Let `R={(a,a),(b,c),(a,b)}` be a relation on a set `A={a,b,c}`. Then the minimum number of ordered pairs which when added to R make it an equivalence realtion are ...

Answer

To determine the minimum number of ordered pairs that need to be added to the relation \( R = \{(a,a), (b,c), (a,b)\} \) on the set \( A = \{a,b,c\} \) 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
A relation is reflexive if every element in the set is related to itself. For the set \( A = \{a, b, c\} \), we need the pairs \( (a,a) \), \( (b,b) \), and \( (c,c) \).

- We already have \( (a,a) \) in \( R \).
- We need to add \( (b,b) \) and \( (c,c) \).

So, we need to add 2 pairs for reflexivity.

### Step 2: Check for Symmetry
A relation is symmetric if for every pair \( (x,y) \) in the relation, the pair \( (y,x) \) is also in the relation.

- We have \( (a,b) \) in \( R \), so we need to add \( (b,a) \).
- We have \( (b,c) \) in \( R \), so we need to add \( (c,b) \).

Thus, we need to add 2 pairs for symmetry.

### Step 3: Check for Transitivity
A relation is transitive if whenever \( (x,y) \) and \( (y,z) \) are in the relation, then \( (x,z) \) must also be in the relation.

After adding the pairs for reflexivity and symmetry, we will check for transitivity:

1. After adding \( (b,b) \) and \( (c,c) \), our relation becomes:
\[ R = \{(a,a), (b,b), (c,c), (a,b), (b,a), (b,c), (c,b)\} \]

2. Now we check:
- From \( (a,b) \) and \( (b,c) \), we need \( (a,c) \).
- From \( (b,a) \) and \( (a,b) \), we need \( (b,b) \) (already included).
- From \( (b,c) \) and \( (c,b) \), we need \( (b,b) \) (already included).
- From \( (c,b) \) and \( (b,a) \), we need \( (c,a) \).
- From \( (c,a) \) and \( (a,b) \), we need \( (c,b) \) (already included).

Thus, we need to add \( (a,c) \) and \( (c,a) \) to ensure transitivity.

### Summary of Ordered Pairs to Add
- For Reflexivity: \( (b,b) \), \( (c,c) \) → **2 pairs**
- For Symmetry: \( (b,a) \), \( (c,b) \) → **2 pairs**
- For Transitivity: \( (a,c) \), \( (c,a) \) → **2 pairs**

### Total Pairs to Add
In total, we need to add \( 2 + 2 + 2 = 6 \) pairs.

Thus, the minimum number of ordered pairs which when added to \( R \) make it an equivalence relation is **6**.