Let R be the relation in the set N given by R = `{(a,b):|a-b|` is odd}. Then:

Options

A. `(0,1) in R`
B. `(2,3) in R`
C. `(-1,4) in R`
D. `(3,7) in R`

Answer

To solve the problem, we need to analyze the relation \( R \) defined on the set of natural numbers \( \mathbb{N} \) as follows:

\[ R = \{(a, b) : |a - b| \text{ is odd} \} \]

### Step 1: Understanding the condition for the relation
The condition \( |a - b| \) being odd means that the difference between \( a \) and \( b \) must be an odd number.

### Step 2: Analyzing the parity of numbers
To have an odd difference, one of the numbers must be even and the other must be odd. This is because:
- Even - Even = Even
- Odd - Odd = Even
- Even - Odd = Odd
- Odd - Even = Odd

Thus, for \( |a - b| \) to be odd, one of \( a \) or \( b \) must be even, and the other must be odd.

### Step 3: Testing specific pairs
Let's test some pairs of natural numbers to see if they satisfy the relation:

1. **Test Pair (2, 3)**:
\[ |2 - 3| = | -1 | = 1 \quad (\text{odd}) \]
So, \( (2, 3) \in R \).

2. **Test Pair (3, 7)**:
\[ |3 - 7| = | -4 | = 4 \quad (\text{even}) \]
So, \( (3, 7) \notin R \).

### Step 4: Conclusion
From our analysis, we can conclude that the relation \( R \) consists of pairs where one number is even and the other is odd.

### Final Answer
Thus, the pairs that satisfy the relation \( R \) are those where one element is even and the other is odd.

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