Set of Cuts under Addition forms Abelian Group
Theorem
Let $\CC$ denote the set of cuts.
Let $\struct {\CC, +}$ denote the algebraic structure formed from $\CC$ and the operation $+$ of addition of cuts.
Then $\struct {\CC, +}$ forms an abelian group.
Proof
In the below, $\alpha$, $\beta$ and $\gamma$ denote arbitrary cuts.
Taking the abelian group axioms in turn:
Group Axiom $\text G 0$: Closure
From Sum of Cuts is Cut:
- $\forall \alpha, \beta \in \CC: \alpha + \beta \in \CC$
Thus $\struct {\CC, +}$ is closed.
$\Box$
Group Axiom $\text G 1$: Associativity
From Addition of Cuts is Associative:
- $\paren {\alpha + \beta} + \gamma = \alpha + \paren {\beta + \gamma}$
Thus the operation $+$ of addition of cuts is associative on $\struct {\CC, +}$.
$\Box$
$\text C$: Commutativity
From Addition of Cuts is Commutative:
- $\alpha + \beta = \beta + \alpha$
Thus the operation $+$ of addition of cuts is commutative on $\struct {\CC, +}$.
$\Box$
Group Axiom $\text G 2$: Existence of Identity Element
Consider the rational cut $0^*$ associated with the (rational) number $0$:
- $0^* = \set {r \in \Q: r < 0}$
From Identity Element for Addition of Cuts:
- $\alpha + 0^* = \alpha$
for all $\alpha \in \CC$.
From Addition of Cuts is Commutative it follows that:
- $0^* + \alpha = \alpha$
That is, $0^*$ is the identity element of $\struct {\CC, +}$.
$\Box$
Group Axiom $\text G 3$: Existence of Inverse Element
We have that $0^*$ is the identity element of $\struct {\CC, +}$.
From Existence of Unique Inverse Element for Addition of Cuts, there exists a unique cut $-\alpha$ such that:
- $\alpha + \paren {-\alpha} = 0^*$
From Addition of Cuts is Commutative it follows that:
- $\paren {-\alpha} + \alpha = 0^*$
Thus every element of $\struct {\CC, +}$ has an inverse $-\alpha$.
$\Box$
All the abelian group axioms are thus seen to be fulfilled, and so $\struct {\CC, +}$ is a group.
$\blacksquare$
Sources
- 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.21$. Remark