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Homework Help: Zakon : What is the problem?

  1. Dec 29, 2011 #1
    I am learning mathematics on my own, self-study type. Currently following Zakon's first book in his 3 part series.

    In chapter 2, section 6, I have been successfully able to solve all the problem till I encountered 11'. It seems simple enough, but I am unable to understand the problem in the first place. I believe, Once I understand the problem itself, I will be in a position to chalk out a solution.

    Please help me out in actually understanding the problem.


    Here is the problem:

    Chapter 2, Section 6, Problem 11’

    11. Show by induction that each natural element x of an ordered field F can be uniquely represented as X=n ∙1', where n is a natural number in E1 (n ∈N) and 1' is the unity in F; that is, x is the sum of n unities.

    Conversely, show that each such n ∙1' is a natural element of F.

    Finally, show that, for m,n ∈N, we have

    m<n iff mx<nx, provided x>0
     
  2. jcsd
  3. Dec 29, 2011 #2
    The question asks you to prove that every natural element x can be written as

    [tex]x=1+1+1+1+....+1[/tex]

    where we sum 1 a number of times. Conversely, (this is very easy) every element of the form

    [tex]1+1+1+1+1+...+1[/tex]

    belongs to F.
     
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