Definition:Ordering of Cuts

Definition

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

$\alpha$ and $\beta$ conventionally have the following ordering imposed on them, as follows:

$\alpha$ is less than or equal to $\beta$, denoted $\alpha \le \beta$

if and only if:

$\alpha = \beta$ or $\alpha < \beta$

where $<$ denotes the strict ordering relation between $\alpha$ and $\beta$.

This can also be expressed as $\beta \ge \alpha$.

Strict Ordering

$\alpha$ is less than $\beta$, denoted $\alpha < \beta$

if and only if:

there exists a rational number $p \in \Q$ such that $p \in \alpha$ but $p \notin \beta$.

Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.9$. Definition