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

We need to determine which ordered pair \((a, b)\) satisfies the given condition for the relation \(R\) defined on the set of natural numbers \(N\).

Step 1: Understand the definition of the relation.
The relation is defined as:
\[ R = \{(a, b) : a = b - 1, \ b \geq 3\} \]
This means:
- \(a\) and \(b\) are natural numbers (\(N\) typically means positive integers \(1, 2, 3, \ldots\))
- The condition \(a = b - 1\) must be satisfied
- Additionally, \(b \geq 3\)

Step 2: Rearrange the condition.
From \(a = b - 1\), we can write \(b = a + 1\).
Also, \(b \geq 3\) means \(a + 1 \geq 3\) or \(a \geq 2\).
So, the relation consists of pairs \((a, b)\) where:
- \(a\) and \(b\) are natural numbers
- \(a \geq 2\)
- \(b = a + 1\) and \(b \geq 3\) (which is automatically satisfied if \(a \geq 2\))

Step 3: Check each option.
- (A) \( (2, 4) \in R \)
Here, \(a = 2\), \(b = 4\).
Check \(a = b - 1\): \(4 - 1 = 3\), but \(a = 2\), so \(2 \neq 3\).
Therefore, \((2, 4)\) does not satisfy the condition. Incorrect.
- (B) \( (4, 5) \in R \)
Here, \(a = 4\), \(b = 5\).
Check \(a = b - 1\): \(5 - 1 = 4\), so \(4 = 4\). ✓
Check \(b \geq 3\): \(5 \geq 3\) ✓
Therefore, \((4, 5)\) satisfies both conditions. Correct.
- (C) \( (4, 6) \in R \)
Here, \(a = 4\), \(b = 6\).
Check \(a = b - 1\): \(6 - 1 = 5\), but \(a = 4\), so \(4 \neq 5\).
Therefore, \((4, 6)\) does not satisfy the condition. Incorrect.
- (D) \( (1, 3) \in R \)
Here, \(a = 1\), \(b = 3\).
Check \(a = b - 1\): \(3 - 1 = 2\), but \(a = 1\), so \(1 \neq 2\).
Therefore, \((1, 3)\) does not satisfy the condition. Incorrect.

Step 4: Conclusion.
Only \((4, 5)\) satisfies the condition \(a = b - 1\) with \(b \geq 3\).
Final Answer: (B) \( (4, 5) \in R \)