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 which when added to $R$ 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 $(1,1)(2,2)(3,3)$

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

$\therefore R$ is symmetric if $(2,1),(3,2)\in R$

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 $(3,1);(1,3)\in R$. Thus, $R$ becomes and

equivalence relation by adding $(1,1)(2,2)(3,3)(2,1)(3,2)(1,3)(1,2)$. Hence,

the total number of ordered pairs is $7$.