Definition:Negative of Cut

Definition

Let $\alpha$ be a cut.

Let $0^*$ be the rational cut associated with the (rational) number $0$:

$0^* = \set {r \in \Q: r < 0}$

Let $\beta$ be the unique cut such that:

$\alpha + \beta = 0^*$

where $+$ denotes the operation of addition of cuts.

Then $\beta$ is referred to as the negative of $\alpha$.

It is usually denoted $-\alpha$.

Also see

• Existence of Unique Inverse Element for Addition of Cuts, which proves both existence and uniqueness of $-\alpha$ for a given $\alpha$

Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.17$. Definition