Supremum/infinum/max/min concerns x axis or y axis?

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Discussion Overview

The discussion revolves around the concepts of supremum and infimum in the context of mathematical sets, particularly focusing on their definitions and applications to specific examples. Participants explore the implications of these concepts on both the domain and range of functions, as well as the distinction between upper and lower bounds.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants question whether supremum and infimum refer to values on the x-axis or y-axis, using examples such as x^2 < 2 and 1/n to illustrate their points.
  • One participant explains that the supremum of the set A = {1, 1/2, 1/3, 1/4, ...} is 1, while the infimum is 0, emphasizing the need to consider all elements of the set.
  • Another participant clarifies that the supremum of the set {x ∈ Q: x^2 < 2} is √2, not 2, because 2 is an upper bound but not the least upper bound.
  • There is a discussion about whether the supremum of {x^2 < 2} would simply be 2, leading to further clarification on the nature of supremum as a least upper bound.
  • One participant proposes an example involving the function f(x) = x^2 over the interval (-1, 3), suggesting that the supremum is 9 and the minimum is 0, which is confirmed by another participant.

Areas of Agreement / Disagreement

Participants express varying interpretations of supremum and infimum, particularly regarding their definitions and applications to specific sets. While some points are clarified, the discussion does not reach a consensus on all aspects, particularly regarding the implications of these concepts on different sets.

Contextual Notes

The discussion highlights the importance of understanding the definitions of supremum and infimum in relation to the elements of a set and the distinction between upper bounds and least upper bounds. There are also nuances regarding whether the supremum must be an element of the set.

kahwawashay1
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When we talk about supremum/infinum/etc, does it mean the largest/smallest number on the x-axis or y axis??

Ok so first my professor was explaining how when x^2 < 2 , the supremum is root of 2 (so he was talking about how the domain has a least upper bound).

But then he said that when we look at 1/n, where n is 1,2,3,... , the infinum is 0...so in this case i guess he meant that the range is bounded from below (since 1/n approaches 0 as n->infinity)...but if you look at the domain, the infinum is actually a minimum and equals 1...

help me clear up this ambiguity?
 
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kahwawashay1 said:
When we talk about supremum/infinum/etc, does it mean the largest/smallest number on the x-axis or y axis??

Ok so first my professor was explaining how when x^2 < 2 , the supremum is root of 2 (so he was talking about how the domain has a least upper bound).

But then he said that when we look at 1/n, where n is 1,2,3,... , the infinum is 0...so in this case i guess he meant that the range is bounded from below (since 1/n approaches 0 as n->infinity)...but if you look at the domain, the infinum is actually a minimum and equals 1...

help me clear up this ambiguity?

An infimum is the greatest lower bound and the supremum is the least upper bound. When you're looking at the set:

A = {1,1/2,1/3,1/4,...}

sup(A) = 1. This is because 1 is an upper bound for A (every element is less than 1) and there is no upper bound that is lower than 1.

inf(A) = 0. This is because 0 is a lower bound for A (every element in A is greater than 0) and there is no lower bound that is greater than 0. If there was, call it a, you could always find an n such that 1/n < a.

I think the confusion you were having is that the infimum would be the first element in the set, but this isn't true. You need to consider all elements of the set and see what's going on.
 
gb7nash said:
An infimum is the greatest lower bound and the supremum is the least upper bound. When you're looking at the set:

A = {1,1/2,1/3,1/4,...}

sup(A) = 1. This is because 1 is an upper bound for A (every element is less than 1) and there is no upper bound that is lower than 1.

inf(A) = 0. This is because 0 is a lower bound for A (every element in A is greater than 0) and there is no lower bound that is greater than 0. If there was, call it a, you could always find an n such that 1/n < a.

I think the confusion you were having is that the infimum would be the first element in the set, but this isn't true. You need to consider all elements of the set and see what's going on.

Oh ok so i understand about the 1/n...but what my professor wrote exactly was that supremum of
\left\{x\in Q: x^2 &lt; 2\right\}
is root of two...but shouldn't it be just 2?
 
kahwawashay1 said:
Oh ok so i understand about the 1/n...but what my professor wrote exactly was that supremum of
\left\{x\in Q: x^2 &lt; 2\right\}
is root of two...but shouldn't it be just 2?

ohh nvm the supremum is root of two in this case because we are talking about x in Q that satisfy the condition x^2 < 2...but if we just ask for supremum of {x^2 < 2}, it would be 2?
 
kahwawashay1 said:
Oh ok so i understand about the 1/n...but what my professor wrote exactly was that supremum of
\left\{x\in Q: x^2 &lt; 2\right\}
is root of two...but shouldn't it be just 2?

Think about it. \left\{x\in Q: x^2 &lt; 2\right\} means the set of rational numbers such that each rational number squared is less than 2. 22 = 4. 2 is definitely an upper bound for the set, but it's not the least upper bound. (note that the supremum does not need to be in the set. In this case, the supremum doesn't need to be a rational number. It only needs to be a least upper bound) What's something else you could plug into x such that there it is an upper bound for the set, and also that it is the least upper bound?
 
gb7nash said:
Think about it. \left\{x\in Q: x^2 &lt; 2\right\} means the set of rational numbers such that each rational number squared is less than 2. 22 = 4. 2 is definitely an upper bound for the set, but it's not the least upper bound. (note that the supremum does not need to be in the set. In this case, the supremum doesn't need to be a rational number. It only needs to be a least upper bound) What's something else you could plug into x such that there it is an upper bound for the set, and also that it is the least upper bound?

kk thanks i get that,

so as another example just so i kno for sure i get it, if

f: (-1, 3)->R
f(x)=x^2

then the supremum is 9 and minimum is 0 ?
 
kahwawashay1 said:
kk thanks i get that,

so as another example just so i kno for sure i get it, if

f: (-1, 3)->R
f(x)=x^2

then the supremum is 9 and minimum is 0 ?

If the set you're considering is f(x) = x2 for x in (-1,3), then yes.
 

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