Question

Check whether the relation \( S \) in the set of real numbers \( \mathbb{R} \), defined by \(S = \{(a, b) : a - b + \sqrt{2}\) \(\text{ is an irrational number}\), is reflexive, symmetric, or transitive.} 
 

Hint:

To test reflexivity, verify if \( (a, a) \in S \) for all \( a \). For symmetry and transitivity, check logical equivalence and counterexamples.

Answer 1

Step 1: Reflexivity Check
For any real number \( a \), consider the expression \( a - a + \sqrt{2} \). This simplifies to \( \sqrt{2} \), which is an irrational number. Therefore, \( (a, a) \) satisfies the condition for being in \( S \), proving that \( S \) is reflexive.

Step 2: Symmetry Check
Assume \( (a, b) \in S \). This means \( a - b + \sqrt{2} \) is irrational. To check for symmetry, we must determine if \( (b, a) \in S \), which requires \( b - a + \sqrt{2} \) to be irrational. However, this is not always the case. For instance, if \( a = \sqrt{2} \) and \( b = 1 \), then \( a - b + \sqrt{2} = \sqrt{2} - 1 + \sqrt{2} = 2\sqrt{2} - 1 \) is irrational. But, \( b - a + \sqrt{2} = 1 - \sqrt{2} + \sqrt{2} = 1 \), which is rational. Consequently, \( S \) is not symmetric.

Step 3: Transitivity Check
Suppose \( (a, b) \in S \) and \( (b, c) \in S \). This implies that \( a - b + \sqrt{2} \) is irrational, and \( b - c + \sqrt{2} \) is irrational. We need to ascertain if \( (a, c) \in S \), meaning \( a - c + \sqrt{2} \) must be irrational. The expression \( a - c + \sqrt{2} \) can be rewritten as \( (a - b + \sqrt{2}) + (b - c + \sqrt{2}) - \sqrt{2} \). The irrationality of \( a - c + \sqrt{2} \) is not guaranteed. Consider the example where \( a = 1 \), \( b = \sqrt{3} \), and \( c = \sqrt{3} - \sqrt{2} \). In this case, \( a - c + \sqrt{2} = 1 - (\sqrt{3} - \sqrt{2}) + \sqrt{2} = 1 - \sqrt{3} + 2\sqrt{2} \), which is irrational. However, other choices of \( a, b, c \) could result in \( a - c + \sqrt{2} \) being rational, demonstrating that \( S \) is not transitive.

Final Assessment
The relation \( S \) exhibits reflexivity but lacks both symmetry and transitivity.