Understanding Dedekind Cuts: Exploring the Negative of a Dedekind Cut

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In summary, Dedekind cuts are sets where every element is equal to or to the left of some other element. This example shows how a cut can be constructed by taking the complement of the set of rationals of elements in the cut.
  • #1
cateater2000
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Hi I'm having trouble understanding dedekind cut.

Suppose ScQ is a Dedekind cut.

i) One might try to define the negative of S as a set

-S={-s in Q : s in S}

Explain why this is not a Dedekind cut.

I have no idea how Dedekind cuts work even after reading over the defn of it 3 or 4 times. Any tips would be great
 
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  • #2
Well, to begin, can you say the definition of a Dedekind cut?
 
  • #3
The definition of "Dedekind cut" is usually given in 3 parts. Which of those parts is violated by {-s | s is in S}? (By the way, it is not necssary to say "-s in Q". Since s is in S, s and -s must be in Q.)
 
  • #4
Definition of dedekind cut:

A set ScQ is a Dedekind set if

1)S is not 0, S is not Q
2)r<s in S -> r in S
3) S doesn't contain it's upper bound.
ie: if s in S, then there exists r in S with r>s


I think I just cannot see which one it disobeys, I might confused with the upper bound part. Anyways any thing will help
 
  • #5
Next step: can you give a couple examples of Dedekind cuts?
 
  • #6
cat: so your definition of a cut basically says it is all rational numbers equal to or to the left of some real number, does that seem right?
 
  • #7
Nvm I understand the question now, thanks for all your help guys
 
  • #8
You "understand" the question. Does that mean you can answer it? ;)

One very simple example of a Dedekind cut is the set of all negative rational numbers. The set of all {-s} where s is in the set of all negative rational numbers is simply the set of all positive integers. That set is not a Dedeking cut because it does not satisfy "if x is in the set and y< x then y is in the set". 1/2 is in the set of all positive rational numbers, 0< 1/2 but 0 is not in the set.
 
  • #9
I haven't taught this topic in a long time and just now I keep thinking " there must be a better way!"

But anyway: suppose we take your suggestion and look at all negatives of elements in our dedekind cut. then that seems to give us the opposite of what we want. so we could then take the complement in the set of rationals of those numbers. now unfortunately that could give us a set containing its lub, so if so we must throw that out. kind of clumsy.


lets try another way. presumably addition is easier, i.e. to add two dedekind cuts probably you just add all their pairs of elements. so we know that the negative of a dedekind cut X should be the solution of the equation X+Y = 0. so maybe we should just take for -X, where X is a dedekind cut, the set of all rationals y such that for all x in X, x+y is negative.

What does that give? Shoot, that also gives a set with a lub in it. Ok, last try:

given a cut X, take the set of all those y in the rationals, such that for each y there exists a positive K, such that for all x in X, x+y is less than -K.

gee this is awful. it would probably be bettter to just notice that you only need to use this construction to construct the positive reals. after that just construct the negative reals formally as a copy of the positive reals with a minus sign attached.

i.e. it is unnecessary to carry the cumbersome construction around for the whole process.
 

1. What are Dedekind cuts?

Dedekind cuts are a mathematical concept developed by German mathematician Richard Dedekind in the late 19th century. They are a way of defining real numbers in terms of rational numbers and are used to construct the real number system, which includes both rational and irrational numbers.

2. How do Dedekind cuts work?

Dedekind cuts work by dividing a set of rational numbers into two subsets: the lower set and the upper set. The lower set contains all rational numbers less than the real number being defined, while the upper set contains all rational numbers greater than or equal to the real number being defined. Together, these two subsets form a Dedekind cut, which represents a unique real number.

3. What is the significance of Dedekind cuts?

Dedekind cuts are significant because they provide a rigorous way of defining the real numbers. Before their development, there was no clear understanding of what exactly real numbers were or how to define them. Dedekind cuts helped to bridge this gap and lay the foundation for modern analysis and calculus.

4. How are Dedekind cuts related to other mathematical concepts?

Dedekind cuts are closely related to other mathematical concepts, such as limits, continuity, and completeness. They can also be used to define operations on real numbers, such as addition, multiplication, and division. Additionally, Dedekind cuts are essential in understanding the concept of irrational numbers, as they provide a way to define them in terms of rational numbers.

5. What are some applications of Dedekind cuts?

Dedekind cuts have various applications in mathematics, including in the study of analysis, topology, and number theory. They are also used in physics and engineering, particularly in the fields of calculus and differential equations. Additionally, Dedekind cuts have practical applications in computer science, such as in the development of algorithms and data structures.

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