Question

Let \( \mathbb{N} \) be the set of all natural numbers. Let \( R \) be a relation defined on \( \mathbb{N} \) given by \( aRb \) if and only if \( a + 2b = 11 \). Then the relation \( R \) is:

• A. reflexive but not symmetric
• B. not reflexive but symmetric
• C. reflexive and symmetric
• D. neither reflexive nor symmetric
• E. an equivalence relation

Hint:

To check if a relation is reflexive, verify if \( aRa \) holds for all elements in the set. To check for symmetry, verify if \( aRb \) implies \( bRa \) for all pairs \( a, b \).

Answer 1

The Correct Option is D

We are given a relation \( R \) on the set of natural numbers \( \mathbb{N} \) defined by \( aRb \) if and only if \( a + 2b = 11 \). We need to determine the properties of this relation: whether it is reflexive, symmetric, or both.

Step 1: Check if the relation is reflexive.

A relation \( R \) is reflexive if for all \( a \in \mathbb{N} \), we have \( aRa \), i.e., \( a + 2a = 11 \). Simplifying this:

\[ a + 2a = 3a = 11. \] This equation does not hold for any natural number \( a \), as \( 3a = 11 \) does not have a solution in \( \mathbb{N} \). Therefore, the relation is not reflexive.

Step 2: Check if the relation is symmetric.

A relation \( R \) is symmetric if for all \( a, b \in \mathbb{N} \), whenever \( aRb \) (i.e., \( a + 2b = 11 \)), we also have \( bRa \) (i.e., \( b + 2a = 11 \)).

Let’s assume \( a + 2b = 11 \). For the relation to be symmetric, we need \( b + 2a = 11 \). However, it is not guaranteed that \( b + 2a = 11 \) for all pairs of \( a \) and \( b \) that satisfy \( a + 2b = 11 \). For example, for \( a = 7 \) and \( b = 2 \), we have:

\[ a + 2b = 7 + 2(2) = 11, \] but:

\[ b + 2a = 2 + 2(7) = 16 \neq 11. \] Thus, the relation is not symmetric.

Step 3: Conclusion.

Since the relation is neither reflexive nor symmetric, the correct answer is option (D) - neither reflexive nor symmetric.

Thus, the correct answer is option (D).