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Q.
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 :
JEE MainJEE Main 2023Relations and Functions - Part 2
Solution:
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 $=7$