Supremum and Infimum of a Set Containing Rational Numbers

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In summary, the conversation is discussing finding the supremum and infimum of a set, specifically the set A= \{ \frac{m}{n}+4\frac{m}{n} : m,n \in \mathbb{N}^*\}. The person trying to solve the problem initially suggests that the minimum and maximum of the set is 5, but the other person corrects them and explains that the set includes all possible combinations of m and n from the natural numbers. They suggest graphing the function f(x)=4x+\frac{1}{x} to better understand the set. The conversation then moves on to discussing a more complicated set, B= \{ \frac{mn}{4m^
  • #1
Felafel
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Homework Statement



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]

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 ##
 
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  • #2
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.
 
  • #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?
 
  • #4
Felafel said:
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?

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?
 
  • #5
micromass said:
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?

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?
 
  • #6
Felafel said:
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?

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].
 
  • #7
enlightning, thank you!
 
  • #8
In particular, suppose n= 1 and m= 100000. What member of the set would that give?
 

1. What is the difference between infimum and maximum of a set?

The infimum of a set is the greatest lower bound, meaning it is the largest number that is less than or equal to all the numbers in the set. The maximum of a set is the largest number in the set. The main difference is that the infimum may or may not be an actual number in the set, while the maximum must be an element of the set.

2. How do you find the infimum and maximum of a set?

To find the infimum of a set, you need to first determine the lower bounds of the set. Then, find the largest number among the lower bounds. This number will be the infimum. To find the maximum of a set, simply find the largest number in the set.

3. Can a set have more than one infimum or maximum?

No, a set can only have one infimum and one maximum. This is because the infimum and maximum are unique values that satisfy specific properties of the set.

4. What happens if a set has no infimum or maximum?

If a set has no infimum, it means that the set is unbounded below, and there is no largest lower bound. Similarly, if a set has no maximum, it is unbounded above, and there is no largest element in the set.

5. How are infimum and maximum used in mathematical analysis?

In mathematical analysis, infimum and maximum are used to define the boundaries of a set. They are also used to prove the existence of certain values in a set, such as the greatest lower bound or largest element. In addition, infimum and maximum are used to define the convergence of sequences and series in calculus.

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