Question

Given the relation on $R=\left\{(a,b),(b,c)\right\}$ in the set $A=\left\{a,b,c\right\}$Then the minimum number of ordered pairs which 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:

Find the minimum number of ordered pairs:

Given,

$R=\left\{(a,b),(b,c)\right\}$ in the set

$A=\left\{a,b,c\right\}$

$R$ is reflexive if it is contains $\left(a,a\right),\left(b,b\right),\left(c,c\right)\in R$

if $\left(b,a\right),\left(c,b\right)\in R$

Therefore $R$ is symmetric if $\left(a,b\right),\left(b,c\right)\in R$

Now, $R=\left\{\left(a,a\right),\left(b,b\right),\left(c,c\right),\left(b,a\right),(c,b),\left(a,c\right),\left(a,b\right)\right\}$

$R$ will be transitive $\left(c,a\right),\left(a,c\right)\in R$

Then $R$ become and equivalence relation by adding

$\left(a,a\right),\left(b,b\right),\left(c,c\right),\left(b,a\right),\left(c,b\right),\left(a,c\right),\left(a,b\right)$.

$\therefore$Hence the total number of ordered pairs is $\mathbf{7}\mathbf{.}$

Hence, option(C) is correct.