Infimum & Supremum: Learn the Difference!

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Discussion Overview

The discussion centers on the differences between infimum and minimum, as well as supremum and maximum, particularly in the context of various sets in mathematical analysis. Participants explore definitions, examples, and implications of these concepts.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Colleen seeks clarification on the differences between infimum and minimum, and supremum and maximum.
  • One participant explains that for the set 0
  • Colleen presents various sets and attempts to determine their suprema, infima, maxima, and minima, including the natural numbers, integers, and specific intervals.
  • Another participant notes that if the supremum is in the set, it is the maximum, and similarly for the infimum and minimum, providing examples with different types of intervals.
  • A participant challenges Colleen's interpretation of the supremum of the integers and the reals, suggesting that the definition of supremum may depend on convention.

Areas of Agreement / Disagreement

Participants express differing views on the definitions and existence of suprema and maxima for certain sets, indicating that multiple competing views remain. The discussion does not reach a consensus on some points, particularly regarding the treatment of infinity in relation to supremum.

Contextual Notes

Some statements rely on specific definitions and conventions that may vary among different mathematical contexts. The discussion includes unresolved aspects regarding the treatment of certain sets and the existence of supremum and infimum.

cmurphy
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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
 
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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.
 
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
 
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.
 
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.
 

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