Given a relation R ={ 7,8),(8,3) on the set A= {3,7,8} the least number of ordered pairs which when added to R make it an equivalence relations is

Options

A. 5
B. 6
C. 8
D. none of these

Answer

To determine the least number of ordered pairs that need to be added to the relation \( R = \{(7, 8), (8, 3)\} \) on the set \( A = \{3, 7, 8\} \) to make it an equivalence relation, we need to ensure that the relation satisfies three properties: reflexivity, symmetry, and transitivity.

### Step 1: Check for Reflexivity
An equivalence relation must be reflexive, meaning every element in the set must relate to itself. For the set \( A = \{3, 7, 8\} \), we need the pairs:
- \( (3, 3) \)
- \( (7, 7) \)
- \( (8, 8) \)

Currently, \( R \) does not include any of these pairs, so we need to add all three pairs to ensure reflexivity.

### Step 2: Check for Symmetry
A relation is symmetric if for every pair \( (a, b) \) in the relation, the pair \( (b, a) \) is also in the relation. In our current relation \( R \):
- From \( (7, 8) \), we need to add \( (8, 7) \) to satisfy symmetry.
- From \( (8, 3) \), we need to add \( (3, 8) \) to satisfy symmetry.

Thus, we need to add two pairs: \( (8, 7) \) and \( (3, 8) \).

### Step 3: Check for Transitivity
A relation is transitive if whenever \( (a, b) \) and \( (b, c) \) are in the relation, then \( (a, c) \) must also be in the relation. We need to check the pairs we currently have:
- From \( (7, 8) \) and \( (8, 3) \), we can derive \( (7, 3) \) which needs to be added to ensure transitivity.

### Summary of Pairs to Add
To summarize, we need to add the following pairs to make \( R \) an equivalence relation:
1. Reflexive pairs: \( (3, 3), (7, 7), (8, 8) \) - 3 pairs
2. Symmetric pairs: \( (8, 7), (3, 8) \) - 2 pairs
3. Transitive pair: \( (7, 3) \) - 1 pair

### Total Ordered Pairs to Add
Adding these together, we have:
- Reflexive: 3 pairs
- Symmetric: 2 pairs
- Transitive: 1 pair

However, we notice that some pairs may overlap in their requirements. The pairs \( (3, 3), (7, 7), (8, 8) \) are necessary for reflexivity, while \( (8, 7) \) and \( (3, 8) \) are necessary for symmetry, and \( (7, 3) \) is necessary for transitivity.

Thus, the least number of ordered pairs that need to be added to make \( R \) an equivalence relation is **7 pairs**.