Number of relations, on set {1,2,3} containing (1,2) and (2,3) , which are reflexive and transitive

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1.Relation and Function

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The number of relations, on the set $\{1,2,3\}$ containing $(1,2)$ and $(2,3)$, which are reflexive and transitive but not symmetric, is

A

$0$

B

$1$

C

$2$

D

$3$

(JEE MAIN-2023)

Solution

$A =\{1,2,3\}$

For Reflexive $(1,1)(2,2),(3,3) \in R$

For transitive : $(1,2)$ and $(2,3) \in R \Rightarrow(1,3) \in R$

Not symmetric : $(2,1)$ and $(3,2) \notin R$

$R _1=\{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)\}$

$R _2=\{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)(2,1)\}$

$R _3=\{(1,1),(2,2),(3,3),(1,2),(2,3),(1,3)(3,2)\}$

Standard 12

Mathematics

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