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  4. The minimum number of elements that must be added to the relation R= (1,2),(2,3) on the sub set 1,2,3 of natural numbers so that it is an equivalence relation is

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Q. The minimum number of elements that must be added to the relation $R=\{(1,2),(2,3)\}$ on the sub set $\{1,2,3\}$ of natural numbers so that it is an equivalence relation is

Relations and Functions - Part 2

Solution:

$R$ is reflexive if $(x, x) . \in R$ for all $x \in\{1,2,3\}$
$\Rightarrow (1,1),(2,2),(3,3) \in R$
$R$ is symmetric if $(1,2),(2,3) \in R$
$\Rightarrow (2,1),(3,2) \in R \text {. }$
$R$ is transitive if $(1,2),(2,3) \in R$
$\Rightarrow (1,3) \in R$ also $(3,1) \in R$ as $R$ is symmetric.
So the total numbers of elements is 9 .