Real Numbers under Addition form Monoid

Theorem

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

Proof

Taking the monoid axioms in turn:

S0: Closure

$\Box$

S1: Associativity

$\Box$

S2: Identity

From Real Addition Identity is Zero, we have that the identity element of $\struct {\R, +}$ is the real number $0$.

$\Box$

Hence the result.

$\blacksquare$