# Transitive Relation over Set - Feedbacks on proofwriting skills

1. Dec 10, 2012

### Kolmin

1. The problem statement, all variables and given/known data

Assume a relation $P$ that is negatively transitive on a set $X$ that is not empty.
Define the binary relation $R$ on $X$ by $xRy$ iff $y P x$ is false.

Prove that $R$ is transitive.

2. Relevant equations

Negative Transitivity: $xPz \rightarrow xPy \vee yPz$

Like in the previous thread (Complete Relation), I think I have a proof, but I am really looking forward to any feedback on the style of this proof. I tried to incorporate the stylistic feedbacks of Michael Redei and pasmith who kindly gave me feedbacks on the previous attempt.

3. The attempt at a solution

Proof:
Let $x$, $y$ and $z$ be arbitrary and assume $xRy$ and $yRz$. By definition of $R$ it follows that $yPx$ and $zPy$ are both false. Since $P$ is negatively transitive, this implies that $zPx$ is false. Thus, again by definition of $R$, the falsity of $zPx$ implies $xRz$, which proves that the relation $R$ is transitive.

Last edited: Dec 11, 2012
2. Dec 10, 2012

### haruspex

I think you mean xPy → xPz V zPy, or somesuch.

3. Dec 11, 2012

### Kolmin

I hate typos...

I edited the previous post: it is $xPz \rightarrow xPy \vee yPz$

4. Dec 11, 2012

### haruspex

With that correction, your proof looks fine.