Talk:Supremum

Difficult to Parse 2nd condition

Did you mean:

for any v in P such that x ≤ v then for all x in S it holds that u ≤ v.

or

for any v in P such that x ≤ v for all x in S then it holds that u ≤ v. — Preceding unsigned comment added by 68.164.6.9 (talk) 00:42, 20 March 2012 (UTC)

Difference between sup and max

What's the difference between 'sup' and 'max'? It seems like 'sup' is 'max', plus definitions about what the value is for an empty set, as well as an unbounded set. I think it'd be nice to compare sup to max for other naive readers like me (and I'd like to know). Zashaw 21:15, 28 Aug 2003 (UTC)

given a set S, max S is a member of that set. sup S might not be. Also, max S might not exist. Eg, suppose S = { -1, -1/2, -1/3, -1/4 , ........ }. Then there is no max S -- any element of S you select as a candidate for max, I can pick one that beats it. But there IS a sup S, it's zero. -- Tarquin 21:19, 28 Aug 2003 (UTC)

I'm new to this, but the line "An important property of the real numbers is that every set of real numbers has a supremum." appears completely false to me. Isn't there only a supremum when there's an upper bound? (ie The set of all real numbers doesn't have a supremum since it's not bounded) -- Muso 19:02, 6 Sep 2003 (UTC)

What is meant is that there is one, perhaps + or - infinity. See also infimum, where I added a phrase to clarify, we can do that here also. - Patrick 19:41, 6 Sep 2003 (UTC)

I've reworded things in an attempt to make it clearer. --Zundark 19:54, 6 Sep 2003 (UTC)

Small correction to last section

In the example: "Let $S$ be the set of all rational numbers $q$ such that $q^{2}<2$. Then $S$ has an upper bound (1000, for example, or 6) but no least upper bound. For suppose $p$ is an upper bound for $S$, so $p^{2}>2$."

shouldn't the last expression be closed i.e. $p^{2}\geq 2$ since $q^{2}<2$ is open.

p is supposed to be rational, so equality is impossible. I've modified the paragraph to make this a bit clearer. --Zundark 14:41, 11 Mar 2004 (UTC)

Questions for clarification

The definition says you have two sets S and T then gives an example with one set (1 2 3). Where has set T gone? There is something here not bein clearly explained to non-mathematicians. And how algorithmically do you distinguish between sup negatives being 0, which is greater than set S, when the definition says it can equal S... giving sup negatives equal Littlest Negative. Which doesnt exist mathematically, but does in most computer languages?

I want to make the definition more clear, but since I don't know much about supremum, I need to ask a couple things:

• Is it true that every finite set has a supremum and infimum that is in that set?
  • True in a total order (or indeed anny join lattice), not true in general. -lethe ^{talk +}
• Would it be correct to say that the supremum of a set with no upper bound is the limit as n->∞ of the set element a_n ?
  • This would be fairly inaccurate. If the set is countable then there exists such a sequence, but there exist many other sequences as well which do not have that limit. And if the set is not first countable, then sequences can't get you there. -lethe ^{talk +}
• What does the set being a subset of something else have anything to do with?
  • The most interesting examples are the ones where the supremum is not in the set, otherwise our supremum is actually a greatest element. This can only occur when we consider the supremum of a subset of some other set. -lethe ^{talk +} 12:38, 29 March 2006 (UTC)

Thanks, Fresheneesz 11:52, 29 March 2006 (UTC)

So what finite sets don't have a supremum and infimum inside itself? Fresheneesz 05:48, 31 March 2006 (UTC)

The set {{1},{2}}, ordered by containment, has supremum {1,2} (the smallest set which contains both {1} and {2} as subsets), and infimum ∅ (the largest set which is contained in both {1} and {2} as a subset). Neither of these is an element of the set. -lethe ^{talk +} 12:59, 31 March 2006 (UTC)

Slight clarification in introduction

I clarified in the introduction that S is a subset of an ordered set T, and that the supremum of S is an element of T. Previously the language was slightly confused and referred to the supremum being an element of S, then switching and saying it isn't always an element of S. Clarifying that S is a subset of a possibly larger ordered set T makes the introduction (hopefully) more accurate. Dugwiki 21:13, 9 November 2006 (UTC)

Illogical proof and possible merge with infimum

I just made a bold move to remove a proof (of the Approximation Property) that did not seem correct, as well as subsequent proofs. Perhaps it was just worded poorly, and someone can present sound reasoning for it.

Also, this article should be merged with infimum so they're not essentially mirror articles with opposite definitions, properties, examples, etc. Least upper bound has already been redirecting to Supremum since 2002. –Pomte 05:15, 15 March 2007 (UTC)

Were you objecting to the statement that the supremum of the empty set is −∞? If so, I disagree with your objection. Michael Hardy 01:38, 23 April 2007 (UTC)

Why are we interested in supremums?

Every subset of the real numbers has an upper bound. Why is it that we are interested at all in l.u.bs when in most cases we cannot find these? For instance, the square root of 2 has an upper bound that we refer to as the square root of 2. Isn't it obvious that any number greater than square root 2 will by default be an upper bound? I cannot understand the importance of the least upper bound property. Could someone please explain as the article does not convince me entirely. 65.28.94.67 14:57, 7 September 2007 (UTC)

The property is essential in distinguishing the rationals from the reals. 142.150.205.244 (talk) 16:35, 17 January 2008 (UTC)

interest in suprema

As I was taught in mathematics courses, the notion of a supremum is very useful in mathemetical proofs, particularly when dealing with sets of infinitely many elements. You may not be able to pick a "largest" or "smallest" element but you are guaranteed to have a supremum or infimum whose properties could be analyzed. My calculus book defined "limits" in terms of suprema and infima, and limits are crucial to calculus. It has to do with mathematical rigor.

