Dedekind Cuts

Hi,

Question:

Prove that the following subsets of Q are Dedekind cuts, or show which of thedefining properties of a Dedekind cut is violated. If it is a Dedekind cut decidewhether it is rational.
(a) C = {x ∈ Q : x − x^2 − 1 ≤ 0}.
(b) D = {x ∈ Q : x^3 − 2x^2 + x < 0} ∩ {x ∈ Q : x ≤1/2}.

My attempt:

I know that:
A subset X ⊂ Q is called a Dedekind cut if
1) X is not equal to ∅ (empty set)
2) X is not equal to Q
3) For all x ∈ X and x' ∈ Q: x' < x implies x' ∈ A.
4) A has no maximal element.

a) C is not a Dedekind cut. The inequality holds for x being a rational number in the interval ( -∞, ∞), so property 2 is violated as Q ⊂ ( -∞, ∞). So C=Q

b) D is not a Dedekind cut. The set has a maximal element 1/2, so property 3 is violated.

Is my deduction for a) correct?

Thanks for any help

I had never heard of this topic before so I can't be of too much help but for the part a) I think this.

You want to show X!=Q i.e that Q\X is empty.

To do this by definition of equality of sets you want to show Q is a subset of X and X is a subset of U.

To show X is subset of Q take an arbitrary x in X then x must be in Q (by definition of X).

Now to show Q is a subset of X take some arbitrary q in Q and then you need to show that the equality holds. You have just stated it in your answer but I think more justification would be required you actually need to show this holds for all q in Q.

If you can do this then indeed X=Q and hence X is not a Dedekind cut.

For the second part can't you just show that 1/2 is in D then you are done. (Since anything greater would not be in D because of the second set in the intersection). I think you would need to show that the maximal element is actually in the set though.

(Maybe someone else can confirm this?)

Just my thoughts anyway

good luck.