Question

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

• A. $5$
• B. $6$
• C. $7$
• D. $8$

Answer 1

The correct option is C $7$


$R$ is reflexive if it contains $(1,1),(2,2),(3,3)$.
$\because (1,2)\in R$ and $(2,3)\in R$
$\therefore R$ is symmetric if $(2,1),(3,2)\in R.$

As $(1,2),(2,3)\in R$

Then, $R$ is transitive is $(1,3)\in R$

Also, $(3,1)\in R$ for symmetric property.

Now, $R=\left\{\right(1,1),(2,2),(3,3),(2,1),(3,2),(2,3),(1,2),(1,3),(3,1\left)\right\}$
Thus, $R$ becomes an equivalence relation by adding

$(1,1),(2,2),(3,3),(2,1),(3,2),(1,3),(3,1)$

Hence, the total number of ordered pairs is $7$.