Real Numbers under Multiplication form Monoid

Theorem

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

Proof

Taking the monoid axioms in turn:

Monoid Axiom $\text S 0$: Closure

Real Multiplication is Closed.

$\Box$

Monoid Axiom $\text S 1$: Associativity

Real Multiplication is Associative.

$\Box$

Monoid Axiom $\text S 2$: Identity

Real Multiplication Identity is $1$.

$\Box$

Hence the result.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.4$. Gruppoids, semigroups and groups: Example $77$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids