Rational Numbers under Multiplication form Commutative Monoid

Theorem
The set of rational numbers under multiplication $\struct {\Q, \times}$ forms a countably infinite commutative monoid.

Proof
From Rational Numbers under Multiplication form Monoid, $\struct {\Q, \times}$ is a monoid.

Then:
 * from Rational Multiplication is Commutative we have that $\times$ is commutative on $\Q$
 * from Rational Numbers are Countably Infinite we have that $\Q$ is a countably infinite set.