Talk:Infimum

The relation at the end was m < x, not m <= x

I fixed it. Priestoferis (talk) 18:28, 16 December 2009 (UTC)

Any bounded nonempty subset has infimum

Hmmm... I can't fix this myself because I'm not 100% sure of the answer, but I wish that the entry made it 100% clear that any bounded nonempty subset of the real numbers has an infimum in the non-extended real numbers.

I added this. - Patrick 08:38, 25 Aug 2003 (UTC)

Wrong?

The infimum and supremum of S are related via

inf(S) = − sup( − S).

? Depends on the set. --Abdull 12:04, 2 December 2005 (UTC)

That sounds correct assuming S is a set of real numbers and −S is defined as the set {s|−s∈S}, that is the set found by negating the elements of S. Then this is simply stating that inf and sup are symmetric in a sign sense. —BenFrantzDale 12:25, 2 December 2005 (UTC)

Confusion

So is it true that to find an infimum, one must know a set and a subset that you want to find the infimum of? And in what cases would the infimum *not* belong to the said subset? The example given on this page:

$\inf \{ (-1)^n + 1/n \mid n = 1, 2, 3, \dots \} = -1$

doesn't seem to work, since :$\{n = 1, 2, 3, \dots \}$ doesn't seem to be a subset of $\{(-1)^n + 1/n \} \!\$. Can anyone help? Fresheneesz 01:58, 22 March 2006 (UTC)

I think the $\{n = 1, 2, 3, \dots \}$ is merely specifying that n must be a natural number in {( − 1)ⁿ + 1 / n}. The infimum specified is the infimum of the set of all ( − 1)ⁿ + 1 / n with natural n – 147.188.225.242 10:18, 15 May 2006 (UTC)

Yes, to find an infimum of a set you need an "enclosing set" to look within. The notation $\{ (-1)^n + 1/n \mid n = 1, 2, 3, \dots \}$ means just one set: "The quantity ( − 1)ⁿ + 1 / n evaluated in turn for each of $n = 1, 2, 3, \dots$. Since that example is in the section "Infima of Real Numbers", it's assumed that the "enclosing set" is $\mathbb{R}$. —The preceding unsigned comment was added by 65.57.245.11 (talk) 01:02, 6 February 2007 (UTC).

Instead of "that is smaller than all other elements of the subset", shouldn't it be "that is not bigger than all other elements of the subset"

Is it me or is this wrong?

   In mathematics the infimum of a subset of some set is the greatest element, not necessarily in the subset

The "greatest element" is defined as a the largest member of the subset, here it says it doesn't have to be .. isn't this a contradiction? —Preceding unsigned comment added by 217.44.0.55 (talk) 14:38, 9 April 2008 (UTC)

The statement is correct. "Greatest element" is the largest member of "some set" (consider, T) that is not bigger than all members of the "subset" (S). It is not necessary that this element belong to the subset (S). This is illustrated by several of the Examples. --Loresayer (talk) 02:35, 1 November 2009 (UTC)

Supremum really a dual to infimum?

Currently, the article says

Infima are in a precise sense dual to the concept of a supremum.

In Wikipedia, the article Duality (order theory) says every partially ordered set P gives rise to a dual partially ordered set. For one subset, I can see that there is exactly one infinum. How does a single infinum lead to a dual "supremum" poset? Thanks, --Abdull (talk) 21:23, 21 February 2010 (UTC)

Yes, they are really dual. Given a poset P and a subset S, we can reverse the ordering and form a dual poset P*, containing S*. (S* is the dual of S, the same subset of elements, but with the ordering reversed as it is in P*).

Now, the statement that infimum and supremum are dual just means

 inf(S)  =  sup(S*)
 sup(S)  =  inf(S*)

Cgwaldman (talk) 23:23, 27 January 2012 (UTC)

Example figure

I think the example in the figure is not very illustrative because it shows a very special case. The one in the supremum page seems slightly better, though I would change the color of the supremum itself. 212.126.224.100 (talk) 11:25, 5 October 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 12:35, 9 March 2011 (UTC)

I agree, both with the proposal to merge these article and with the proposed merge target. Compare for example the articles greatest element, upper and lower bounds and semilattice which discuss dual notions in a single article. — Tobias Bergemann (talk) 08:46, 21 July 2011 (UTC)

Absolutely. —Mark Dominus (talk) 15:26, 21 July 2011 (UTC)

I think a merge would be a great idea, as the two concept are the inverse of one and other and could be explained better together. — Preceding unsigned comment added by 86.44.146.52 (talk) 18:55, 13 January 2012 (UTC)

Question

Talk:Supremum#arg?. Rinconsoleao (talk) 12:04, 20 July 2011 (UTC)