User:Ascii/Coprime Relation for Integers is Non-Transitive

Theorem

The relation "is coprime to" on the integers is non-transitive.

That is, it is neither transitive:

$\exists n \in \Z: \neg n \perp n$

nor antireflexive:

$\exists n \in \Z: n \perp n$

where $\perp$ denotes "is coprime to".

Proof

Coprime is Not Transitive

2 perp 1 and 1 perp 2 but 2 perp 2? I think not

Coprime is Not Antitransitive

1 perp -1 and -1 perp 1 and 1 perp 1! So, not antitransitive