I'm trying to come up with an example of a relation that is quasi-transitive but not transitive. The relation $ x R y $ is a subset of the cartesian product $XxX$, and the asymmetric relation is $xPy \iff xRy $ and $\neg (yRx) $.
The example I'm trying to create is a quasi-transitive relation that is not transitive.
For any $x,y, z \in X$, a binary relation R is
transitive: if $xRy, yRz $ then $xRz$
quasi-transitive: if $xPy, yPz $ then $xPz$
So, in $R^2$ I define the following relation:
$$xRy \iff x\text{ is not larger than } y.$$
If we have $$x=(3,1), y=(0,2) \text{ and } z=(1,1)$$, then $xRy$ and $yRz$, but $ \neg(xRz)$ i.e. $R$ is not transitive.
On the other hand, I also need to show that $R$ is quasi-transitive i.e: $$xPy, yRz \text{ then }xPz$$ and here is where I have problems as I'm failing to arrive to an example where $xPy$ and $yRz$ are also true.
EDIT1
Here are the same definitions but with different notations Relations:
https://en.wikipedia.org/wiki/Preference_%28economics%29#Notation
Transitivity quasi-transitivity:
https://en.wikipedia.org/wiki/Preference_%28economics%29#Meaning_in_decision_sciences