Question

A relation \( R = \{(a, b) : a = b - 1, b \geq 3\ \) is defined on set \( N \), then}

• A. \( (2, 4) \in R \)
• B. \( (4, 5) \in R \)
• C. \( (4, 6) \in R \)
• D. \( (1, 3) \in R \)

Hint:

When checking ordered pairs in a relation, substitute the values directly into the given condition. Here, the relation requires \(a = b - 1\) and \(b \geq 3\). Always verify both parts.

Answer 1

The Correct Option is B

To determine which of the given pairs belongs to the relation \( R \), let's analyze the definition and each option one by one.

The relation \( R \) is defined as:

\( R = \{(a, b) : a = b - 1, b \geq 3\} \) on the set of natural numbers \( N \).

This means that for each pair \((a, b)\) to be in the relation \( R \), the following two conditions must be satisfied:

1. \(a = b - 1\)
2. \(b \geq 3\)

Let's evaluate each option:

• Option 1: \((2, 4)\)
  • Check: \(a = b - 1 \rightarrow 2 = 4 - 1\)
  • Condition 1: True, since \(2 = 3\).
  • Condition 2: Also True, since \(4 \geq 3\).
  • Result: The condition that \(a = b - 1\) holds for these values.
• Option 2: \((4, 5)\)
  • Check: \(a = b - 1 \rightarrow 4 = 5 - 1\)
  • Condition 1: True, since \(4 = 4\).
  • Condition 2: True, since \(5 \geq 3\).
  • Result: Meets both conditions of the relation \( R \).
• Option 3: \((4, 6)\)
  • Check: \(a = b - 1 \rightarrow 4 = 6 - 1\)
  • Condition 1: False, since \(4 \neq 5\).
  • Condition 2: True, since \(6 \geq 3\).
  • Result: Does not satisfy \(a = b - 1\).
• Option 4: \((1, 3)\)
  • Check: \(a = b - 1 \rightarrow 1 = 3 - 1\)
  • Condition 1: False, since \(1 \neq 2\).
  • Condition 2: True, since \(3 \geq 3\).
  • Result: Does not satisfy \(a = b - 1\).

From the analysis above, the only pair that satisfies both conditions is:

\((4, 5) \in R\)

This concludes that the correct answer is Option 2: \( (4, 5) \in R \).