What is actual difference between transitivity and quasitransitivity?

Question

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.

Answer 1

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 $9\not\le 7$.

Answer 2

Consider the relation $\approx$. To be concrete, consider $S = \{0, 1, 2\}$ with $a \approx b :\equiv (|a - b| \leq 1)$. Then $\approx$ is quasitransitive but not transitive.

Answer 3

There are 3 quasitranitive relations for a set of 2 elements.

{(0, 1), (1, 0)}
{(0, 1), (1, 0), (0, 0)}
{(0, 1), (1, 0), (1, 1)}

There are 277 quasitranitive relations for a set of 3 elements.

[(0, 1), (1, 0)]
[(0, 2), (2, 0)]
[(1, 2), (2, 1)]
[(0, 0), (0, 1), (1, 0)]
[(0, 0), (0, 2), (2, 0)]
[(0, 0), (1, 2), (2, 1)]
[(0, 1), (0, 2), (1, 0)]
[(0, 1), (0, 2), (2, 0)]
[(0, 1), (1, 0), (1, 1)]
[(0, 1), (1, 0), (1, 2)]
[(0, 1), (1, 0), (2, 0)]
[(0, 1), (1, 0), (2, 1)]
[(0, 1), (1, 0), (2, 2)]
[(0, 1), (1, 2), (2, 1)]
[(0, 2), (1, 0), (2, 0)]
[(0, 2), (1, 1), (2, 0)]
[(0, 2), (1, 2), (2, 0)]
[(0, 2), (1, 2), (2, 1)]
[(0, 2), (2, 0), (2, 1)]
[(0, 2), (2, 0), (2, 2)]
[(1, 0), (1, 2), (2, 1)]
[(1, 1), (1, 2), (2, 1)]
[(1, 2), (2, 0), (2, 1)]
[(1, 2), (2, 1), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 0)]
[(0, 0), (0, 1), (0, 2), (2, 0)]
[(0, 0), (0, 1), (1, 0), (1, 2)]
[(0, 0), (0, 1), (1, 0), (2, 0)]
[(0, 0), (0, 1), (1, 0), (2, 1)]
[(0, 0), (0, 1), (1, 0), (2, 2)]
[(0, 0), (0, 1), (1, 2), (2, 1)]
[(0, 0), (0, 2), (1, 0), (2, 0)]
[(0, 0), (0, 2), (1, 1), (2, 0)]
[(0, 0), (0, 2), (1, 2), (2, 0)]
[(0, 0), (0, 2), (1, 2), (2, 1)]
[(0, 0), (0, 2), (2, 0), (2, 1)]
[(0, 0), (1, 0), (1, 2), (2, 1)]
[(0, 0), (1, 1), (1, 2), (2, 1)]
[(0, 0), (1, 2), (2, 0), (2, 1)]
[(0, 0), (1, 2), (2, 1), (2, 2)]
[(0, 1), (0, 2), (1, 0), (1, 1)]
[(0, 1), (0, 2), (1, 0), (1, 2)]
[(0, 1), (0, 2), (1, 0), (2, 0)]
[(0, 1), (0, 2), (1, 0), (2, 1)]
[(0, 1), (0, 2), (1, 0), (2, 2)]
[(0, 1), (0, 2), (1, 1), (2, 0)]
[(0, 1), (0, 2), (1, 2), (2, 0)]
[(0, 1), (0, 2), (1, 2), (2, 1)]
[(0, 1), (0, 2), (2, 0), (2, 1)]
[(0, 1), (0, 2), (2, 0), (2, 2)]
[(0, 1), (1, 0), (1, 1), (1, 2)]
[(0, 1), (1, 0), (1, 1), (2, 0)]
[(0, 1), (1, 0), (1, 1), (2, 1)]
[(0, 1), (1, 0), (1, 1), (2, 2)]
[(0, 1), (1, 0), (1, 2), (2, 0)]
[(0, 1), (1, 0), (1, 2), (2, 1)]
[(0, 1), (1, 0), (1, 2), (2, 2)]
[(0, 1), (1, 0), (2, 0), (2, 1)]
[(0, 1), (1, 0), (2, 0), (2, 2)]
[(0, 1), (1, 0), (2, 1), (2, 2)]
[(0, 1), (1, 1), (1, 2), (2, 1)]
[(0, 1), (1, 2), (2, 0), (2, 1)]
[(0, 1), (1, 2), (2, 1), (2, 2)]
[(0, 2), (1, 0), (1, 1), (2, 0)]
[(0, 2), (1, 0), (1, 2), (2, 0)]
[(0, 2), (1, 0), (1, 2), (2, 1)]
[(0, 2), (1, 0), (2, 0), (2, 1)]
[(0, 2), (1, 0), (2, 0), (2, 2)]
[(0, 2), (1, 1), (1, 2), (2, 0)]
[(0, 2), (1, 1), (1, 2), (2, 1)]
[(0, 2), (1, 1), (2, 0), (2, 1)]
[(0, 2), (1, 1), (2, 0), (2, 2)]
[(0, 2), (1, 2), (2, 0), (2, 1)]
[(0, 2), (1, 2), (2, 0), (2, 2)]
[(0, 2), (1, 2), (2, 1), (2, 2)]
[(0, 2), (2, 0), (2, 1), (2, 2)]
[(1, 0), (1, 1), (1, 2), (2, 1)]
[(1, 0), (1, 2), (2, 0), (2, 1)]
[(1, 0), (1, 2), (2, 1), (2, 2)]
[(1, 1), (1, 2), (2, 0), (2, 1)]
[(1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 2)]
[(0, 0), (0, 1), (0, 2), (1, 0), (2, 0)]
[(0, 0), (0, 1), (0, 2), (1, 0), (2, 1)]
[(0, 0), (0, 1), (0, 2), (1, 0), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 1), (2, 0)]
[(0, 0), (0, 1), (0, 2), (1, 2), (2, 0)]
