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The domain of a relation R is the set of real numbers. x is related to y under relation R if |x+y[>2. Select the description that accurately describes relation R. Anti-reflexive Neither reflexive nor anti-reflexive O Transitive O Reflexive

The domain of a relation R is the set of real numbers. x is related to y under relation R if |x+y[>2. Select the description that accurately describes relation R. Anti-reflexive Neither reflexive nor anti-reflexive O Transitive O Reflexive

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Understanding Relations** The domain of a relation \( R \) is the set of real numbers. The relation states that \( x \) is related to \( y \) under \( R \) if \( |x + y| \geq 2 \). ### Question: Select the description that accurately describes relation \( R \). - ○ Anti-reflexive - ○ Neither reflexive nor anti-reflexive - ○ Transitive - ○ Reflexive ### Explanation: To determine which property the relation has, consider the conditions under which \( R \) either holds or does not hold: - **Reflexive**: For a relation to be reflexive, every element must relate to itself. Here, \( |x + x| = |2x| \geq 2 \) must be true for all \( x \). This is not true for all \( x \). For instance, if \( x = 0 \), then \( |x + x| = 0 \), which does not satisfy the condition \( \geq 2 \). - **Anti-reflexive**: The relation is anti-reflexive if no element relates to itself. For \( x \), \( |x + x| = |2x| \geq 2 \) can be true for some \( x \), so this relation is not strictly anti-reflexive. - **Transitive**: To be transitive, if \( x \) is related to \( y \) and \( y \) is related to \( z \), then \( x \) must also relate to \( z \). Testing this with specific numbers can determine transitivity. - **Neither reflexive nor anti-reflexive**: Given the nature of the relation, it may not satisfy the strict criteria for either reflexivity or anti-reflexivity broadly across all real numbers.
Transcribed Image Text:**Understanding Relations** The domain of a relation \( R \) is the set of real numbers. The relation states that \( x \) is related to \( y \) under \( R \) if \( |x + y| \geq 2 \). ### Question: Select the description that accurately describes relation \( R \). - ○ Anti-reflexive - ○ Neither reflexive nor anti-reflexive - ○ Transitive - ○ Reflexive ### Explanation: To determine which property the relation has, consider the conditions under which \( R \) either holds or does not hold: - **Reflexive**: For a relation to be reflexive, every element must relate to itself. Here, \( |x + x| = |2x| \geq 2 \) must be true for all \( x \). This is not true for all \( x \). For instance, if \( x = 0 \), then \( |x + x| = 0 \), which does not satisfy the condition \( \geq 2 \). - **Anti-reflexive**: The relation is anti-reflexive if no element relates to itself. For \( x \), \( |x + x| = |2x| \geq 2 \) can be true for some \( x \), so this relation is not strictly anti-reflexive. - **Transitive**: To be transitive, if \( x \) is related to \( y \) and \( y \) is related to \( z \), then \( x \) must also relate to \( z \). Testing this with specific numbers can determine transitivity. - **Neither reflexive nor anti-reflexive**: Given the nature of the relation, it may not satisfy the strict criteria for either reflexivity or anti-reflexivity broadly across all real numbers.
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