Question

The relation \( R \) in the set of integers \( \mathbb{Z} \) is given by \( R = \{(a, b) : b = 2a + 3 \} \). Then the relation \( R \) is:

• A. reflexive, symmetric and transitive
• B. neither reflexive nor symmetric nor transitive
• C. not reflexive but symmetric and transitive
• D. reflexive and symmetric but not transitive
• E. reflexive but not symmetric and transitive

Hint:

To check the properties of a relation, carefully verify the conditions for reflexivity, symmetry, and transitivity.

Answer 1

The Correct Option is B

To analyze the relation \( R \), we will check its properties: 
1. Reflexivity: For \( R \) to be reflexive, \( (a, a) \) must be in \( R \) for all \( a \in \mathbb{Z} \). For \( (a, a) \), we need \( a = 2a + 3 \), which is not possible for any integer \( a \). 
Hence, the relation is not reflexive. 
2. Symmetry: For \( R \) to be symmetric, if \( (a, b) \in R \), then \( (b, a) \) must also be in \( R \). If \( b = 2a + 3 \), then \( a = 2b + 3 \) is not satisfied. 
Hence, the relation is not symmetric. 
3. Transitivity: For \( R \) to be transitive, if \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \) must also be in \( R \). If \( b = 2a + 3 \) and \( c = 2b + 3 \), then we have \( c = 2(2a + 3) + 3 = 4a + 9 \), which does not equal \( 2a + 3 \), so the relation is not transitive. 
Thus, the relation \( R \) is neither reflexive, nor symmetric, nor transitive, and the correct answer is (B).