Question

The relation \( R \) defined on the set of natural numbers as \( (a, b) : a \) differs from \( b \) by 3 is given by:

• A. \( \{(1, 4), (2, 5), (3, 6), \dots\} \)
• B. \( \{(4, 1), (5, 2), (6, 3), \dots\} \)
• C. \( \{(1, 3), (2, 5), (3, 9), \dots\} \)
• D. None of the above

Hint:

In problems involving relations, carefully consider the condition defined for the relation and verify it with the pairs given in the options.

Answer 1

The Correct Option is B

We are given the relation \( R \) defined on the set of natural numbers \( \mathbb{N} \) as: \[ (a, b) \in R \quad \text{if and only if} \quad a \text{ differs from } b \text{ by } 3. \] This means that \( a = b + 3 \) or \( b = a + 3 \), i.e., the difference between \( a \) and \( b \) is exactly 3. Step 1: Understand the condition for the relation. The relation pairs numbers where the difference between them is exactly 3. So, for example: - If \( a = 4 \), then \( b = 1 \) (since \( 4 - 1 = 3 \)). - If \( a = 5 \), then \( b = 2 \) (since \( 5 - 2 = 3 \)). - If \( a = 6 \), then \( b = 3 \) (since \( 6 - 3 = 3 \)). Step 2: Match the given options. From the given options: - Option (a): \( \{(1, 4), (2, 5), (3, 6), \dots\} \) shows pairs where \( a \) is smaller than \( b \) by 3, but the problem specifies that \( a \) is larger than \( b \), so this is incorrect. - Option (b): \( \{(4, 1), (5, 2), (6, 3), \dots\} \) correctly reflects that \( a \) differs from \( b \) by 3 in the right order (since \( 4 - 1 = 3 \), \( 5 - 2 = 3 \), and \( 6 - 3 = 3 \)). - Option (c): \( \{(1, 3), (2, 5), (3, 9), \dots\} \) does not satisfy the condition that \( a \) differs from \( b \) by exactly 3. Thus, the correct answer is (b).