Let `A={a , b , c)` and the relation R be defined on A as follows: `R={(a , a),(b , c),(a , b)}dot` Then, write minimum number of ordered pairs to be added in R to make it reflexive and transitive.

Explanation

To solve the problem, we need to ensure that the relation \( R \) defined on the set \( A = \{a, b, c\} \) is both reflexive and transitive. Let's go through the steps to determine the minimum number of ordered pairs that need to be added to \( R \).

### Step 1: Identify the current relation
The current relation \( R \) is given as:
\[ R = \{(a, a), (b, c), (a, b)\} \]

### Step 2: Check for reflexivity
A relation is reflexive if every element in the set \( A \) is related to itself. Therefore, we need to check if the pairs \( (b, b) \) and \( (c, c) \) are present in \( R \).

- Currently, we have \( (a, a) \) in \( R \).
- We need to add \( (b, b) \) and \( (c, c) \) to make it reflexive.

### Step 3: Add pairs for reflexivity
To make \( R \) reflexive, we need to add:
- \( (b, b) \)
- \( (c, c) \)

So, we have added 2 pairs so far.

### Step 4: Check for transitivity
A relation is transitive if whenever \( (x, y) \) and \( (y, z) \) are in \( R \), then \( (x, z) \) must also be in \( R \).

Currently, we have the following pairs in \( R \):
1. \( (a, a) \)
2. \( (b, c) \)
3. \( (a, b) \)

Now, let's check for transitivity:
- From \( (a, b) \) and \( (b, c) \), we can derive \( (a, c) \). This pair is not in \( R \), so we need to add \( (a, c) \).

### Step 5: Add pairs for transitivity
To make \( R \) transitive, we need to add:
- \( (a, c) \)

### Step 6: Count the total pairs added
Now, we summarize the pairs added:
1. \( (b, b) \) for reflexivity
2. \( (c, c) \) for reflexivity
3. \( (a, c) \) for transitivity

Thus, the total number of ordered pairs that need to be added to make \( R \) reflexive and transitive is:
\[ 3 \]

### Final Answer
The minimum number of ordered pairs to be added in \( R \) to make it reflexive and transitive is **3**.

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