Let X be the set of all persons living in Delhi. The persons a and b in X are said to be related if the difference in their ages is at most 5 years. The relation is

The correct option is D Reflexive and symmetric but not transitive

Given,

$X=\left\{\text{Set of persons living in Delhi}\right\}$

$R=\left\{\right(a,b):|a-b|\le 5\}$

For, reflexive $(a,a)\in R$

$(a,a)=|a-a|=0,0\le 5,(a,a)\le 5$

Hence, R is reflexive.

For, symmetric

Let $(a,b)\in R$

$(a,b)=|a-b|\le 5\Rightarrow |b-a|\le 5)$

$⟹(b,a)\in R$

$\therefore (a,b)\in R⟹(b,a)\in R$

Hence, $R$ is symmetric.

For transitive

Let $(a,b)\in R$ and $(b,c)\in R$

$(a,b)=|a-b|\le 5,(b,c)=|b-c|\le 5$

$|a-c|\le |a-b|+|b-c|\le 10$

$\Rightarrow |a-c|\le 10$

$\therefore (a,c)\notin R$

Hence, $R$ is not transitive.

So, $R$ is reflexive and symmetric but not transitive.

Hence, D is the correct option.