# Real Numbers under Addition form Abelian Group

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## Contents

## 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:

### G0: Closure

$\Box$

### G1: Associativity

$\Box$

### G2: Identity

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

$\Box$

### G3: Inverses

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

$\Box$

### C: Commutativity

$\Box$

### Infinite

Real Numbers are Uncountably Infinite.

$\blacksquare$

## Sources

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*Sets and Groups*... (previous) ... (next): $\S 4.5$. Examples of groups: Example $81$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 7$: Example $7.1$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 34$. Examples of groups: $(1)$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Examples of groups $\text{(i)}$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $1$: Definitions and Examples: Example $1.4$

- 1974: Thomas W. Hungerford:
*Algebra*... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups