Understanding Dedekind Cuts: How to Recognize a Cut

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• mikeyBoy83
In summary, a Dedekind cut is a way of splitting the set of all rationals into two nonempty pieces, where every rational is either in the "left" set or the "right" set. The real number associated with the cut is the unique number that is greater than or equal to every element of the left set and less than or equal to every element of the right set. This allows for the construction of real numbers through cuts, where the left set is considered an extended real number.
mikeyBoy83
I'm trying to wrap my head around these Dedekind cuts. The definition is straightforward but I'm a little confused about the downward closure part.

##x \in Q## and ##y<x \Longrightarrow y \in Q##

Does that mean that this is not a cut because it is bounded below?

{## x \in Q : x>1 \wedge x<2 ##}

Clear this up for me please.

mikeyBoy83 said:
I'm trying to wrap my head around these Dedekind cuts. The definition is straightforward but I'm a little confused about the downward closure part.

##x \in Q## and ##y<x \Longrightarrow y \in Q##

Does that mean that this is not a cut because it is bounded below?

{## x \in Q : x>1 \wedge x<2 ##}

Clear this up for me please.

No, it's not a cut. Another way to think of it is that a cut splits the set of all rationals into two nonempty pieces: A "left" set, $L$, and a "right" set, $R$, where
• Every rational $x$ is either in $L$ or $R$.
• If $x$ is in $L$, and $y$ is in $R$, then $x < y$
• The real associated with the pair $L,R$ is the unique number $r$ that is greater than or equal to every element of $L$ and less than or equal to every element of $R$
A Dedekind cut is just such an $L$.

stevendaryl wrote:

The real associated with the pair ##L,R## is the unique number ##r## that is greater than or equal to every element of ##L## and less than or equal to every element of ##R##

Since Dedekind cuts are used to construct the reals I think it would be better to say that the real number ##r## is the cut, or preferably that ##L## as defined by OP is an extended real number.

1. What is a Dedekind cut?

A Dedekind cut is a mathematical concept introduced by German mathematician Richard Dedekind. It is a partition of the rational numbers into two non-empty sets, usually denoted as A and B, such that all the numbers in set A are less than any number in set B. This cut separates the rational numbers into two distinct subsets, one containing all numbers less than a certain value and the other containing all numbers greater than or equal to that value.

2. How do you recognize a Dedekind cut?

A Dedekind cut can be recognized by looking for a point on the number line where all the numbers to the left are less than the point and all the numbers to the right are greater than or equal to the point. In other words, the cut separates the rational numbers into two distinct subsets with no gaps or overlaps.

3. What is the purpose of Dedekind cuts?

Dedekind cuts are used in mathematics to construct the real number system. They provide a way to define irrational numbers, which cannot be expressed as a ratio of two integers, by using the rational numbers as a reference point. This allows for a more complete and precise understanding of the real numbers.

4. Are Dedekind cuts limited to rational numbers?

No, Dedekind cuts can also be applied to other sets of numbers, such as irrational numbers or complex numbers. The basic principle remains the same, where the cut separates the numbers into two distinct subsets with no gaps or overlaps.

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

Dedekind cuts are related to other mathematical concepts, such as equivalence relations and order relations. In fact, Dedekind cuts can be used to define both of these concepts. Equivalence relations are used to classify objects into different categories, while order relations are used to compare objects and determine their relative position. Dedekind cuts provide a way to create order relations between rational numbers and ultimately construct the real number system.

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