# Additive inverse of Dedekind cut

• math771
In summary, the conversation is about trying to prove the existence of an additive inverse for every Dedekind cut A, using the definitions of addition of cuts and 0*. The conversation also discusses the Archimedean property and its relation to the additive inverse, and seeks advice on how to proceed with the proof.
math771
Hi there! I'm trying to prove that every Dedekind cut A has an additive inverse -A such that A + (-A)=0*. We've defined addition of cuts A and B to mean A + B= {a + b|a belongs to A and b belongs to B} and 0*={x|x<0 and x belongs to Q}.
Now, it seems fairly clear that the additive inverse of A will be -A={-a'|a' belongs to Q\A}. Furthermore, it's relatively easy to show that A + (-A) is a subset of 0*. For a belonging to cut A and a' belonging to cut Q\A, a < a'. Thus, a - a' < 0. Because A + (-A)={a - a'|a belongs to A and a' belongs to Q\A}, A + (-A) is a subset of 0*.
Now, I need show that 0* is a subset of A + (-A). To do that, I was thinking that it would suffice to show that for any x<0, there exist a belonging to A and -a' belonging to -A such that x $\leq$ a - a' < 0. Because A + (-A) is itself a cut (something we proved earlier) x $\leq$ a - a' < 0 would imply that all members of 0* belong to A + (-A).
I'll give my start at the proof. x $\leq$ a - a' < 0 is equivalent to -x $\geq$ a' - a > 0. Suppose, then, that 0 < -x < a' - a for all a and -a' beloning to A and -A respectively. At this point, perhaps we could use something like the Archimedian property but for rational numbers so that a' - a < k(-x) and (a' - a)/k < -x for some natural number k. Though, it seems we would have to show that -a'/k and a/k belong to -A and A respectively (correct me if I'm wrong), which would be odd if k were a natural number (and difficult (for me) to prove even if k were not natural). One question I would like to pose, then, is whether the Archimedean property requires that k be a natural number. In most proofs I've seen, it's stated that k is natural, but I fail to see the necessity of this requirement.
Even more troubling however is the fact that I've been led to believe that the Archimedean property is a consequence of what I'm trying to prove here. But I don't see how this could be so. The proof of the Archimedean property that I am familiar with relies on the connectedness of real numbers, but, if I'm not mistaken, it is possible to prove that the real numbers (taken to be Dedekind cuts) are connected without ever talking of the additive identities of cuts.
Any advice would be much appreciated. Thanks!

Hi math771!

You're not using the correct definition of the additive inverse here. Indeed, we expect the additive inverse of 0* to be 0*, but in this case it is

$$\{x\in\mathbb{Q}~\vert~x\leq 0\}$$

But we want a strict inequality there... So you'll need to eliminate the largest element everytime...

## 1. What is the definition of additive inverse of Dedekind cut?

The additive inverse of a Dedekind cut is a number that when added to the original number, results in zero. In other words, it is the opposite of the original number.

## 2. How is the additive inverse of Dedekind cut represented?

The additive inverse of a Dedekind cut is represented by a negative sign (-) placed in front of the original number. For example, the additive inverse of the Dedekind cut (1,3) would be (-1,-3).

## 3. How is the additive inverse of Dedekind cut calculated?

To calculate the additive inverse of a Dedekind cut, you must first determine the complement of the original cut. This is done by subtracting each element in the original cut from the rational number 1. The resulting cut is the additive inverse.

## 4. How does the additive inverse of Dedekind cut relate to zero?

The additive inverse of a Dedekind cut is closely related to zero because when added together, they result in zero. This is because the Dedekind cut represents all rational numbers greater than the cut, while its additive inverse represents all rational numbers less than the cut.

## 5. What is the significance of the additive inverse of Dedekind cut in mathematics?

The additive inverse of a Dedekind cut is an important concept in mathematics, specifically in the study of real numbers. It allows for the representation of negative numbers and is essential in operations such as subtraction and division. It also plays a role in understanding the completeness and order of the real numbers.

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