Talk:Supremum

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[edit] 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)

[edit] Small correction to last section

In the example: "Let S be the set of all rational numbers q such that q2 < 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 p2 > 2."

shouldn't the last expression be closed i.e. p^{2} \ge 2 since q2 < 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)

[edit] Questions for clarification

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 any 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 an ?
    • 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)