The number of relations on the set {1,2,3} co | vedclass

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(B) Let the set be $A = \{1, 2, 3\}$.For a relation $R$ to be reflexive,it must contain $(1,1), (2,2), (3,3)$.Given that $(1,2) \in R$ and $(2,3) \in

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(B) Let the set be $A = \{1, 2, 3\}$.For a relation $R$ to be reflexive,it must contain $(1,1), (2,2), (3,3)$.Given that $(1,2) \in R$ and $(2,3) \in R$,for $R$ to be transitive,it must contain $(1,3)$ because $(1,2) \in R$ and $(2,3) \in R \implies (1,3) \in R$.Thus,the minimal relation $R$ containing these elements is $R_0 = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\}$.This relation $R_0$ is reflexive and transitive. It is not symmetric because $(1,2) \in R_0$ but $(2,1) 
otin R_0$.If we add any other element to $R_0$,such as $(2,1)$,the relation becomes symmetric with respect to $(1,2)$ and $(2,1)$. If we add $(3,2)$,it becomes symmetric with respect to $(2,3)$ and $(3,2)$.Therefore,there is only $1$ such relation,which is $R_0$ itself.

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

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