Results 1 to 4 of 4
Like Tree2Thanks
  • 1 Post By Plato
  • 1 Post By Plato

Math Help - Completeness Axiom in R (Real numbers)

  1. #1
    Member
    Joined
    Jun 2013
    From
    Israel
    Posts
    158

    Completeness Axiom in R (Real numbers)

    So I am going over the Completeness Axiom, and I have a question, First I will explain what I think my answer is and my logic , and I am waiting for feedback thank you.

    So the question is about the Examples that the lecturer gave,

    As you can see in the picture it says for integers, not dense but without gaps and for rationals it says dense but with gaps.

    I am having trouble with understanding this,

    I understand the fact that rationals have gaps, because there are numbers like square root of something or number e or pi, numbers that can't be represented with rationals, therefore they are not rationals so there are a lot of gaps on the number line.

    But how come Z is without gaps and not dense. I mean I understand the fact that it's not dense because you can't put a number between 1 and 2 or any two integers.
    But why is it without gaps? Is it because numbers like 1/2 are not defined in Z? According to the definition that they gave us an integer is in the form of n-m when n and m is naturals, so you can't really talk about fractions in Z because it has no meaning, I guess. Is my logic correct?

    Thank you.
    Attached Thumbnails Attached Thumbnails Completeness Axiom in R (Real numbers)-q3.png  
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,966
    Thanks
    1785
    Awards
    1

    Re: Completeness Axiom in R (Real numbers)

    Quote Originally Posted by davidciprut View Post
    So I am going over the Completeness Axiom,
    So the question is about the Examples that the lecturer gave,
    As you can see in the picture it says for integers, not dense but without gaps and for rationals it says dense but with gaps.
    I am having trouble with understanding this,
    I understand the fact that rationals have gaps, because there are numbers like square root of something or number e or pi, numbers that can't be represented with rationals, therefore they are not rationals so there are a lot of gaps on the number line.
    But how come Z is without gaps and not dense. I mean I understand the fact that it's not dense because you can't put a number between 1 and 2 or any two integers.
    But why is it without gaps? Is it because numbers like 1/2 are not defined in Z? According to the definition that they gave us an integer is in the form of n-m when n and m is naturals, so you can't really talk about fractions in Z because it has no meaning, I guess. Is my logic correct? .
    Have you studied Dedekind cuts ?
    That is what your attachment suggests you are working with. I have a small complaint about the handout. It should made it clear that in all cases c\in S
    From that web page in the link that S=\{r\in\mathbb{Q}|~r<\sqrt{2}\text{ or }r>\sqrt{2}\}=\mathbb{Q}.

    Note that in the example, we can define L~\&~U but there is no greatest or least element in one or the the other set. Hence a gap.
    Between any two rational there is a third rational number, density.
    But you have shown that there is no integer between n~\&~n+1 so \mathbb{Z} cannot be dense. By the same logic it cannot have gaps.

    Does that help?
    Last edited by Plato; January 31st 2014 at 09:31 AM.
    Thanks from davidciprut
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jun 2013
    From
    Israel
    Posts
    158

    Re: Completeness Axiom in R (Real numbers)

    Quote Originally Posted by Plato View Post
    Note that in the example, we can define L~\&~U but there is no greatest or least element in one or the the other set. Hence a gap.
    That depends on our set though, I mean If we define L=[-infinity,2] and U=[2,infinity], this case 2 is the greatest , the least element, respectively in U and L, no? At this point it's not a dedekind cut, but still, in the definition that the lecturer gave it doesn't specify it as a dedekind cut, and that is confusing. Maybe there is a point that I am missing here? I understood that we are working with dedekind cuts but the teacher didn't even define what a dedekind is and he is using it, I get so confused sometimes by the lecturer notes...

    Quote Originally Posted by Plato View Post
    But you have shown that there is no integer between n~\&~n+1 so \mathbb{Z} cannot be dense. By the same logic it cannot have gaps.
    What do you mean by ''the same logic''? It's still not clear why it doesn't have gaps, can you explain?

    Thank you.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,966
    Thanks
    1785
    Awards
    1

    Re: Completeness Axiom in R (Real numbers)

    Quote Originally Posted by davidciprut View Post
    What do you mean by ''the same logic''? It's still not clear why it doesn't have gaps, can you explain?
    It is really simple: S=\{r\in\mathbb{Z}|~n<0.5\text{ or }n>0.5\}=\mathbb{Z}.

    L=\{n\in\mathbb{Z}: n<0.5\}\text{ and }U=\{n: n>0.5\}

    0=\sup(L)~\&~0\in L and 1=\inf(U)~\&~1\in U but as you proved (0,1)\cap\mathbb{Z}=\emptyset so no gaps.
    Thanks from davidciprut
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. completeness of real numbers
    Posted in the Algebra Forum
    Replies: 0
    Last Post: September 3rd 2011, 10:39 AM
  2. The Completeness Axiom
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: February 2nd 2010, 02:32 PM
  3. Completeness Axiom
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: November 7th 2009, 09:13 AM
  4. completeness axiom
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 18th 2009, 03:30 PM
  5. Completeness Axiom
    Posted in the Calculus Forum
    Replies: 5
    Last Post: January 21st 2009, 05:05 PM

Search Tags


/mathhelpforum @mathhelpforum