### Sets And Relations Question 4

#### Question 4 - 30 January - Shift 1

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:

(1) 4

(2) 7

(3) 5

(4) 3

## Show Answer

#### Answer: (2)

#### Solution:

#### Formula: Symmetric relation (v), Transitive relation (vi)

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

- Symmetric

$\therefore(a, c) \in R \Rightarrow(c, a) \in R$

- Transitive

$\therefore(a, b),(b, a) \in R$

$\Rightarrow(a, a) \in R ; and ; (b, c),(c, b) \in R$

$\Rightarrow(b, b) ; and ; (c, c) \in R$

$\therefore$ Elements to be added

$ {(b, a),(c, b),(a, c),(c, a),(a, a),(b, b),(c, c)} $

Number of elements to be added $=7$