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.

$\blacksquare$

Sources

• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups