1.Relation and Function
hard

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

$4$

B

$7$

C

$5$

D

$3$

(JEE MAIN-2023)

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

$\left\{\begin{array}{r}(b, a),(c, b),(a, c),(c, a) \\,(a, a),(b, b),(c, c)\end{array}\right\}$

Number of elements to be added $=7$

Standard 12
Mathematics

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