I have been trying to construct q.t. relation, but always get transitive relation.
It seems to me, that transitivity includes q.t. Ok, but how would look like pure q.t. relation?
Examples and definitions seem to me ( in my opinion, which might be wrong) to contracict each other.
Here:
https://en.wikipedia.org/wiki/Quasitransitive_relation I cant belp myself but it seems to me there is contradiction in definition and example given in properties. Further search on the internet didnt clear it up for me.
Please give me explicit examples.
Thank you all kindly.
Best Answer
What is wrong with their example of people being indifferent between $7g$ and $8g$ of sugar, and also indifferent between $8g$ and $9g$ of sugar, but preferring $9g$ to $7g$? The relation on $X=\{7,8,9\}$ is given as $\le=\{(7,7),(7,8),(7,9),(8,7),(8,8),(8,9),(9,8),(9,9)\}=X^2\setminus\{(9,7)\}$. I will leave to you to prove that the implication in the definition of quasitransitivity:
$$(a\text{ T }b)\land\lnot(b\text{ T }a)\land(b\text{ T }c)\land\lnot(c\text{ T }b)\implies(a\text{ T }c)\land\lnot(c\text{ T }a)$$
is always true because the left side of it is always false (as the first two terms imply $a=7,b=9$ but the second two term imply $b=7,c=9$. However, the relation is not transitive as $9\le 8$ and $8\le 7$ but $9\not\le 7$.