Real Numbers under Addition form Group

Theorem

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

The structure $\struct {\R, +}$ is a group.

Proof

Taking the group axioms in turn:

Group Axiom $\text G 0$: Closure

Real Addition is Closed.

$\Box$

Group Axiom $\text G 1$: Associativity

Real Addition is Associative.

$\Box$

Group Axiom $\text G 2$: Existence of Identity Element

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

$\Box$

Group Axiom $\text G 3$: Existence of Inverse Element

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

$\blacksquare$

Sources

• 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): $\S 1$: Some examples of groups: Example $1.03$
• 1974: Robert Gilmore: Lie Groups, Lie Algebras and Some of their Applications ... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $2$. GROUP: Example $5$

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