Sibling Relation Transitivity and a Little More

Alexandre Pierre — Fri, 30 May 2025 23:05:38 +0000

I have a small number of friends with whom I discuss any craziness my brain is capable of producing. Some days ago I sent a voice note to one of them (shout out to Iago, I've linked his blog which is way better than mine) saying something along the lines: "I was thinking about transitivity and sibling relations are intransitive if you consider half-siblings as siblings". He replied: "Yes, and it's transitive if you only consider full siblings". Them he went a step ahead and said the same in more mathematical terms, which I explain below:

Intransitive Sibling Relation

\text{Let $S_I$ be the sibling relation as written above, then:} \\ \text{$x ~ S_I ~ y \iff \exists p_{x,y}$ is parents for $x$ and $y$}

I don't think it's necessary to write "parents to" in terms of relations too. But if you, dear reader, disagree you may comment and I'll consider putting it there or to use it to answer the comment itself.

It's possible to prove that the relation is intransitive if we consider that $z$ can be siblings with $y$ but not with $x$ . Even though $x$ and $y$ are siblings to each other.

\text{Let $p_{x,y}$ be the parent to $x$ and $y$} \\ \text{$x ~ S_I ~ y$ because $p_{x,y}$ exists} \\~\\ \text{Let $p_{y,z}$ be the parent to $y$ and $z$} \\ \text{$y ~ S_I ~ z$ because $p_{y,z}$ exists}

But why is this relation intransitive?

Answer: because in the context of half-siblings $x$ and $z$ may not have a common parent.

Transitive Sibling Relation

Exploring mathematically the other observation Iago made:

\text{Let $S_T$ be the full-sibling relation:} \\ \text{$x ~ S_T ~ y \iff \forall p_{x,y}$ are parents for $x$ and $y$}

But how is it different from what we've seen before?

If we are talking about full-siblings, they have to share all parents. And it makes impossible the condition that made $S_I$ intransitive. You can see that in the diagram below, which is minimal for full-sibling relations.

What I've Learned Today: Anti-Transitive Relation

While doing the research for this post I've got acquainted with the concept of Anti-Transitive Relations in the site Geeks for Geeks. Which means a relation $A_T$ that is made impossible between $x$ and $z$ if $x ~ A_T ~ y$ and $y ~ A_T ~ z$ .

Wikipedia article on Transitivity has a very good example that I've used as a base to make my own:

The antitransitive nature of the cycle makes it so Fire beats Leaf, Leaf beats Water but Fire can't beat Water. (If you know enough about pokémon you may know that this model has its limits, but consider only the starter types and not all of pokémon).

Conclusion

To end this text about the examples I really recommend the Geeks for Geeks article, the Wikipedia article on Transitivity and the Wikipedia article on Intransitivity.

DEV Community: kerthump

Sibling Relation Transitivity and a Little More

Intransitive Sibling Relation

Transitive Sibling Relation

What I've Learned Today: Anti-Transitive Relation

Conclusion