Thread: The Supremum

I am just finding that I have a couple questions about the definitions of supremum and infimum, that I am having a hard time answering. Perhaps it is a possiblity that I am just overthinking these things a bit...

The definition of supremum is the least upper bound of a set, or it is the least element that is greater than or equal to every other element of the set. Why is it necessary to consider the "least element"? Is it possible to have multiple different upper bounds of a set? And if that is true, then why look for the least one and not the greatest? The way I understand this concept is that the supremum of a set, say {1,2,4} would be 4. I am just having difficulty even trying to come up with a set that there would be confusion in the upper bound...

And another question that I have... is it possible to have multiple distinct suprema for a set?

Of course it is easy to consider a set such as {1,2,4,4} where the supremum (4) occurs multiple times. But my interpretation of the definition of supremum, that is, the supremum is the least upper bound (so there might be multiple upper bounds), leads me to ponder whether there could be multiple suprema for one set.

Or am I possibly just overthinking this?

Originally Posted by pbandjay

The definition of supremum is the least upper bound of a set, or it is the least element that is greater than or equal to every other element of the set. Why is it necessary to consider the "least element"? Is it possible to have multiple different upper bounds of a set?

Note that an "upper bound" of a set A is defined as an element of the entire space (such as the set of real numbers), not just of set A, that is greater than or equal to every element of A.

In this sense, 4, 5, 6, 7 ... , 10000 , ... , 1000000, .... are all upper bounds of your set {1,2,4}. The list goes on and on, there is no "greatest" upper bound. But only 4 is the least upper bound, and therefore it is the supremum. And there is never more than one supremum.

The supremum of a set A may or may not be an element of A. If it is an element of A, it is the greatest one. If it is not, it is the smallest element in the set of upper bounds (of numbers greater or equal to all the numbers in A). In this case, the "or equal" part is void, there is no number Z such that Z is equal to some element of A but not smaller than any other one.

Think of the set S of all real numbers smaller than 1. There is no greatest element in S. If you take any number smaller than 1, say 0.99, there are always infinitely many greater numbers which still are smaller than 1: 0.9900000000001, 0.99001 and 0.995 are just three examples.

Now the number 1 is the supremum (least upper bound) of S: it is the smallest number Y such that for any X in S, X<=Y. And obviously, 1 is not an element of S.

1.5, 1.7 and 2009 are a few of the infinitely many other (not least) upper bounds of S.

You might enjoy reading about the Dedekind cut.

Hope this helps,
Leszek.

Ah... Thank you very much! This helps quite a lot. So my main problem was that I was only considering upper bounds that exist in the set.

This also clears up the following statements:






And thank you for the link on the Dedekind Cut!

Have a good day.

You're very welcome.
However, your notation:

Originally Posted by pbandjay

looks redundant to me. I'd have simply written:






Cheers, L.

Originally Posted by pbandjay

I am just finding that I have a couple questions about the definitions of supremum and infimum, that I am having a hard time answering. Perhaps it is a possiblity that I am just overthinking these things a bit...

The definition of supremum is the least upper bound of a set, or it is the least element that is greater than or equal to every other element of the set. Why is it necessary to consider the "least element"? Is it possible to have multiple different upper bounds of a set? And if that is true, then why look for the least one and not the greatest? The way I understand this concept is that the supremum of a set, say {1,2,4} would be 4. I am just having difficulty even trying to come up with a set that there would be confusion in the upper bound...

And another question that I have... is it possible to have multiple distinct suprema for a set?

Of course it is easy to consider a set such as {1,2,4,4} where the supremum (4) occurs multiple times. But my interpretation of the definition of supremum, that is, the supremum is the least upper bound (so there might be multiple upper bounds), leads me to ponder whether there could be multiple suprema for one set.

Or am I possibly just overthinking this?

consider the set of the open interval (0,1) It has has lots of upper bounds. 10 is an upperbound,. So are 12: 2,975;, 1.4 ; etc. The least upper bound is 1.

Now consider the set (0, ). It has also has lots upper bounds. is the leasts upper bound, but 5; 927 and 16 are also upper bounds. Think a bit further and consider only the rational numbers in that interval within the set of rational numbers. It is still bounded and 5; 927, and 16 are still upperbounds within the rationals. But it has no least upper bound within the rationals, because is not a rational number.

The entire point is that any set of real numbers that has any upper bound has a least upper bound which is is a real number. That is the least upper bound property. It is not true for the rational numbers.

If i will prove Pigreco have a relation with squarerootof2 (and i can),can someone give me some money? need money

p.s. i lost weight...

85.5 4 now

Originally Posted by clearwar

If i will prove Pigreco have a relation with squarerootof2 (and i can),can someone give me some money? need money

p.s. i lost weight...

85.5 4 now

If Piegreco is having relations with the square root of 2, I have a buddy on the vice squad who might be interested. Sometimes good informants are paid a little.

What sort of proof do you have? Pictures that would be admissible in court ?

I have drawings and formulas. I don't know if my knowlege of english is "evolved" enough to debat in a court. Anywhere my calculus talks better of me. I sent you a PM,please read it

my email: cito@email.it