• # 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

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.