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:

Monoid Axiom $\text S 0$: Closure

Real Addition is Closed.

$\Box$

Monoid Axiom $\text S 1$: Associativity

Real Addition is Associative.

$\Box$

Monoid Axiom $\text S 2$: 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.

$\blacksquare$

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