check whether the relation R in …

Let's check the properties of the relation \( R \) defined in \( \mathbb{R} \) (the set of real numbers) by \( R = \{(a, b) : a \times b \text{ is an irrational number}\} \): 1. **Reflexive:** A relation \( R \) is reflexive if for every element \( a \) in the set \( A \), the pair \( (a, a) \) belongs to \( R \). In this case, for any real number \( a \), \( a \times a = a^2 \) is always a real number, not necessarily irrational. So, \( R \) is not reflexive. 2. **Symmetric:** A relation \( R \) is symmetric if for every pair \( (a, b) \) in \( R \), the pair \( (b, a) \) also belongs to \( R \). If \( a \times b \) is irrational, it doesn't imply that \( b \times a \) is irrational. For example, \( \sqrt{2} \times \sqrt{3} \) is irrational, but \( \sqrt{3} \times \sqrt{2} \) is also irrational. So, \( R \) is symmetric. 3. **Transitive:** A relation \( R \) is transitive if for every pair \( (a, b) \) and \( (b, c) \) in \( R \), the pair \( (a, c) \) also belongs to \( R \). If \( a \times b \) and \( b \times c \) are irrational, it doesn't necessarily imply that \( a \times c \) is irrational. For example, \( \sqrt{2} \times \sqrt{3} \) and \( \sqrt{3} \times \sqrt{2} \) are both irrational, but \( \sqrt{2} \times \sqrt{2} = 2 \) is rational. So, \( R \) is not transitive. In summary: - \( R \) is not reflexive. - \( R \) is symmetric. - \( R \) is not transitive.