Real Numbers under Multiplication form Monoid

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Theorem

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


Proof

Taking the monoid axioms in turn:


$\text S 0$: Closure

Real Multiplication is Closed.

$\Box$


$\text S 1$: Associativity

Real Multiplication is Associative.

$\Box$


$\text S 2$: Identity

Real Multiplication Identity is $1$.

$\Box$


Hence the result.

$\blacksquare$


Sources