# Category:Cuts

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This category contains results about Cuts.

Definitions specific to this category can be found in Definitions/Cuts.

Let $\alpha \subset \Q$ be a subset of the set of rational numbers $\Q$ which has the following properties:

- $(1): \quad \alpha \ne \O$ and $\alpha \ne \Q$, that is: $\alpha$ contains at least one rational number but not all rational numbers

- $(2): \quad$ If $p \in \Q$ and $q \in \Q$ such that $q < p$, then $q \in \Q$

- $(3): \quad \alpha$ does not contain a greatest element.

Then $\alpha$ is called a **cut**.

## Pages in category "Cuts"

The following 35 pages are in this category, out of 35 total.

### A

### C

### E

### M

### O

- Ordered Field of Rational Cuts is Isomorphic to Rational Numbers
- Ordering of Rational Cuts preserves Ordering of Associated Rational Numbers
- Ordering on Cuts is Compatible with Addition of Cuts
- Ordering on Cuts is Compatible with Addition of Cuts/Corollary
- Ordering on Cuts is Total
- Ordering on Cuts is Transitive
- Ordering on Cuts satisfies Trichotomy Law