Question

Let \( R \) be the relation in the set \( \mathbb{N} \) given by: \[ R = \{ (a, b) : a = b - 2, b>6 \}. \] The correct answer in the following is:

• A. \( (6, 8) \in R \)
• B. \( (2, 4) \in R \)
• C. \( (3, 8) \in R \)
• D. \( (8, 7) \in R \)

Hint:

In relations, always verify both the condition \( a = b - 2 \) and \( b>6 \) for each option.

Answer 1

The Correct Option is A

We are given the relation \( R = \{ (a, b) : a = b - 2, b>6 \} \). - For option (A), \( a = 6 \) and \( b = 8 \). Check if the condition \( a = b - 2 \) holds: \[ a = b - 2 \quad \Rightarrow \quad 6 = 8 - 2 \quad \Rightarrow \quad 6 = 6. \] Thus, \( (6, 8) \in R \), so option (A) is correct. For the other options: - For option (B), \( a = 2 \) and \( b = 4 \). Check if \( a = b - 2 \): \[ 2 = 4 - 2 \quad \Rightarrow \quad 2 = 2, \] but \( b>6 \) is not satisfied, so \( (2, 4) \notin R \). - For option (C), \( a = 3 \) and \( b = 8 \). Check if \( a = b - 2 \): \[ 3 = 8 - 2 \quad \Rightarrow \quad 3 = 6, \] which is false, so \( (3, 8) \notin R \). - For option (D), \( a = 8 \) and \( b = 7 \). Check if \( a = b - 2 \): \[ 8 = 7 - 2 \quad \Rightarrow \quad 8 = 5, \] which is false, so \( (8, 7) \notin R \). Thus, the correct answer is \( (A) \).