[(0, 0), (0, 1), (0, 2), (1, 2), (2, 1)]
[(0, 0), (0, 1), (0, 2), (2, 0), (2, 1)]
[(0, 0), (0, 1), (0, 2), (2, 0), (2, 2)]
[(0, 0), (0, 1), (1, 0), (1, 1), (1, 2)]
[(0, 0), (0, 1), (1, 0), (1, 1), (2, 0)]
[(0, 0), (0, 1), (1, 0), (1, 1), (2, 1)]
[(0, 0), (0, 1), (1, 0), (1, 2), (2, 0)]
[(0, 0), (0, 1), (1, 0), (1, 2), (2, 1)]
[(0, 0), (0, 1), (1, 0), (1, 2), (2, 2)]
[(0, 0), (0, 1), (1, 0), (2, 0), (2, 1)]
[(0, 0), (0, 1), (1, 0), (2, 0), (2, 2)]
[(0, 0), (0, 1), (1, 0), (2, 1), (2, 2)]
[(0, 0), (0, 1), (1, 1), (1, 2), (2, 1)]
[(0, 0), (0, 1), (1, 2), (2, 0), (2, 1)]
[(0, 0), (0, 1), (1, 2), (2, 1), (2, 2)]
[(0, 0), (0, 2), (1, 0), (1, 1), (2, 0)]
[(0, 0), (0, 2), (1, 0), (1, 2), (2, 0)]
[(0, 0), (0, 2), (1, 0), (1, 2), (2, 1)]
[(0, 0), (0, 2), (1, 0), (2, 0), (2, 1)]
[(0, 0), (0, 2), (1, 0), (2, 0), (2, 2)]
[(0, 0), (0, 2), (1, 1), (1, 2), (2, 0)]
[(0, 0), (0, 2), (1, 1), (1, 2), (2, 1)]
[(0, 0), (0, 2), (1, 1), (2, 0), (2, 1)]
[(0, 0), (0, 2), (1, 2), (2, 0), (2, 1)]
[(0, 0), (0, 2), (1, 2), (2, 0), (2, 2)]
[(0, 0), (0, 2), (1, 2), (2, 1), (2, 2)]
[(0, 0), (0, 2), (2, 0), (2, 1), (2, 2)]
[(0, 0), (1, 0), (1, 1), (1, 2), (2, 1)]
[(0, 0), (1, 0), (1, 2), (2, 0), (2, 1)]
[(0, 0), (1, 0), (1, 2), (2, 1), (2, 2)]
[(0, 0), (1, 1), (1, 2), (2, 0), (2, 1)]
[(0, 0), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 1), (0, 2), (1, 0), (1, 1), (1, 2)]
[(0, 1), (0, 2), (1, 0), (1, 1), (2, 0)]
[(0, 1), (0, 2), (1, 0), (1, 1), (2, 1)]
[(0, 1), (0, 2), (1, 0), (1, 1), (2, 2)]
[(0, 1), (0, 2), (1, 0), (1, 2), (2, 0)]
[(0, 1), (0, 2), (1, 0), (1, 2), (2, 1)]
[(0, 1), (0, 2), (1, 0), (1, 2), (2, 2)]
[(0, 1), (0, 2), (1, 0), (2, 0), (2, 1)]
[(0, 1), (0, 2), (1, 0), (2, 0), (2, 2)]
[(0, 1), (0, 2), (1, 0), (2, 1), (2, 2)]
[(0, 1), (0, 2), (1, 1), (1, 2), (2, 0)]
[(0, 1), (0, 2), (1, 1), (1, 2), (2, 1)]
[(0, 1), (0, 2), (1, 1), (2, 0), (2, 1)]
[(0, 1), (0, 2), (1, 1), (2, 0), (2, 2)]
[(0, 1), (0, 2), (1, 2), (2, 0), (2, 1)]
[(0, 1), (0, 2), (1, 2), (2, 0), (2, 2)]
[(0, 1), (0, 2), (1, 2), (2, 1), (2, 2)]
[(0, 1), (0, 2), (2, 0), (2, 1), (2, 2)]
[(0, 1), (1, 0), (1, 1), (1, 2), (2, 0)]
[(0, 1), (1, 0), (1, 1), (1, 2), (2, 1)]
[(0, 1), (1, 0), (1, 1), (1, 2), (2, 2)]
[(0, 1), (1, 0), (1, 1), (2, 0), (2, 1)]
[(0, 1), (1, 0), (1, 1), (2, 0), (2, 2)]
[(0, 1), (1, 0), (1, 1), (2, 1), (2, 2)]
[(0, 1), (1, 0), (1, 2), (2, 0), (2, 1)]
[(0, 1), (1, 0), (1, 2), (2, 0), (2, 2)]
[(0, 1), (1, 0), (1, 2), (2, 1), (2, 2)]
[(0, 1), (1, 0), (2, 0), (2, 1), (2, 2)]
[(0, 1), (1, 1), (1, 2), (2, 0), (2, 1)]
[(0, 1), (1, 1), (1, 2), (2, 1), (2, 2)]
[(0, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 2), (1, 0), (1, 1), (1, 2), (2, 0)]
[(0, 2), (1, 0), (1, 1), (1, 2), (2, 1)]
[(0, 2), (1, 0), (1, 1), (2, 0), (2, 1)]
[(0, 2), (1, 0), (1, 1), (2, 0), (2, 2)]
[(0, 2), (1, 0), (1, 2), (2, 0), (2, 1)]
[(0, 2), (1, 0), (1, 2), (2, 0), (2, 2)]
[(0, 2), (1, 0), (1, 2), (2, 1), (2, 2)]
[(0, 2), (1, 0), (2, 0), (2, 1), (2, 2)]
[(0, 2), (1, 1), (1, 2), (2, 0), (2, 1)]
[(0, 2), (1, 1), (1, 2), (2, 0), (2, 2)]
[(0, 2), (1, 1), (1, 2), (2, 1), (2, 2)]
[(0, 2), (1, 1), (2, 0), (2, 1), (2, 2)]
[(0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]
[(1, 0), (1, 1), (1, 2), (2, 0), (2, 1)]
[(1, 0), (1, 1), (1, 2), (2, 1), (2, 2)]
[(1, 0), (1, 2), (2, 0), (2, 1), (2, 2)]
