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Infimum&maximum of a set

  1. Dec 8, 2012 #1
    1. The problem statement, all variables and given/known data

    Hello. I think I've managed to solve the exercise, but I'd like it to be checked and I'd also like to kow whether it is well-written and explained enough.

    The problem is: find the supremum and infimum of this set:
    [itex] A= \{ \frac{m}{n}+4\frac{m}{n} : m,n \in \mathbb{N}^*\} [/itex]

    3. The attempt at a solution

    if m=n then 5 is the only element of the set. Therefore it is its minimum and maximum and also its supremum and infimum.
    if m<n the infimum (and also the minimum) is ## \frac{17}{2} ## because the minimum values for m and n are 1 e 2.
    if n<m the infimum is 4, for the same reason.
    either when m<n or n<m the supremum is ## +\infty ##
     
  2. jcsd
  3. Dec 8, 2012 #2

    micromass

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    I think you are severely misunderstanding the notation. The notation

    [tex]A=\{\frac{m}{n}+4\frac{m}{n} | m,n\in \mathbb{N}^*\}[/tex]

    means that we take all m and n in [itex]\mathbb{N}^*[/itex]

    For example:
    • 5 is an element of A since we can take m=n=1
    • 5/2 is an element of A since we can take m=1 and n=2
    • 10 is an element of A since we can take m=2 and n=1

    These are three examples of elements in A. There are others.

    We are letting m and n vary among the natural numbers. They can be truly anything.
     
  4. Dec 8, 2012 #3
    ok, should I put the three cases m>n,m<n,m=n together then, and call infimum and supremum the highest and lowest values I find in the union?
     
  5. Dec 8, 2012 #4

    micromass

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    I don't really know why you are so set in separating the problem in cases m>n, m<n and n=m. That doesn't seem necessary here.

    What elements of A do you get if m=1 and n varies?
    What elements of A do you get if n=1 and m varies?
    What does that tell you?
     
  6. Dec 8, 2012 #5
    hahaha don't know why. but what if the set were more complicated? like
    ## B= \{ \frac{mn}{4m^2+n^2} : m \in \mathbb{Z}, n \in \mathbb{N}^* \} ##
    wouldn't it be a good idea to separate the cases?
     
  7. Dec 8, 2012 #6

    micromass

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    It really depends on which sets you are given. Right now, I would be inclined to write

    [tex]\frac{mn}{4m^2 + n^2} = 4\frac{n}{m} + \frac{m}{n}[/tex]

    So it would be a good idea to graph the function [tex]f(x)=4x+\frac{1}{x}[/tex].
     
  8. Dec 8, 2012 #7
    enlightning, thank you!
     
  9. Dec 9, 2012 #8

    HallsofIvy

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    In particular, suppose n= 1 and m= 100000. What member of the set would that give?
     
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