I have been struggling to think of an example of a relation that is neither transitive nor intransitive, does anyone have any tips? I ended up finding one website that described this as non transitive, but they're description just confused me even more.
The website defines:
$Rxy$ is intransitive just if there is no broken journey in its graph with a short cut. That is: $Rxy$ is intransitive just if $$\forall x\forall y \forall z\ [[Rxy \wedge Ryz] \implies \neg Rxz]$$
Let $S$ be the set of all countries on Earth and consider the "is a neighbour of" relation on $S$.
Notice that this relation is not transitive. To see this, observe that Canada is a neighbour of USA and USA is a neighbour of Mexico, but Canada is not a neighbour of Mexico.
Furthermore, notice that this relation is not intransitive. To see this, observe that Norway is a neighbour of Sweden and Sweden is a neighbour of Finland, but Norway is a neighbour of Finland.