Question

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

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

Hint:

In questions involving relations defined by multiple conditions, methodically check each condition for every option. Do not stop after finding one condition is true; all conditions must hold for the element to be in the relation.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The problem defines a relation R on the set of natural numbers \( \mathbb{N} \). An ordered pair \( (a, b) \) belongs to this relation R if it satisfies two conditions simultaneously:
\( a = b - 1 \)
\( b \geq 3 \)
We need to check which of the given options satisfies both these conditions.
Step 2: Detailed Explanation:
Let's check each option against the two conditions.
(A) (2, 4) \( \in \) R:
Here, \( a = 2 \) and \( b = 4 \).
Condition 2: Is \( b \geq 3 \)? Yes, \( 4 \geq 3 \).
Condition 1: Is \( a = b - 1 \)? Is \( 2 = 4 - 1 \)? No, \( 2 \neq 3 \).
Since one condition is not met, (2, 4) \( \notin \) R.
(B) (4, 5) \( \in \) R:
Here, \( a = 4 \) and \( b = 5 \).
Condition 2: Is \( b \geq 3 \)? Yes, \( 5 \geq 3 \).
Condition 1: Is \( a = b - 1 \)? Is \( 4 = 5 - 1 \)? Yes, \( 4 = 4 \).
Since both conditions are met, (4, 5) \( \in \) R.
(C) (4, 6) \( \in \) R:
Here, \( a = 4 \) and \( b = 6 \).
Condition 2: Is \( b \geq 3 \)? Yes, \( 6 \geq 3 \).
Condition 1: Is \( a = b - 1 \)? Is \( 4 = 6 - 1 \)? No, \( 4 \neq 5 \).
Since one condition is not met, (4, 6) \( \notin \) R.
(D) (1, 3) \( \in \) R:
Here, \( a = 1 \) and \( b = 3 \).
Condition 2: Is \( b \geq 3 \)? Yes, \( 3 \geq 3 \).
Condition 1: Is \( a = b - 1 \)? Is \( 1 = 3 - 1 \)? No, \( 1 \neq 2 \).
Since one condition is not met, (1, 3) \( \notin \) R.
Step 3: Final Answer:
The only option that satisfies both conditions for the relation R is (4, 5).