Definition:Addition of Cuts
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Definition
Let $\alpha$ and $\beta$ be cuts.
Let the operation of addition be defined on $\alpha$ and $\beta$ as:
- $\gamma := \alpha + \beta$
where $\gamma$ is the set of all rational numbers $r$ such that:
- $\exists p \in \alpha, q \in \beta: r = p + q$
In this context, $\gamma$ is known as the sum of $\alpha$ and $\beta$.
Also see
- Sum of Cuts is Cut where it is shown that $\gamma$ is also a cut.
Sources
- 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.13$. Definition