Coprimality Relation is not Antisymmetric

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 not antisymmetric.

Proof

Proof by Counterexample:

We have:

$\gcd \set {3, 5} = 1 = \gcd \set {5, 3}$

and so:

$3 \perp 5$ and $5 \perp 3$

However, it is not the case that $3 = 5$.

The result follows by definition of antisymmetric relation.

$\blacksquare$