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
Let SI be the sibling relation as written above, then:x SI y ⟺ ∃px,y is parents for x and y
\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 $y$}
Let SI be the sibling relation as written above, then:x SI y⟺∃px,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
zzz
can be siblings with
yyy
but not with
xxx
. Even though
xxx
and
yyy
are siblings to each other.
Let px,y be the parent to x and yx SI y because px,y exists Let py,z be the parent to y and zy SI z because py,z exists
\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}
Let px,y be the parent to x and yx SI y because px,y exists Let py,z be the parent to y and zy SI z because py,z exists
But why is this relation intransitive?
Answer: because in the context of half-siblings
xxx
and
zzz
may not have a common parent.
Transitive Sibling Relation
Exploring mathematically the other observation Iago made:
Let ST be the full-sibling relation:x ST y ⟺ ∀px,y are parents for x and y
\text{Let $S_T$ be the full-sibling relation:}
\\
\text{$x ~ S_T ~ y \iff \forall p_{x,y}$ are parents for $x$ and $y$}
Let ST be the full-sibling relation:x ST y⟺∀px,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
SIS_ISI
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
ATA_TAT
that is made impossible between
xxx
and
zzz
if
x AT yx ~ A_T ~ yx AT y
and
y AT zy ~ A_T ~ zy AT 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.