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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!

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    $\begingroup$ Suppose $x$ and $y$ only share a mother, and $y$ and $z$ only share a father. $\endgroup$ Apr 3, 2015 at 1:25
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    $\begingroup$ People don't often think that they are their own brothers. $\endgroup$ Apr 3, 2015 at 1:26
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    $\begingroup$ @Polichinelle I think you nailed it. $\endgroup$ Apr 3, 2015 at 1:27
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    $\begingroup$ @Polichinelle Why not make that an answer? $\endgroup$ Apr 3, 2015 at 1:28
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    $\begingroup$ The problem with this question is that it is not about math, but about the interpretation of the word "brother ". If you replace "brother" with "share both parents" then the relation becomes reflexive and transitive. $\endgroup$ Apr 6, 2015 at 7:12

4 Answers 4

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

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    $\begingroup$ And why is being your own brother a contradiction? Note: I would define a brother of Fred as a son of his parents. $\endgroup$
    – A.P.
    Apr 3, 2015 at 9:35
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    $\begingroup$ @A.P.: And I could equally define each person to be that same person's only brother -- but if I did that, you would probably object that that is not how the word is used by people in the real world. Do you see how that same objection applies to your definition? $\endgroup$ Apr 3, 2015 at 10:49
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    $\begingroup$ @A.P. - I'm an only child. How many brothers do I have? None. Yes, if you squint just right, you can pretend that the right answer is one, but you'll have a hard time getting anyone else to agree. $\endgroup$ Apr 3, 2015 at 14:00
  • $\begingroup$ @PeteBecker so in your case its transitive right? cause the definition is if (a,b) and (b,c) exists we need (a,c) for transitivity... there's no (a,b) or (b,c) here... $\endgroup$ Dec 26, 2021 at 9:57
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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.

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

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

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    $\begingroup$ This means that the "brotherhood" relation isn't symmetric, not that it isn't transitive. $\endgroup$
    – A.P.
    Apr 3, 2015 at 22:36

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