Minimum number of elements that must be added to relation R ={( a , b ),( b , c )} on set {a, b, c}

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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:

$4$

$7$

$5$

$3$

(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

The number of symmetric relations defined on the set $\{1,2,3,4\}$ which are not reflexive is

medium

(JEE MAIN-2024)

If $R$ is an equivalence relation on a set $A$, then ${R^{ – 1}}$ is

normal

Show that the relation $R$ in the set $\{1,2,3\}$ given by $R =\{(1,1),\,(2,2),$ $(3,3)$, $(1,2)$, $(2,3)\}$ is reflexive but neither symmetric nor transitive.

easy

Let $A=\{1,3,4,6,9\}$ and $B=\{2,4,5,8,10\}$. Let $R$ be a relation defined on $A \times B$ such that $R =$ $\left\{\left(\left(a_1, b_1\right),\left(a_2, b_2\right)\right): a_1 \leq b_2\right.$ and $\left.b_1 \leq a_2\right\}$. Then the number of elements in the set $R$ is