Question

Given the relation $R=\left\{\right(1,2),(2,3\left)\right\}$ on the set $A=\{1,2,3\}$, the minimum number of ordered pairs required to make $R$ an equivalence relation is

Answer 1

$R$ is reflexive if it contains $(1,1)(2,2)(3,3)$

$\because (1,2)\in R,(2,3)\in R$
$\therefore \text{For}R$ to be symmetric $(2,1)\text{and}(3,2)\text{should}\in R$

Now, $R=\left\{\right(1,1),(2,2),(3,3),(2,1),(3,2),(2,3),(1,2\left)\right\}$
$R$ will be transitive if $(3,1)\text{and}(1,3)\in R$.

Thus, $R$ becomes an equivalence relation by adding $(1,1)(2,2)(3,3)(2,1)(3,2)(1,3)(3,1).$

Hence, minimum number of ordered pairs required to make the given relation equivalance is $7.$