Existence of Unique Difference between Cuts

Theorem

Let $\alpha$ and $\beta$ be cuts.

Then there exists exactly one cut $\gamma$ such that:

$\alpha + \gamma = \beta$

Proof

From Ordering on Cuts is Compatible with Addition of Cuts:

$\gamma_1 \ne \gamma_2 \implies \alpha + \gamma_1 \ne \alpha + \gamma_2$

That demonstrates uniqueness.

Let:

$\gamma = \beta + \paren {-\alpha}$

where $-\alpha$ is the negative of $\alpha$.

Then by Identity Element for Addition of Cuts:

\(\ds \alpha + \gamma\)

\(=\)

\(\ds \alpha + \paren {\beta + \paren {-\alpha} }\)

Definition of $\gamma$

\(\ds \)

\(=\)

\(\ds \alpha + \paren {\paren {-\alpha} + \beta}\)

Addition of Cuts is Commutative

\(\ds \)

\(=\)

\(\ds \paren {\alpha + \paren {-\alpha} } + \beta\)

Addition of Cuts is Associative

\(\ds \)

\(=\)

\(\ds 0^* + \beta\)

Existence of Unique Inverse Element for Addition of Cuts

\(\ds \)

\(=\)

\(\ds \beta\)

Identity Element for Addition of Cuts

The result follows.

$\blacksquare$

Also see

• Definition:Subtraction of Cuts: $\gamma$ is defined as $\beta - \alpha$, the difference between $\beta$ and $\alpha$

Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.19$. Theorem