• # 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

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.

You need to login to perform this action.
You will be redirected in 3 sec