Rational Numbers under Multiplication form Monoid

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Theorem

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


Proof

Taking the monoid axioms in turn:


Monoid Axiom $\text S 0$: Closure

Rational Multiplication is Closed.

$\Box$


Monoid Axiom $\text S 1$: Associativity

Rational Multiplication is Associative.

$\Box$


Monoid Axiom $\text S 2$: Identity

Rational Multiplication Identity is $1$.

$\Box$


Hence the result.

$\blacksquare$


Sources