Talk:Upper and lower bounds

How does S & K relate to M?

Hi. I'm trying to solve for P versus NP problem, and I don't know how subset S and partially ordered set K relate to the various bounds of M. I would appreciate clarification about this, for both myself and others who visit here, to thereby understand and use the concepts on this page. Thank you for reading this. Somethingsea (talk) 02:37, 15 September 2016 (UTC)

Comment

If lower bound is going to redirect here then the page should probably be renamed. One does not expect to end up at upper bound if one follows a link to lower bound... Evercat 02:42, 25 Mar 2004 (UTC)

can i get an example?

this stuff is very complicated. there should be an example so ppl like me can figure out how to solve a problem --Jaysscholar 22:19, 12 October 2005 (UTC)

I contest that this article should either initially redirect to Supremum or explicitly mention the distinctions between upper bounds, least upper bounds/supremums, and maximal elements. I doubt that many non-specialists would immediately accept 704 as an upper bound for the set of all negative numbers; they might not realise they need to prefix "upper bound" with "least" in order to find the article they expect. It might solve Jaysscholar's problems, too. Endomorphic 23:43, 13 November 2006 (UTC)

Upper-bounded

Some scholars use the expressions "upper-bounded" and "lower-bounded" (a Google search provides several examples). Yet it does not feel like good English. I would appreciate it if somebody can comment on the usage of such expressions and to what degree it is accepted in the professional jargon (it clearly exists there). AmirOnWiki (talk) 14:47, 25 September 2011 (UTC)

Bounds of functions

This subsection is confusing. It is titled "bounds of functions" but it talks about bounds for sets of functions. It also uses a comparison of functions, which may confuse the reader as it has not been defined in the current article (it is defined in partially ordered set). If one knows how functions are compared, the case of a set of functions is not different from any other set, so I don't see why this subsection is at all needed. AmirOnWiki (talk) 15:06, 25 September 2011 (UTC)

What am I missing here?

In regards to:

  2 and 5 are both lower bounds for the set { 5, 10, 34, 13934 }, but 8 is not.

How can 2 be the lower bound of the given set, when it is NOT part of the set? I can see that '8' is not the lower bound, but it seems that something is missing - how can I test the bounds of the given set against a number that is not part of the set? (an example from computer programing is when testing a list or vector to determine if it contains a given element, the method will return a -1 to indicate that the element does not exist, rather than the position in the list - I don't want to confuse the issue, I just saying...).

If there is some rule that allows '2' to be the lower bounds of the given set, even though '2' is not part of that set, then couldn't it be said (applying the same rule) that '2' is also the lower bound of the other sample set, to wit: {42}?

Lastly, I belive that is it poor grammer to start a sentance with a number (i.e. "2 and 5 are both...")

— Preceding unsigned comment added by 192.136.15.178 (talk) 16:49, 14 January 2013 (UTC) 

You are missing that there are really two sets involved: The partially ordered set, and the subset. In the example, the partially ordered set is not explicitly stated, but from the explanation afterwards it is obviously intended to be the set of real numbers with the usual order. The subset is the set {5, 10, 34, 13934}. Now the lower bound needs only to be in the partially ordered set, not in the subset. Now clearly 2 is a real number, that is, an element of the set of real numbers, and therefore it is a lower bound of {5, 10, 34, 13934}. --132.199.99.50 (talk) 10:18, 14 September 2016 (UTC)

Preorders?

Can we say this about all preorders, not just posets? ciphergoth (talk) 12:11, 7 December 2015 (UTC)

misleading

one can also have an upper bound in a set that's more complicated than the picture — Preceding unsigned comment added by 2610:130:102:800:18FA:ED36:6C57:8F83 (talk) 12:27, 5 April 2017 (UTC)

Requested move 12 September 2017

The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

Not moved as to the first item; moved as proposed as to the second and third items. bd2412 T 17:43, 22 October 2017 (UTC)

• Upper and lower bounds → Lower and upper bounds
• Greatest element → Greatest and least elements
• Maximal element → Maximal and minimal elements

– I like to conform these pages to the title style of Infimum and supremum (i.e., greatest lower bound and least upper bound). At any rate, Greatest element and Maximal element should definitely move to include respectively least and minimal in their titles regardless of the order (no pun intended). Ailenus (talk) 19:59, 12 September 2017 (UTC) --Relisting. bd2412 T 13:27, 29 September 2017 (UTC) --Relisting. DrStrauss talk 21:11, 11 October 2017 (UTC)

