Existence of Unique Difference between Cuts
Jump to navigation
Jump to search
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