Let $A=\{1,2,3\} .$ Then show that the number of relations containing $(1,2) $ and $(2,3)$ which are reflexive and transitive but not symmetric is four.
Let $A=\{1,2,3\} .$ Then show that the number of relations containing $(1,2) $ and $(2,3)$ which are reflexive and transitive but not symmetric is four.
The smallest relation $R_{1}$ containing $(1,2)$ and $(2,3)$ which is reflexive and transitive but not symmetric is $\{(1,1),\,(2,2),\,(3,3),$ $(1,2),\,(2,3),\,(1,3)\} .$ Now, if we add the pair $(2,1)$ to $R_{1}$ to get $R_{2}$, then the relation $R_{2}$ will be reflexive, transitive but not symmetric. Similarly, we can obtain $R _{3}$ and $R _{4}$ by adding $(3,2)$ and $(3,1)$ respectively, to $R_{1}$ to get the desired relations. However, we can not add any two pairs out of $(2,1),$ $(3,2)$ and $(3,1)$ to $R_{1}$ at a time, as by doing so, we will be forced to add the remaining third pair in order to maintain transitivity and in the process, the relation will become symmetric also which is not required. Thus, the total number of desired relations is four.
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