Real Numbers under Multiplication form Monoid

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


Taking the monoid axioms in turn:

Monoid Axiom $\text S 0$: Closure

Real Multiplication is Closed.


Monoid Axiom $\text S 1$: Associativity

Real Multiplication is Associative.


Monoid Axiom $\text S 2$: Identity

Real Multiplication Identity is $1$.


Hence the result.