JEE Main & Advanced Sample Paper JEE Main Sample Paper-5

  • question_answer
    Let R be the relation on N defined by  \[R=\{(x,y)|x={{2}^{n}}y;n\in N\}\]Then, R is

    A)  equivalence relation  

    B)  not symmetric          

    C)  reflexive only         

    D)  transitive only

    Correct Answer: A

    Solution :

     Idea A relation R on set A is reflexive if (a, a) \[\in R\forall a\in A,\], symmetric if (a, b) \[\in R\Rightarrow \] (b, a) \[\in R\forall (a,b)\in A,\], transitive if (a, b) \[\in R\], (b, c) \[\in R\Rightarrow (a,c)\in R\forall (a,b,c)\in A\] if above three conditions are satisfied, then R is an equivalence relation. The given relation R on N is \[R=\{(x,y)|x={{2}^{n}}y,n\in Z\}\] Now,     \[x={{2}^{0}}\cdot x\] \[\Rightarrow \]\[xRx\] \[\Rightarrow \]R is reflexive. Now, let \[x,y\in R\] such that xRy \[\Rightarrow \]               \[x={{2}^{n}}y\] \[\Rightarrow \]               \[{{2}^{-n}}x=y,-n\in Z\] \[\Rightarrow \]               \[yRz\] R is transitive. Let \[x,y,\in R\] such that xRy and y R Z \[\Rightarrow \]               \[x={{2}^{n}}y\]and        \[y={{2}^{n}}y\] Now,           \[x={{2}^{n}}y\] \[={{2}^{n}}({{2}^{n}}Z)\] \[={{2}^{2n}}Z,2n\in Z\]                 \[\Rightarrow \]                               \[x={{2}^{2n}}Z,2n\in Z\] \[\Rightarrow \]                               \[xRZ\] \[\Rightarrow \]is transitive. \[\Rightarrow \] is an equivalence relation. TEST Edge Application of relation based questions are asked. To solve these types of questions, students are advised to understand the concept of relation.

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