# Definition:Cut (Analysis)

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## Definition

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**.

### Lower Number

Let $\alpha$ be a cut.

Let $p \in \alpha$.

Then $p$ is referred to as a **lower number** of $\alpha$.

### Upper Number

Let $\alpha$ be a cut.

Let $q \in \Q$ such that $q \notin \alpha$.

Then $p$ is referred to as an **upper number** of $\alpha$.

### Rational Cut

Let $r \in \Q$ be a rational number.

Let $\alpha$ be the cut consisting of all rational numbers $p$ such that $p < r$.

Then $\alpha$ is referred to as a **rational cut**.

## Also see

- Results about
**cuts**can be found here.

## Sources

- 1964: Walter Rudin:
*Principles of Mathematical Analysis*(2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.4$. Definition