My course also mentioned that the statement "every bounded set with an upper bound has a supremum" is an AXIOM of set theory; it cannot be proved from the rest of the theory but is useful to assume. The article did not mention that.

For my part, I'm curious to know who invented the idea, and the article doesn't say. The only mathematician cited by name is Dedekind, so was he the originator? CharlesTheBold (talk) 10:55, 9 March 2008 (UTC)

"every bounded set with an upper bound has a supremum" is not an axiom of set theory. It is one of the axioms used to characterize the real numbers. From the point of view of set theory, the real numbers are defined to be a certain set (the operations of addition and multiplication and the order relation are also defined to be certain sets) and the fact that this set obeys the axioms for the reals is a theorem of set theory. In any case, the axiomatic status of this statement is already discussed in its proper place, at real number. Algebraist 16:41, 18 May 2008 (UTC)

How to begin an entry like this

I'm thinking it would be really useful if this started with something reasonably comprehensible by non-specialists. The explanation under the first sub-heading seems to make sense without bringing in any more abstruse mathematical concepts, though presumably it loses some degree of generality. It looks like it works for the context that led me to look up the term, while after twenty minutes of looking up other things to try to make sense of the first paragraph, I still have very little clue what that's saying.

I'd love to fix this myself, but this is probably one of those entries which needs to be written and edited by the non-ignorant with suggestions from the ignorant regarding how to make it accessible... --Oolong (talk) 14:31, 4 June 2008 (UTC)

This is probably because I'm too used to this stuff already, but I can't see how the first sentence of section 1 is any clearer than the first sentence of the lede. It's exactly the same except it says 'smallest' instead of 'least' and talks about the real numbers instead of an arbitrary partially ordered set. Can you explain further what makes the first sentence hard to understand? Algebraist 21:50, 4 June 2008 (UTC)

What would really help the ignorant are drawings (like the box for supremum) or concrete examples of sets of numbers, to illustrate the distinction between supremum, maximal vs. greatest element, etc. Wadh27NK (talk) 22:51, 31 July 2008 (UTC)

Only if the diagrams match the words of the definitions. Having a description that uses subset S of T and a picture with A subset A of A and blue balls and red diamonds and no commas or periods in the sub-text, well, that doesn't really help anyone. —Preceding unsigned comment added by 99.251.240.54 (talk) 17:24, 30 October 2009 (UTC)

Mistake in diagram?

A set A of real numbers (shown as blue balls), a set of upper bounds of A (red balls), and the smallest such upper bound, that is, the supremum of A (shown as a red diamond).

As much as I understand this diagram, it shows a set "N" consisting of elements of the real numbers. There is a set A (subset of "N") which is all the blue elements, and there is a set N\A (all the red elements). Why isn't it the blue dot left to the red diamond that is the supremum?
Besides, this article says that the supremum needn't be part of the subset, while least element says that a least element (a supremum is a least element) must be part of the subset. I'm confused :-) --Abdull (talk) 21:40, 21 February 2010 (UTC)

A supremum is a least upper bound of a set S. That is not the same as saying the supremum is the least element of S. It is true that the supremum is the least element of the set of upper bounds, but that is not the same as saying it is in S, because the set of upper bounds for S is not the same as the set S. For example, in the case of the blue dots here, S may not contain any element that is an upper bound for the entire set. — Carl (CBM · talk) 13:11, 22 February 2010 (UTC)

Merge

It seems to me to be a sensible idea to merge infimum and supremum, perhaps as infimum and supremum. The concepts are so related that separate articles are bound to be redundant to some degree. Thoughts? —Anonymous Dissident^Talk 12:34, 9 March 2011 (UTC)

Please discuss this merge proposal at Talk:Infimum#Merge so that the discussion is not spread to two places. — Tobias Bergemann (talk) 08:49, 21 July 2011 (UTC)

arg?

Is it correct to use the notation arg sup, in analogy with arg max? I don't see 'arg sup' used in this article.

Perhaps the issue is this: arg max is well-defined if max is well-defined. But sometimes sup exists for a maximization problem even though the max is not defined. If so, it is because the 'argument' at which the sup occurs is outside of the permitted choice set. Therefore the 'arg sup' would refer to a point outside the choice set.

Comments appreciated. Rinconsoleao (talk) 12:02, 20 July 2011 (UTC)

"any"

I found this:

is the least element of T that is greater than or equal to any element of S.

I changed it to this:

is the least element of T that is greater than or equal to every element of S.

"any" is absolutely the worst possible word that could be used here. Reasonable readers could see "x is the least element that is is greater than or equal to any element of S" and think it means x is the least element for which there is any element of S that x is greater than or equal to. That is obviously not what is intended. Michael Hardy (talk) 15:56, 9 October 2011 (UTC)

Yes, that's a good edit. — Carl (CBM · talk) 19:41, 9 October 2011 (UTC)

Union of elements?

In the example of supremum of a partially ordered set it says "the union (set theory) of the elements of S."

AFAIK this has no sense, the union operator is defined for sets, not for elements, and the elements of S need not be sets. Am I wrong? What is the supremum in this example? — Preceding unsigned comment added by 200.89.153.3 (talk) 22:33, 24 October 2011 (UTC)

It says:

The supremum of a subset S of (P, ⊆), where P is the power set of some set, is the supremum with respect to ⊆ (subset) of a subset S of P is the union (set theory) of the elements of S.

Since S is a subset of P, the power set of some set, the elements of S are indeed sets, and it makes sense to take their union. —Mark Dominus (talk) 06:54, 25 October 2011 (UTC)