Question

Let $R$ be the relation on the set of positive integers such that $aRb$ if and only if $a$ and $b$ are distinct and have a common divisor other than 1. Which one of the following statements about R is True?

• A. $R$ is symmetric and reflexive but not transitive
• B. $R$ is reflexive but not symmetric and not transitive
• C. $R$ is transitive but not reflexive and not symmetric
• D. $R$ is symmetric but not reflexive and not transitive

Answer 1

The correct option is D $R$ is symmetric but not reflexive and not transitive

$aRb$ iff $a$ and $b$ are distinct $a$ and $b$ have a common divisor other than 1.
(i) $R$ is not reflexive since $a$ and $b$ are distinct
i.e. $(a,a)\text{/}ϵR$
(ii) $R$ is symmetric
If $a$ and $b$ are distinct and have a common divisor other than 1, then $b$ and $a$ also are distinct and have a common divisor other than 1.
(iii) $R$ is not transitive
If $(a,b)ϵR$ and $(b,c)ϵR$ then (a, c) need not be in R.
Example: (2,6) and $(6,2)ϵR$, but $(2,2)\text{/}ϵR$
$\therefore R$ is symmetric but not reflexive and not transitive.