Real Numbers under Addition form Abelian Group

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Theorem

Let $\R$ be the set of real numbers.

The structure $\struct {\R, +}$ is an infinite abelian group.


Proof

Taking the group axioms in turn:


$\text G 0$: Closure

Real Addition is Closed.

$\Box$


$\text G 1$: Associativity

Real Addition is Associative.

$\Box$


$\text G 2$: Identity

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

$\Box$


$\text G 3$: Inverses

From Inverses for Real Addition, we have that the inverse of $x \in \struct {\R, +}$ is $-x$.

$\Box$


$\text C$: Commutativity

Real Addition is Commutative.

$\Box$


Infinite

Real Numbers are Uncountably Infinite.

$\blacksquare$


Sources