JEE Main & Advanced Mathematics Sets Question Bank Critical Thinking

  • question_answer Let R be the relation on the set R of all real numbers defined by a R b iff \[|a-b|\le 1\]. Then R is [Roorkee 1998]

    A) Reflexive and Symmetric

    B) Symmetric only

    C) Transitive only

    D) Anti-symmetric only

    Correct Answer: A

    Solution :

    \[|a-a|=0<1\] \[\therefore \,a\,R\,a\,\forall \,a\in R\] \[\therefore \] R is reflexive.  Again a R b Þ \[|a-b|\le 1\Rightarrow |b-a|\le 1\Rightarrow bRa\] \[\therefore \] R is symmetric, Again \[1R\frac{1}{2}\] and \[\frac{1}{2}R1\] but \[\frac{1}{2}\ne 1\] \[\therefore \] R is not anti-symmetric. Further, 1 R 2 and 2 R 3 but \[1\,\not{R}\,3\],  [\[\because \,|1-3|=2>1\]] \[\therefore \] R is not transitive.

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