Question

Given 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

$7$

Explanation for the correct option:

Given ,

relation $R=\left\{\right(1,2),(2,3\left)\right\}$ on the set

$A=\{1,2,3\}$

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

Therefore, $\left(1,1\right)\in R,\left(2,3\right)\in R$

Therefore $R$ is symmetric if $\left(2,1\right),\left(1,2\right)\in R,$

Now, $R=\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(2,1\right),\left(3,2\right),\left(2,3\right),\left(1,2\right)$

$R$ will be transitive if,$\left(3,1\right),\left(1,3\right)\in R$.

Thus $R$ becomes and equivalence Relation

by adding $\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(2,1\right),\left(3,2\right),(1,3),\left(3,1\right).$

Hence ,the total number of ordered pairs is 7.

Hence, option(C) is correct.