# Proving a property of a Dedekind cut

• Hall
In summary, a Dedekind cut is a pair ##(A,B)## of subsets of rationals that satisfy certain properties. In particular, for any given cut, Property 5 holds, which states that if ##b \not\in B##, then either ##b=0## or ##b^2 \leq 2##. The given proof for this property is correct, but it should specify that ##b## is positive and consider both positive and negative cases for ##a##.
Hall
Homework Statement
##A = \{x: x\lt 0 or x^2 \lt 2\}##
##B = \{x: x \gt 0 ~and~x^2 \gt 2\}##

Prove that ##(A,B)## is a Dedekind cut.
Relevant Equations
There are some properties which a cut must have.
A Dedekind cut is a pair ##(A,B)##, where ##A## and ##B## are both subsets of rationals. This pair has to satisfy the following properties
1. A is nonempty
2. B is nonempty
3. If ##a\in A## and ##c \lt a## then ##c \in A##
4. If ##b \in B## and ## c\gt b## then ##c \in B##
5. If ##b \not\in B## and ## a\lt b##, then ##a \in A##
6. If ##a \not\in A## and ##b \gt a##, then ##b \in B##
7. For each ##a \in A## there is some ## b \gt a ## so that ##b \in A##
8. For each ##b \in B## there is some ## a \lt b## so that ##a \in B##

For the given cut, I tried to prove Property 5. Here is my attempt:

If ## b \not \in B## then we have:

Case 1: ## b \leq 0##
Sub-Case (I): ##b=0##. If ##a \lt b \implies a \lt 0##, well then ##a \in A##.
Sub-Case (II): ##b \lt 0##. If ## a \lt b \implies b \lt 0##, again ## a \in A##

Case 2: ## b^2 \leq 2##
Sub-Case (I): ##b^2 = 2##. If ## a \lt b implies a^2 \lt b^2 = 2##, well then ## a \in A##
Sub-Case (II): ## b^2 \lt 2##. If ## a \lt b \implies a^2 \lt b^2 \lt 2##, again ## a \in A##

That proves Property 5.

Am I right? I had a hard time in understanding why would ##b \not\in B## and ##b \not\in A## when all ## a## and ##b## in th properties listed above has to be rational numbers.

It looks ok to me except that your Case 2 hypothesis and proofs should specify that ##b## is positive and consider both positive and negative cases for ##a##. (In fact, maybe you should just change Case 1 to the case of ##a \le 0##.)

## 1. What is a Dedekind cut?

A Dedekind cut is a mathematical concept used to define real numbers in a rigorous way. It is a partition of the rational numbers into two non-empty subsets, where all elements in the first subset are less than any element in the second subset.

## 2. How do you prove a property of a Dedekind cut?

To prove a property of a Dedekind cut, you must first define the property you want to prove. Then, you can use the definition of a Dedekind cut to show that the property holds for all elements in the cut. This can be done through logical reasoning and using the properties of rational numbers.

## 3. What are some common properties of Dedekind cuts?

Some common properties of Dedekind cuts include the completeness property, which states that every cut corresponds to a unique real number, and the density property, which states that between any two distinct real numbers, there is always another real number.

## 4. Can a Dedekind cut represent an irrational number?

Yes, a Dedekind cut can represent an irrational number. In fact, most real numbers are irrational, and can only be represented by Dedekind cuts.

## 5. How are Dedekind cuts used in real analysis?

Dedekind cuts are used in real analysis to provide a rigorous foundation for the real numbers. They allow for the definition of basic operations on real numbers, such as addition and multiplication, and can be used to prove important theorems in analysis, such as the intermediate value theorem and the mean value theorem.

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