\begin{definition}[Definition:Addition of Cuts]
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 $\beta$'''.
\end{definition}