Definition:Addition of Cuts

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.