Infimum & Supremum: Learn the Difference!

[cmurphy]

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

 

[mathman]

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.

 

[cmurphy]

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

 

[HallsofIvy]

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.

 

[matt grime]

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.