Definition:Ordering of Cuts/Strict
Jump to navigation
Jump to search
Definition
Let $\alpha$ and $\beta$ be cuts.
$\alpha$ and $\beta$ conventionally have the following strict ordering imposed on them, as follows:
- $\alpha$ is less than $\beta$, denoted $\alpha < \beta$
- there exists a rational number $p \in \Q$ such that $p \in \alpha$ but $p \notin \beta$.
This can also be expressed as $\beta > \alpha$.
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