• Support in principle per WP:AND, but can we please disregard the alphabetic ordering and apply min-max order consistently, i.e. Least and greatest elements, Minimal and maximal elements; Infimum and supremum and Lower and upper bounds are already in that order. That would require some tweaking of lead sections as well. No such user (talk) 14:20, 13 September 2017 (UTC)
  • Thanks for your comment. IMHO, however, the consistent min–max ordering would be what I nominated, because the infimum is the greatest element of the set of lower bounds, and the supremum is the least element of the set of upper bounds. Therefore if we accept the order of infimum and supremum, we should probably accept the order of greatest and least elements and lower and upper bounds. I choose the order of maximal and minimal elements so that it is analogous to that of greatest and least elements. Ailenus (talk) 15:40, 13 September 2017 (UTC)
    • That's fine with me as well (I admittedly got confused about definitions of infimum and supremum), so if we align on that, only Upper and lower bounds need not be moved. Right? No such user (talk) 08:35, 14 September 2017 (UTC)
      • I think the alignment would make it Lower and upper bounds, as in nom. Ailenus (talk) 11:58, 15 September 2017 (UTC)
        • @Ailenus: Maybe I'm dense, but I don't see the analogy – from Maximal element: Zorn's lemma states that every partially ordered set for which every totally ordered subset has an upper bound contains at least one maximal element. Therefore, maximal element correlates with an upper bound, and minimal element with a lower bound. Right? No such user (talk) 08:33, 19 September 2017 (UTC)
          • @No such user: That's one type of relation, indeed, and we may rename the page "Minimal and Maximal elements", to which I have no objection. The relation I had in mind is that in total orders the maximal element and the greatest element coincide and also the minimal and least element. Ailenus (talk) 02:58, 20 September 2017 (UTC)

        I think I'm in agreement with No Such User here. The supremum, while technically a "least upper bound", is still an upper bound. It still coincides with teh maximal/greatest element of the set, while the lower bound is something down by the least element of the set. I think all three should be ordered as "max/min" (as stated in my !vote below).  — Amakuru (talk) 10:24, 6 October 2017 (UTC)

• Comment: Could someone please close this RM already? I don’t see anyone objecting; No such user’s multiple comments don’t seem to me to be objections. I admit I should have been bold and moved the pages myself without resorting to Wikipedia’s absurd bureaucratic procedure, but I have been made chronically frightened by all those “senior” Wikipedians who spend hours each day reverting edits while citing some entry hidden in the labyrinth of rules and “guidelines” and offering few sound explanations. Ailenus (talk) 17:49, 23 September 2017 (UTC)
• Support this rename, per WP:AND; I don't think the order particularly matters here. But Oppose renaming of Greatest element and Maximal element for the same reason that Colimit simply redirects to Limit (category theory); that is, especially in more advanced articles, we don't really need the name of both the notion and its dual in the article title. --Deacon Vorbis (talk) 14:33, 25 September 2017 (UTC)
• Comment. On a side note, I think Greatest element should really be merged into Maximal element. The former is just a very simple special case of the latter. Having separate articles here seems like overkill. --Deacon Vorbis (talk) 14:36, 25 September 2017 (UTC)
• Suggestion. However, upon closer inspection, I see we have Initial and terminal objects as an article title. So even there, we seem to have this same sort of inconsistency. So, seeing as how this same issue seems to cover more than just these couple articles, it might be a good idea to start a discussion at WT:WPM and let others weigh in, especially those who aren't necessarily watching these pages. --Deacon Vorbis (talk) 14:42, 25 September 2017 (UTC)
• Question. You make a big point of changing "upper and lower" to "lower and upper" so that the "lower" one comes first, but then in the next two articles you name after that you put the "upper" one first. What then is the point of the first of the three proposed moves? Michael Hardy (talk) 23:47, 2 October 2017 (UTC)
• Oppose first move, support second and third - I think the way most people think about these things has the "maximum" concept first and then the "minimum" concept, so all three should be ordered that way. This matches maxima and minima, which is based on usage in textbooks and other reliable sources.  — Amakuru (talk) 10:19, 6 October 2017 (UTC)
• Support 2 and 3 but lightly oppose 1. Big and small. Red Slash 16:07, 10 October 2017 (UTC)

The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Sharp and tight bounds of functions

The text says "The upper bound is called sharp if equality holds for at least one value of x." but the referenced source does not support that. As an example of sharp (or tight) bounds that never achieve equality, see my preprint at math arXiv, in which some of the "tight" bounds only approach equality in a limit. I'm no mathematician, so not sure what's the best way to state this. And is there a difference between sharp and tight? Dicklyon (talk) 03:21, 19 April 2021 (UTC)