Question

Let \( R \) be a relation on the set \( \mathbb{N} \) defined by \[ \{(x, y) \mid x, y \in \mathbb{N}, \, 2x + y = 41\}. \] {Then, \( R \) is:}

• A. Reflexive
• B. Symmetric
• C. Transitive
• D. None of these

Hint:

To determine properties of a relation, verify whether the conditions for reflexivity, symmetry, and transitivity are satisfied.

Answer 1

The Correct Option is D

Step 1: A relation \( R \) is reflexive if \( (x, x) \in R \) for all \( x \in \mathbb{N} \).
For \( R \) to be reflexive, we must have \( 2x + x = 41 \), which simplifies to \( 3x = 41 \), but this has no solution in natural numbers. Thus, \( R \) is not reflexive.
Step 2: A relation \( R \) is symmetric if \( (x, y) \in R \) implies \( (y, x) \in R \). For \( R \) to be symmetric, if \( 2x + y = 41 \), then \( 2y + x = 41 \) must also hold, which does not generally happen. Hence, \( R \) is not symmetric.
Step 3: A relation \( R \) is transitive if \( (x, y) \in R \) and \( (y, z) \in R \) imply \( (x, z) \in R \).
However, transitivity does not hold for the given relation. Hence, \( R \) is not transitive.