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 $=7$
