• Aug 10th 2009, 04:14 PM
justchillin
Sorry i have no teacher so this is the only place i can get help. Btw i got another question about the Dedekind cut, i don't really understand what i need to do. It says : Using properties (A)-(H)
(In my book this is the basic properties of real numbers like commutative, associative,distributive laws.... the inverse of a number and the tricotomy property and transitive property ) of the real numbers and taking Dedekind theorem ( As stated in the previous question i posted) as given, show that every nonempty set U of real numbers that is bounded above has a supremum.

The hint is : Let T be the set of upper bounds of U and S be the set of real numbers that are not upper bounds of U.
What do i do? I mean i guess i can't use the completeness axiom. So how to i arrive at this result using only the basic properties of the real numbers and the Dedekind cut ? Btw Siclar
Quote:

Originally Posted by siclar
It's still in reference to the OP so its not really a new question.

justchillin, I'm not sure what you are getting at with density. As far as the $\beta$ being unique, the idea behind these cuts is to construct the real numbers partitions of the rational numbers. In other words for each real $\beta$ the corresponding &quot;cut&quot; can be defined to be the rational numbers less than $\beta$. We can then define addition and multiplication and get something isomorphic to the real numbers as we know them. Then the uniqueness property now makes sense. We want each cut to uniquely represent each real number. I hope that clarifies things somewhat.

I don't understand what you mean when you say " these cuts is to construct the real numbers partitions of the rational numbers". What do you mean by partitions of rational numbers ? From what i understand we cut an interval of real numbers into two and then show that they are both bounded ( at the cut location) either from above or below . Is that what you mean ?
• Aug 10th 2009, 04:32 PM
siclar
Quote:

Originally Posted by justchillin
Btw Siclar

I don't understand what you mean when you say " these cuts is to construct the real numbers partitions of the rational numbers". What do you mean by partitions of rational numbers ? From what i understand we cut an interval of real numbers into two and then show that they are both bounded ( at the cut location) either from above or below . Is that what you mean ?

Sorry, perhaps I was confusing the issue by being too informal about a nonsimple idea. I would recommend looking up this construction if you want to understand its details, since the minutia of it gets off topic here. The intuition though is that cuts correspond to real numbers.
• Aug 10th 2009, 04:33 PM
justchillin
Okay, can you help me out with the proof question then ?
It says : Using properties (A)-(H) (In my book this is the basic properties of real numbers like commutative, associative,distributive laws.... the inverse of a number and the tricotomy property and transitive property ) of the real numbers and taking Dedekind theorem (As stated in the previous question i posted) as given, show that every nonempty set U of real numbers that is bounded above has a supremum.

The hint is : Let T be the set of upper bounds of U and S be the set of real numbers that are not upper bounds of U.
• Aug 10th 2009, 06:07 PM
Plato
Quote:

Originally Posted by justchillin
I created a new thread so you can post in it. Sorry plato.

To do density properly you need to properties: Archimedean property & the well ordering property.

The well ordering property: any subset of the positive integers has a least element.
Archimedean property: If a & b are positive numbers then there is a positive integer K such that b<Ka.

Then there are three steps in proving density of the rational.
If a is a real number then $\left( {\exists n \in \mathbb{Z}} \right)\left[ {n \leqslant a < n + 1} \right]$.

$y - x > 1\; \Rightarrow \;\left( {\exists n \in \mathbb{Z}} \right)\left[ {x < n < y} \right]
$

Now if $0 < a < b\; \Rightarrow \;\left( {\exists n \in \mathbb{Z}} \right)\left[ {1 < nb - na} \right]$.

Can you finish?
• Aug 10th 2009, 07:30 PM
justchillin
Quote:

Originally Posted by Plato
To do density properly you need to properties: Archimedean property & the well ordering property.

The well ordering property: any subset of the positive integers has a least element.
Archimedean property: If a & b are positive numbers then there is a positive integer K such that b<Ka.

Then there are three steps in proving density of the rational.
If a is a real number then $\left( {\exists n \in \mathbb{Z}} \right)\left[ {n \leqslant a < n + 1} \right]$.

$y - x > 1\; \Rightarrow \;\left( {\exists n \in \mathbb{Z}} \right)\left[ {x < n < y} \right]
$

Now if $0 < a < b\; \Rightarrow \;\left( {\exists n \in \mathbb{Z}} \right)\left[ {1 < nb - na} \right]$.

Can you finish?

I wish i understood what you were trying to do.
I understand know of the archimedean property as well as well as the well orded property.
But i'm confused with your notations . Are your n's all the same ?
Also what are x and y ? I'm assuming they are arbitrary real numbers.
I saw the proof that the rationals are dense in the reals, it was sort of different from what you are doing.
Anyway i can worry about that letter is it possible to help me with my other question in this thread?