Real Numbers under Addition form Monoid
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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
$\Box$
Monoid Axiom $\text S 1$: Associativity
$\Box$
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