Infimum & Supremum: Learn the Difference!

  • Thread starter cmurphy
  • Start date
  • Tags
    Supremum
In summary, the difference between infimum and minimum is that the infimum is the greatest lower bound of a set, while the minimum is the smallest number in the set. Similarly, the supremum is the smallest upper bound of a set, while the maximum is the largest number in the set. However, the supremum and infimum may not be in the set itself. In terms of specific sets, the supremum and infimum of the intervals (0,1), [0,1), (0,1], and [0,1] are 0 and 1 respectively for all four intervals. The maximum and minimum of these intervals may or may not exist depending on the set. Additionally, for the set of natural numbers
  • #1
cmurphy
30
0
Hi,

It has been awhile since I have taken calculus, and now I am in analysis. I need to know what is the difference between the infimum and minimum and what is the difference between supremum and maximum?

I know there is a difference, I just don't understand how they could be.

Thanks -
Colleen
 
Physics news on Phys.org
  • #2
The difference is slightly technical. Example, consider the set 0<x<1. This has no maximum or minimum, however 0 is the infimum and 1 is the supremum.
 
  • #3
Ok, so I want to find the sup, inf, max, and min of some sets. Would this be on the right track?

Let E = N. Then it has no max, inf = 1, min = 1. For sup E, would that be infinity?

If E = Z, then no max or min, but sup = infinity and inf = -infinity?

If E = {-3, 2, 5, 7}, would sup = max = 7 and inf = min = -3?

If E = {x : x^2 < 2}, the set would have no max, but the sup = 2, and inf = -root 2? Would it have a min?

If E = R, then there should be no sup, inf, max, or min?

Colleen
 
  • #4
If the supremum is IN the set, then it is the maximum of the set.
If the infimum in IN the set, then it is the minimum of the set.

But the supremum does not have to be in a set in which case the set would not have a maximum.

The supremum and infimum of the intervals (0,1), [0,1), (0,1], and [0,1] are 0 and 1 respectively for all four intervals. The maximum (largest number in the set) of (0,1] and [0,1] is 1 but (0,1), [0,1) do not have a maximum. The minimum (smallest number in the set) of [0,1) and [0,1] is 0 but (0,1] and (0,1) do not have a minimum.
 
  • #5
the x^2>2 one is wrong (it is symmetric, in the sense of changing x to -x leaves it unchanged) you may have just missed the root out of the description of sup though. there is no min.

i don't see why you say that the sup of Z is infinity, but the sup of R is not defined. In any case this is matter of convention, i think. some people would say that the sup does not exist. some may say it is infinity, you'd have to check the convention you're working with.
 

1. What is the difference between infimum and supremum?

The infimum and supremum are both mathematical terms used to describe the boundaries of a set of numbers. The infimum, or greatest lower bound, is the smallest number that is greater than or equal to all the numbers in the set. The supremum, or least upper bound, is the largest number that is less than or equal to all the numbers in the set. In simpler terms, the infimum is the lowest possible value in a set, while the supremum is the highest possible value.

2. Why are infimum and supremum important?

Infimum and supremum are important because they help us understand the behavior and limits of a set of numbers. They provide a concise way to describe the boundaries of a set and can be used to prove the existence or uniqueness of certain values. Additionally, they are fundamental concepts in real analysis and other branches of mathematics.

3. How are infimum and supremum used in calculus?

In calculus, infimum and supremum are used to define the concepts of limits and continuity. The limit of a function is the value that the function approaches as the input approaches a certain value, and the infimum and supremum help us determine the existence and behavior of this limit. In continuity, the infimum and supremum play a role in determining if a function is continuous at a certain point.

4. Can a set have multiple infimum or supremum?

Yes, a set can have multiple infimum and supremum if the set is not bounded. For example, the set of all negative real numbers has no infimum, as there is no smallest negative number. Similarly, the set of all positive real numbers has no supremum, as there is no largest positive number. However, if a set is bounded, it will have a unique infimum and supremum.

5. How can infimum and supremum be calculated?

The calculation of infimum and supremum depends on the set of numbers. For a finite set, we can simply determine the smallest and largest values. For an infinite set, we can use mathematical techniques such as the method of exhaustion or the nested interval property to approximate the infimum and supremum. In some cases, the infimum and supremum can also be calculated using calculus techniques such as differentiation and optimization.

Similar threads

MHB Finding the Infimum and Supremum
    • Topology and Analysis
Replies
4
Views
749
MHB Infimum and Supremum of a Set (Need Help Finding Them)
    • Topology and Analysis
Replies
2
Views
1K
Find the infimum and/or supremum and see if the set is bounded
    • Calculus and Beyond Homework Help
Replies
6
Views
2K
Infimum and Supremum, when they Do not exist in finite sets
    • Calculus and Beyond Homework Help
Replies
8
Views
7K
MHB How can we prove the inequality for the supremum and infimum of f*g and f*g?
    • Topology and Analysis
Replies
2
Views
1K
I Munkres-Analysis on Manifolds: Extended Integrals
    • Calculus
Replies
5
Views
1K
MHB Proof of the Equality of Supremums (Or Something Like That Anyway :) )
    • Topology and Analysis
Replies
2
Views
1K
B Nature of Limits: Learn Calculus Basics w/o Limits Distraction
    • Calculus
Replies
13
Views
2K
I am taking issue with the Riemann Integral
    • Calculus
Replies
4
Views
2K
Supremum and infimum of specific sets
    • Precalculus Mathematics Homework Help
2
Replies
61
Views
6K

Hot Threads

Recent Insights

Back
Top