Check whether the relation R in the set N of natural numbers given by

R = {(a, b) : a is divisor of b}

is reflexive, symmetric or transitive. Also determine whether R is an equivalence relation.

 

 
 
 
 
 

Answers (1)

\\ $ Check for Reflexive: $ \\ $ if $ (\mathrm{a}, \mathrm{a}) $ for every $ a \in \mathrm{R}$ $\Rightarrow \mathrm{R}$ is reflexive $ \\ $ Here a divided a $ \Rightarrow (\mathrm{a}, \mathrm{a}) \in \mathrm{R} \\ $ \mathrm{R} is reflexive $ \\

\\ $ Check for Symmetric: $ \\ $ if $ (\mathrm{a}, \mathrm{b}) \in \mathrm{R}$ $ then $ (\mathrm{b}, \mathrm{a}) \in \mathrm{R} $\Rightarrow \mathrm{R}$ is Symmetric $ \\ $ Here a divided b $ \Rightarrow (\mathrm{1}, \mathrm{2}) \in \mathrm{R} $but $ (2,1)$\notin \mathrm{R}$ \\ \ \mathrm{R}$ is not symmetric

\\ $ Check For transitive : $ \\ $ If $(a, b),(b, c) \in R $ then $ (a, c) \in R \Rightarrow $ R is transitive $\\ $ Here a divides $\mathrm{b}$ and $\mathrm{b}$ divides $\mathrm{c}$ $\Rightarrow$ a divides $c \\ (a, c) \in R$ \Rightarrow $\mathrm{R}$ is transitive

$\mathrm{R} is not symmetric. Hence it is not an equivalence relation.

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