[(1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 0)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 1)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 2), (2, 0)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 2), (2, 1)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 2), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 0), (2, 0), (2, 1)]
[(0, 0), (0, 1), (0, 2), (1, 0), (2, 0), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 0), (2, 1), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 1), (1, 2), (2, 0)]
[(0, 0), (0, 1), (0, 2), (1, 1), (1, 2), (2, 1)]
[(0, 0), (0, 1), (0, 2), (1, 1), (2, 0), (2, 1)]
[(0, 0), (0, 1), (0, 2), (1, 1), (2, 0), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1)]
[(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 2), (2, 1), (2, 2)]
[(0, 0), (0, 1), (1, 0), (1, 1), (1, 2), (2, 0)]
[(0, 0), (0, 1), (1, 0), (1, 1), (1, 2), (2, 1)]
[(0, 0), (0, 1), (1, 0), (1, 1), (1, 2), (2, 2)]
[(0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 2)]
[(0, 0), (0, 1), (1, 0), (1, 1), (2, 1), (2, 2)]
[(0, 0), (0, 1), (1, 0), (1, 2), (2, 0), (2, 1)]
[(0, 0), (0, 1), (1, 0), (1, 2), (2, 0), (2, 2)]
[(0, 0), (0, 1), (1, 0), (1, 2), (2, 1), (2, 2)]
[(0, 0), (0, 1), (1, 0), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 1), (1, 1), (1, 2), (2, 0), (2, 1)]
[(0, 0), (0, 1), (1, 1), (1, 2), (2, 1), (2, 2)]
[(0, 0), (0, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0)]
[(0, 0), (0, 2), (1, 0), (1, 1), (1, 2), (2, 1)]
[(0, 0), (0, 2), (1, 0), (1, 1), (2, 0), (2, 1)]
[(0, 0), (0, 2), (1, 0), (1, 1), (2, 0), (2, 2)]
[(0, 0), (0, 2), (1, 0), (1, 2), (2, 0), (2, 1)]
[(0, 0), (0, 2), (1, 0), (1, 2), (2, 1), (2, 2)]
[(0, 0), (0, 2), (1, 0), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 2), (1, 1), (1, 2), (2, 0), (2, 1)]
[(0, 0), (0, 2), (1, 1), (1, 2), (2, 0), (2, 2)]
[(0, 0), (0, 2), (1, 1), (1, 2), (2, 1), (2, 2)]
[(0, 0), (0, 2), (1, 1), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 0), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1)]
[(0, 0), (1, 0), (1, 1), (1, 2), (2, 1), (2, 2)]
[(0, 0), (1, 0), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0)]
[(0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 1)]
[(0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 2)]
[(0, 1), (0, 2), (1, 0), (1, 1), (2, 0), (2, 1)]
[(0, 1), (0, 2), (1, 0), (1, 1), (2, 0), (2, 2)]
[(0, 1), (0, 2), (1, 0), (1, 1), (2, 1), (2, 2)]
[(0, 1), (0, 2), (1, 0), (1, 2), (2, 0), (2, 1)]
[(0, 1), (0, 2), (1, 0), (1, 2), (2, 0), (2, 2)]
[(0, 1), (0, 2), (1, 0), (1, 2), (2, 1), (2, 2)]
[(0, 1), (0, 2), (1, 0), (2, 0), (2, 1), (2, 2)]
[(0, 1), (0, 2), (1, 1), (1, 2), (2, 0), (2, 1)]
[(0, 1), (0, 2), (1, 1), (1, 2), (2, 0), (2, 2)]
[(0, 1), (0, 2), (1, 1), (2, 0), (2, 1), (2, 2)]
[(0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 1), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1)]
[(0, 1), (1, 0), (1, 1), (1, 2), (2, 0), (2, 2)]
[(0, 1), (1, 0), (1, 1), (1, 2), (2, 1), (2, 2)]
[(0, 1), (1, 0), (1, 1), (2, 0), (2, 1), (2, 2)]
[(0, 1), (1, 0), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 1), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1)]
[(0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 2)]
[(0, 2), (1, 0), (1, 1), (1, 2), (2, 1), (2, 2)]
[(0, 2), (1, 0), (1, 1), (2, 0), (2, 1), (2, 2)]
