Question

Let $R$ be a relation on $R$, given by $R=\{(a, b): 3 a-3 b+\sqrt{7}$ is an irrational number $\}$Then $R$ is

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

Answer 1

The Correct Option is C

To determine the nature of the relation \( R \) defined on \( \mathbb{R} \), with \( R = \{(a, b) \mid 3a - 3b + \sqrt{7} \text{ is an irrational number}\} \), we need to examine whether this relation is reflexive, symmetric, and transitive.

1. Reflexivity:
   • A relation \( R \) is reflexive if every element is related to itself. Here, for any element \( a \) in \( \mathbb{R} \), we check if \( (a, a) \in R \).
   • Substituting, \( 3a - 3a + \sqrt{7} = \sqrt{7} \), which is indeed an irrational number.
   • Thus, \( R \) is reflexive since \( (a, a) \in R \) for all \( a \).
2. Symmetry:
   • A relation \( R \) is symmetric if whenever \( (a, b) \in R \), then \( (b, a) \in R \).
   • If \( (a, b) \in R \), then \( 3a - 3b + \sqrt{7} \) is irrational.
   • We need to check if \( 3b - 3a + \sqrt{7} \) is also irrational. Consider \( 3a - 3b + \sqrt{7} \) which is irrational. Rearrange to get \( 3b - 3a = (3b - 3a + \sqrt{7}) - \sqrt{7} \).
   • This expression involves the subtraction of one irrational number from another; however, without specific information, \( 3b - 3a + \sqrt{7} \) will not necessarily remain irrational, hence \( (b, a) \notin R \) in general.
   • Therefore, \( R \) is not symmetric.
3. Transitivity:
   • A relation \( R \) is transitive if whenever \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \).
   • Suppose \( (a, b) \in R \) and \( (b, c) \in R \). This implies \( 3a - 3b + \sqrt{7} \) and \( 3b - 3c + \sqrt{7} \) are irrational.
   • Check: \( 3a - 3c + \sqrt{7} = (3a - 3b + \sqrt{7}) + (3b - 3c + \sqrt{7}) - \sqrt{7} \).
   • This expression combines two irrational expressions, leading to a possibility of cancellation of irrational parts, making \( 3a - 3c + \sqrt{7} \) rational in some cases.
   • Therefore, \( R \) is not always transitive.

Given that \( R \) is reflexive but neither symmetric nor transitive, the correct answer is: Reflexive but neither symmetric nor transitive.