Why the homegeneous relation $\mathcal{R}$ over a set $E$ below, isn't transitive? I concluded that it was a transitive property.
A couple has 5 children: Andrew, Billy, Carl, Dariel and Elizabeth: After listing the elements of the $\mathcal{R}$ relation over a set $E= \{a,b,c,d,e\}$ by: $ x\mathcal{R}y \Leftrightarrow x $ is brother of $y$.
Note: is brother of $y$ when $x$ is man. $x \neq y$ and $x$ has the same parents. Show if the transitive property is valid or not. Justify it.
Andrew is a brother of Carl, and Carl is a brother of Andrew. If the relation were transitive, it would follow that Andrew is his own brother. But that is excluded by the definition of $\mathcal R$, which explicitly states that no child can be its own brother.
If there were at most one male child, the relation would indeed be transitive. The same would be the case if irreflexivity ($x\ne y$) were not part of the definition (that is, if a male child were considered his own brother).