8
B. 9
C. 12
D. 16

Answer

To determine the minimum number of ordered pairs that need to be added to the relation on set A = {1, 2, 3, 4} to make it an equivalence relation, we need to ensure that the relation satisfies three properties: reflexivity, symmetry, and transitivity.

### Step-by-Step Solution:

1. **Identify the given pairs**:
We have the pairs (1, 3) and (1, 2) already included in the relation.

2. **Check for Reflexivity**:
For a relation to be reflexive, every element in the set must relate to itself. Therefore, we need to add the following pairs:
- (1, 1)
- (2, 2)
- (3, 3)
- (4, 4)

Total pairs added for reflexivity = 4.

3. **Check for Symmetry**:
For the relation to be symmetric, if (a, b) is in the relation, then (b, a) must also be in the relation. Since we have (1, 3) and (1, 2), we need to add:
- (3, 1) for (1, 3)
- (2, 1) for (1, 2)

Total pairs added for symmetry = 2.

4. **Check for Transitivity**:
For the relation to be transitive, if (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation. We already have:
- (1, 2) and (2, 1) implies (1, 1) which is already added.
- (1, 3) and (3, 1) implies (1, 1) which is already added.
- We need to check pairs involving 2 and 3. Since we have (1, 2) and (1, 3), we need to add:
- (2, 3) and (3, 2) to ensure transitivity.

Total pairs added for transitivity = 2.

5. **Count all pairs added**:
- Reflexive pairs: 4
- Symmetric pairs: 2
- Transitive pairs: 2

Total pairs added = 4 + 2 + 2 = 8.

### Final Answer:
The minimum number of ordered pairs that need to be added to make the relation an equivalence relation is **8**.