Question

The minimum number of elements that must be added to the relation $R=\{(a,b),(b,c)\}$ on the set $\{a,b,c\}$ so that it becomes symmetric and transitive is :

• A. 3
• B. 7
• C. 4
• D. 5

Answer 1

Answer: (B)

For Symmetric $(a,b),(b,c)\in R$
$\Rightarrow (b,a),(c,b)\in R$
For Transitive $(a,b),(b,c)\in R$ $\Rightarrow (a,c)\in R$
Now
1. Symmetric
$\therefore (a,c)\in R\Rightarrow (c,a)\in R$
2. Transitive
$\therefore (a,b),(b,a)\in R$
$\Rightarrow (a,a)\in R\&(b,c),(c,b)\in R$
$\Rightarrow (b,b)\&(c,c)\in R$
$\therefore$ Elements to be added
$\left\{(b,a),(c,b),(a,c),(c,a),(a,a),(b,b),(c,c)\right\}$
Number of elements to be added $=7$