Given the relation `R= {(2,3),(3,4)}` on the set `A = {2, 3, 4}`. The number of minimum ordered pasirs to be added to R so that R is reflexive and symmetric

Options

A. `4`
B. `6`
C. `7`
D. `5`

Explanation

To solve the problem, we need to make the relation \( R \) reflexive and symmetric. Let's go through the steps systematically.

### Step 1: Identify the given relation and the set
We have the relation \( R = \{(2,3), (3,4)\} \) on the set \( A = \{2, 3, 4\} \).

### Step 2: Determine the reflexive pairs
A relation is reflexive if every element in the set \( A \) is related to itself. Therefore, we need to add the following pairs to \( R \):
- \( (2,2) \)
- \( (3,3) \)
- \( (4,4) \)

### Step 3: Add reflexive pairs to the relation
After adding the reflexive pairs, the updated relation becomes:
\[ R = \{(2,3), (3,4), (2,2), (3,3), (4,4)\} \]

### Step 4: Determine the symmetric pairs
A relation is symmetric if for every pair \( (a,b) \) in \( R \), the pair \( (b,a) \) is also in \( R \). We need to check the existing pairs:
- For \( (2,3) \), we need to add \( (3,2) \).
- For \( (3,4) \), we need to add \( (4,3) \).

### Step 5: Add symmetric pairs to the relation
After adding the symmetric pairs, the updated relation becomes:
\[ R = \{(2,3), (3,4), (2,2), (3,3), (4,4), (3,2), (4,3)\} \]

### Step 6: Count the total number of ordered pairs
Now, we count the total number of ordered pairs in the updated relation:
- \( (2,2) \)
- \( (3,3) \)
- \( (4,4) \)
- \( (2,3) \)
- \( (3,2) \)
- \( (3,4) \)
- \( (4,3) \)

Thus, the total number of ordered pairs is 7.

### Step 7: Determine the number of pairs added
Initially, we had 2 pairs in \( R \). We added 3 reflexive pairs and 2 symmetric pairs, totaling 5 pairs added.

### Final Answer
The number of minimum ordered pairs to be added to \( R \) so that \( R \) is reflexive and symmetric is **5**.

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