# Definition:Ordering of Cuts

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## 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 $\alpha \le \beta$

- $\alpha = \beta$ 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 $\alpha < \beta$

- 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