Relations
Let R be the relation on the set R of all real numbers defined by aRb \u21Longleftrightarrow |a - b| ≤ 1. Then R is
Select the correct option:
Solution
Reflexive and symmetric but not transitive
- Reflexive: ∣a−a∣=0≤1. (Yes)
- Symmetric: If ∣a−b∣≤1, then ∣b−a∣=∣a−b∣≤1. (Yes)
- Transitive:
- Let a=0,b=0.6,c=1.2.
- ∣a−b∣=0.6≤1 (aRb is true).
- ∣b−c∣=0.6≤1 (bRc is true).
- Check aRc: ∣0−1.2∣=1.2, which is >1. Thus a is not related to c.
- Conclusion: Reflexive and symmetric but not transitive.
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About This Question
- Subject
- mathematics
- Chapter
- sets, relations and functions
- Topic
- relations
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
Reflexive and symmetric but not transitive
- Reflexive: ∣a−a∣=0≤1. (Yes)
- Symmetric: If ∣a−b∣≤1, then ∣b−a∣=∣a−b∣≤1. (Yes)
- Transitive:
- Let a=0,b=0.6,c=1.2.
- ∣a−b∣=0.6≤1 (aRb is true).
- ∣b−c∣=0.6≤1 (bRc is true).
- Check aRc: ∣0−1.2∣=1.2, which is >1. Thus a is not related to c.
- Conclusion: Reflexive and symmetric but not transitive.
This hard difficulty mathematics question is from the chapter sets, relations and functions, covering the topic of relations. It appeared in the 2025 exam.
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