Real Numbers under Addition form Abelian Group

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

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


Taking the group axioms in turn:

G0: Closure

Real Addition is Closed.


G1: Associativity

Real Addition is Associative.


G2: Identity

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


G3: Inverses

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


C: Commutativity

Real Addition is Commutative.



Real Numbers are Uncountably Infinite.