Coprimality Relation is Non-Transitive

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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:

\(\displaystyle \gcd \set {2, 3}\) \(=\) \(\displaystyle 1\)
\(\displaystyle \gcd \set {3, 4}\) \(=\) \(\displaystyle 1\)
\(\displaystyle \gcd \set {2, 4}\) \(=\) \(\displaystyle 2\)


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:

\(\displaystyle \gcd \set {2, 3}\) \(=\) \(\displaystyle 1\)
\(\displaystyle \gcd \set {3, 5}\) \(=\) \(\displaystyle 1\)
\(\displaystyle \gcd \set {2, 5}\) \(=\) \(\displaystyle 1\)
$2 \perp 3$ and $3 \perp 5$

and also:

$2 \perp 5$

Thus $\perp$ is not antitransitive either.


The result follows by definition of non-transitive relation.

$\blacksquare$