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
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids