Real Numbers under Addition form Monoid

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Proof

Taking the monoid axioms in turn:


S0: Closure

Real Addition is Closed.

$\Box$


S1: Associativity

Real Addition is Associative.

$\Box$


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.

$\blacksquare$


Sources