Coprimality Relation is Non-Transitive

Theorem
Consider the coprimality relation on the set of integers:
 * $\forall x, y \in \Z: x \perp y \iff \gcd \set {x, y} = 1$

where $\gcd \set {x, y}$ denotes the greatest common divisor of $x$ and $y$.

Then:
 * $\perp$ is non-transitive.

Proof
Proof by Counterexample:

We have:

Hence we have:
 * $2 \perp 3$ and $3 \perp 4$

However, it is not the case that $2 \perp 4$.

Thus $\perp$ is not transitive.

Then we have:


 * $2 \perp 3$ and $3 \perp 5$

and also:
 * $2 \perp 5$

Thus $\perp$ is not antitransitive either.

The result follows by definition of non-transitive relation.