Definition:Subtraction of Cuts

Definition

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

Let $\beta - \alpha$ denote the operation on $\alpha$ and $\beta$ defined as $\beta + \paren {-\alpha}$.

Then the operator $-$ is known as subtraction, and $\beta - \alpha$ is itself called the difference between $\beta$ and $\alpha$.

Also see

• Existence of Unique Difference between Cuts, which shows that the operation is well-defined

Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.20$. Definition