[(0, 2), (1, 0), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 2), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 1)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 0), (2, 1)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 0), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 1), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 2), (2, 0), (2, 1)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 2), (2, 0), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 2), (2, 1), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 0), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 1), (1, 2), (2, 0), (2, 1)]
[(0, 0), (0, 1), (0, 2), (1, 1), (1, 2), (2, 0), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 1), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1)]
[(0, 0), (0, 1), (1, 0), (1, 1), (1, 2), (2, 0), (2, 2)]
[(0, 0), (0, 1), (1, 0), (1, 1), (1, 2), (2, 1), (2, 2)]
[(0, 0), (0, 1), (1, 0), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 1), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1)]
[(0, 0), (0, 2), (1, 0), (1, 1), (1, 2), (2, 1), (2, 2)]
[(0, 0), (0, 2), (1, 0), (1, 1), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 2), (1, 0), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 2), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1)]
[(0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 2)]
[(0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 1), (2, 2)]
[(0, 1), (0, 2), (1, 0), (1, 1), (2, 0), (2, 1), (2, 2)]
[(0, 1), (0, 2), (1, 0), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 1), (0, 2), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 1), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 1), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 0), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 1), (0, 2), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 1), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 0), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
[(0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]

These data was created with the following python program.

from  itertools import product, chain, combinations
NumberOfElements=3
def is_tranitive(rel):
    for triple in product(list(range(NumberOfElements)),repeat=3):
        if ((triple[0],triple[1]) in rel 
        and (triple[1],triple[2]) in rel 
        and not (triple[0],triple[2]) in rel):
            return False
    return True

def is_quasitranitive(rel):
    for triple in product(list(range(NumberOfElements)),repeat=3):
        if ((triple[0],triple[1]) in rel 
        and (triple[1],triple[0]) not in rel 
        and (triple[1],triple[2]) in rel 
        and (triple[2],triple[1]) not in rel 
        and  not((triple[0],triple[2]) in rel
        and (triple[2],triple[1]) not in rel)):
            return False
    return True


crossproduct=list(product(list(range(NumberOfElements)),repeat=2))
powerset=chain.from_iterable(combinations(crossproduct, r) for r in range(len(crossproduct)+1))
cnt_quasi=0
for x in powerset:
    y=set(x)
    if is_quasitranitive(y):
        if not is_tranitive(y):
            cnt_quasi+=1
            print(sorted(x))
print('There are ',cnt_quasi,'quasitranitive relations for a set of',NumberOfElements,'elements.')