“$ x $ is a brother of $ y $.” Why is this not transitive?

Question

I am working on a problem set at the moment, and while checking my answers I realized that I have listed "x is a brother of y" as a transitive relation, while the answers say that it is not.

EDIT: I realized I forgot to put some assumptions from the question: i) Domain is people in general. ii) Assume that all siblings share both parents.

But if x is a brother of y, and y is a brother of z, then surely x is a brother of z? Am I missing something here?

Please help a student out. Thanks!

Answer 1

If Fred is Bob's brother, and vice versa, transitivity would imply that Fred is his own brother.

Answer 2

Transitivity says:

$aRb$ and $bRc$ imply $aRc$

$x$ is a brother of $y$, $y$ is a brother of $x$, but $x$ is not a brother of $x$.

Therefore, it's not a transitive relation.

Answer 3

If X is the brother of Y and Y is the brother of X, then X is his own brother, if it were transitive.

Answer 4

What if Y is female. So X is a brother of Y, but Y is a sister of X.