To solve the question, we need to analyze each given relation on the set of real numbers, \( \mathbb{R} \), to determine their properties: reflexivity, symmetry, and transitivity.
Analyzing Each Option:
- Option 1: \( |x - y| < 2 \)
- Reflexivity: A relation is reflexive if every element is related to itself. Here, \( |x - x| = 0 < 2 \), so the relation is reflexive.
- Symmetry: If \( |x - y| < 2 \), it implies \( |y - x| < 2 \) due to the absolute value having symmetric properties. Hence, it is symmetric.
- Transitivity: For it to be transitive, if \( |x - y| < 2 \) and \( |y - z| < 2 \), then \( |x - z| \) should also be less than 2. However, \(|x - z| \leq |x - y| + |y - z| < 4\), which doesn’t guarantee it's less than 2, making it non-transitive.
- Conclusion: This relation is reflexive and symmetric but not transitive. This doesn't fully match any option given.
- Option 2: \( |x| \ge y \)
- Reflexivity: For \( x = y \), \(|x| = |y| \ge y\) is true for all real numbers, thus reflexive.
- Symmetry: If \( |x| \ge y \), it doesn’t imply that \( |y| \ge x \). Therefore, it is not symmetric.
- Transitivity: If \( |x| \ge y \) and \( |y| \ge z \), then \( |x| \ge z \) by combining inequalities, confirming transitivity.
- Conclusion: This relation is reflexive and transitive but not symmetric, matching option 2 perfectly.
- Option 3: \( x > |y| \)
- Reflexivity: This is not reflexive as it cannot be true for all \( x \in \mathbb{R} \), especially for non-positive \( x \).
- Symmetry: If \( x > |y| \), it is not necessarily true that \( y > |x| \). Hence, not symmetric.
- Transitivity: If \( x > |y| \) and \( y > |z| \), it implies \( x > |z| \). Thus, it is transitive.
- Conclusion: This relation is transitive but neither reflexive nor symmetric, matching the description in the option 3 but not the correct choice.
- Option 4: \( x - y < 2 \)
- Reflexivity: For any real number \( x \), \( x - x = 0 < 2 \). Hence, reflexive.
- Symmetry: If \( x - y < 2 \), it is not necessarily true that \( y - x < 2 \), therefore not symmetric as the reverse condition needs to be verified separately (which fails for some real numbers).
- Transitivity: If \( x - y < 2 \) and \( y - z < 2 \), then combining gives \( x - z < 4 \), not strictly less than 2. Hence, not transitive.
- Conclusion: This relation is reflexive and could be symmetric under special conditions but is generally not, and never transitive, misaligning with properties described.
Final Answer: The correct option is Option 2: \( |x| \ge y \) is reflexive and transitive but not symmetric.
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