Help understanding Dedekind cuts

Question

I have an exam this tuesday and our prof gave us these problems to practice. Me and my friend were trying to do it. but I really never understood the concept or if its right.

These are the questions:

1) Let $A_1$ be the set of rational numbers $x$ such that $x^2 < 3$, and $A_2$ be the set of rational numbers $x$ such that $x^2 > 3$. Is $(A_1, A_2)$ a dedekind cut? Prove your answer.

2) Let $(A_1, A_2)$ and $(B_1, B_2)$ be dedekind cuts representing real numbers $\alpha$ and $\beta$. Define what $\alpha < \beta$ means in terms of the Dedekind cuts.

3) Give an example of two Dedekind cuts $(A_1, A_2)$ and $(B_1, B_2)$ be dedekind cuts representing real numbers $\alpha$ and $\beta$ such that $A_1$ strictly contains $B_1$ but $\alpha$ is not strictly larger than $\beta$.

4) Give an example of two "unessentially different" Dedekind cuts.

5) Is it true that a rational number can be represented by two "unesentially different" Dedekind cuts? Explain

Our answers:

1) for this one the teacher told us that is $A_1 = \{x \in \mathbb{Q} | x^2 <3\}$ and $A_2 = \{x \in \mathbb{Q} | x^2 >3\}$ she said is $0 \in A_1$ and $-100 \in A_2$ and that implies its not a dedekind cut. Am I missing something here? like where did she get 0 and -100 from.

2) and 3) we didnt get any help on this would be appreciated.

4) So if we let $(A_1 = (\mathbb{Q} \cap (-\infty , 2], A_2 = (2, \infty) \cap \mathbb{Q})$ and if $(B_1 = (\mathbb{Q} \cap (-\infty , 2), B_2 = [2, \infty) \cap \mathbb{Q})$. I dont know if that is right. please help out.

and need help on 5)

Any help on this would be greatly appreciated.

Thank you very much

Answer 1

(1) If $(A_1,A_2)$ were a Dedekind cut, every element of $A_1$ would have to be less than every element of $A_2$: that’s part of the definition. However, this is not the case, because (for instance) $0\in A_1$, $-100\in A_2$, and $0\not<-100$. She could have used many other examples: $1\in A_1$, $-2\in A_2$, and $1\not<-2$, for instance.

For the remainder I’m going to assume that a Dedekind cut is a pair $(A_1,A_2)$ of disjoint, non-empty subsets of $\Bbb Q$ such that

• $A_1\cup A_2=\Bbb Q$;
• if $a<b\in A_1$, then $a\in A_1$; and
• if $a>b\in A_2$, then $a\in A_2$.

There are other ways to define the notion, but the remaining questions suggest that this is how you’ve done it.

(2) A first approximation is that $\alpha<\beta$ if $A_1\subsetneqq B_1$. That’s the right basic idea, but parts (3) and (4) show that it doesn’t quite work: it has to be modified to take into account the possibility that $\alpha$ and $\beta$ are inessentially different Dedekind cuts. I suggest that you return to this part after you’re sure that you understand (3)-(5): at that point you should see how the condition $A_1\subsetneqq B_1$ has to be strengthened.

(3) Let $A_1=\{q\in\Bbb Q:q\le 0\}$ and $B_1=\{q\in\Bbb Q:q<0\}$; clearly we must have $A_2=\Bbb Q\setminus A_1$ and $B_2=\Bbb Q\setminus B_1$ in order to have Dedekind cuts. In other words, $A_2$ is the set of positive rationals, and $B_2$ is the set of non-negative rationals. Clearly $A_1\supsetneqq B_1$, but if $\alpha=(A_1,A_2)$ and $\beta=(B_1,B_2)$, then $\alpha=\beta$; why?

(4) Your example is correct. (In fact it’s based on exactly the same idea as mine for (3).)

(5) What you did in (4) for the rational number $2$ and I did in (3) for the rational number $0$ can be done for any rational number $q$. Given $q\in\Bbb Q$, what two Dedekind cuts both correspond to $q$?