Question

Assertion (A): The relation \( R = \{(x, y) : (x + y) \text{ is a prime number and } x, y \in \mathbb{N}\} \) is not a reflexive relation.
Reason (R): The number \( 2n \) is composite for all natural numbers \( n \).

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
• B. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

A relation is reflexive if every element relates to itself; check this condition for all elements.

Answer 1

The Correct Option is C

Step 1: {Analyze Assertion (A)}
For \( R \) to be reflexive, \( (x, x) \) must belong to \( R \) for all \( x \in \mathbb{N} \). This means \( x + x = 2x \) must be a prime number. However, for \( x>1 \), \( 2x \) is not a prime number as it is divisible by \( 2 \). Therefore, \( R \) is not reflexive, and Assertion (A) is true. Step 2: {Analyze Reason (R)}
The Reason states that \( 2n \) is composite for all \( n \). This is false because when \( n = 1 \), \( 2n = 2 \), which is a prime number. Therefore, Reason (R) is false. Step 3: {Conclusion}
Since Assertion (A) is true and Reason (R) is false, the correct answer is option (C).