Real Numbers under Addition form Group

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

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


Taking the group axioms in turn:

Group Axiom $\text G 0$: Closure

Real Addition is Closed.


Group Axiom $\text G 1$: Associativity

Real Addition is Associative.


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 $0$.


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 $-x$.