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 of making the relation \( R = \{(2,3), (3,4)\} \) reflexive and symmetric on the set \( A = \{2, 3, 4\} \), we need to follow these steps:

### Step 1: Determine Reflexivity
A relation is reflexive if every element in the set is related to itself. For the set \( A \), we need the following pairs to be present in \( R \):
- \( (2, 2) \)
- \( (3, 3) \)
- \( (4, 4) \)

Currently, \( R \) contains none of these pairs. Therefore, we need to add all three pairs to make \( R \) reflexive.

### Step 2: Determine Symmetry
A relation is symmetric if for every pair \( (a, b) \) in the relation, the pair \( (b, a) \) is also in the relation. The current pairs in \( R \) are:
- \( (2, 3) \)
- \( (3, 4) \)

To make \( R \) symmetric, we need to add:
- \( (3, 2) \) for the pair \( (2, 3) \)
- \( (4, 3) \) for the pair \( (3, 4) \)

### Step 3: Count the Minimum Pairs to Add
Now, we summarize the pairs we need to add:
1. To make it reflexive: \( (2, 2), (3, 3), (4, 4) \) — **3 pairs**
2. To make it symmetric: \( (3, 2), (4, 3) \) — **2 pairs**

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

However, we must ensure that we do not double count any pairs. The pairs \( (3, 2) \) and \( (4, 3) \) do not overlap with the reflexive pairs, so we need to add all of them.

Thus, the total number of minimum ordered pairs to be added to \( R \) is:
\[ 3 + 2 = 5 \]

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

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