Talk:Interval (mathematics)

Doesn't the definition of Interval make as much sense for a poset as a totally ordered set? There is a link to this page from poset that would suggest as much. I'll make the change and expect someone to let me know if I am way off base. ;-> -- Jeff 18:20 Jan 22, 2003 (UTC)

Ah, just changing the definition to use posets allows incomparable elements to be an "Interval" which just wouldn't be right. So I guess the reference from poset is really pointing to the interval notation which can be extended to posets (and just might produce empty sets a lot). Is this used enough to be worth mentioning? -- Jeff

I added [a,b] for a partially ordered set - Patrick 23:00 Jan 22, 2003 (UTC)

In the section about Interval Arithmetic... Division by an interval containing zero is indeed possible if some extensions are made. This makes it possible to get answers such as $[a, \infty ]$.

Divsion with intervals may result in two intervals. Ex. say [1,1]/[-1,1] results in two intervals. [-inf,-1] and [1,inf]

Varying vs. constant interval

Let's say P₁ = Q₁ = 100 and P₅ = Q₅ = 500, but P₂ = 197, P₃ = 301 and P₄ = 404, while Q₂ = 200, Q₃ = 300 and Q₄ = 400——i.e., the "P" and "Q" endpoints are the same, while the "Q"s are equally spaced between each other, and the "P"s aren't. For example (TN = "term number" and UT = upper TN):

$P_{tn}=F(x) G(y_{tn}) H(z)\,\!$

while

$Q_{tn}=Q_1+\frac{TN-1}{UT-1}\Delta Q =P_1+\frac{TN-1}{UT-1}\Delta P\,\!$

Would the proper definition/identification be "the Q's provide auxiliary points between P₁ and P₅ at a constant interval"? Or does this concept have an established name? This article doesn't appear to address this variation.  ~Kaimbridge~20:01, 16 February 2006 (UTC)

fixed improper examples of open and closed intervals

The reason I'm making a change: There are many things wrong with the text I am replacing, but mostly it's because by definition an interval is a set which contains two endpoints so a single valued set can not be an interval. The previous definitions for open and closed intervals also looks more like a discussion of numbers dating back to the turn of the previous millennia where scholars where discussing odd and even numbers more as a philosophical problem. Open and closed intervals have nothing to do with single valued sets nor whether [integers] are open or closed.

Reference: "Calculus With Analytic Geometry" by Earl W. Swokowski, Prindle, Weber & Schmidt, 1979, ISBN: 0-87150-268-2. Pages 5 and 6.

I've removed the following: Intervals of type (1), (5), (7), (9) and (11) are called open intervals (because they are open sets) and intervals (2), (6), (8), (9), (10) and (11) closed intervals (because they are closed sets). Intervals (3) and (4) are sometimes called half-closed (or, not surprisingly, half-open) intervals. Notice that intervals (9) and (11) are both open and closed, which is not the same thing as being half-open and half-closed.

Intervals (1), (2), (3), (4), (10) and (11) are called bounded intervals and intervals (5), (6), (7), (8) and (9) unbounded intervals. Interval (10) is also known as a singleton.

The length of the bounded intervals (1), (2), (3), (4) is b-a in each case. The total length of a sequence of intervals is the sum of the lengths of the intervals. No allowance is made for the intersection of the intervals. For instance, the total length of the sequence {(1,2),(1.5,2.5)} is 1+1=2, despite the fact that the union of the sequence is an interval of length 1.5.

and added the following: Intervals using the round brackets ( or ) as in the general interval (a,b) or specific examples (-1,3) and (2,4) are called open intervals and the endpoints are not included in the set. Intervals using the square brackets [ or ] as in the general interval [a,b] or specific examples [-1,3] and [2,4] are called closed intervals and the endpoints are included in the set. Intervals using both square and round brackets [ and ) or ( and ] as in the general intervals (a,b] and [a,b) or specific examples [-1,3) and (2,4] are called half-closed intervals or half-open intervals.

Rockn-Roll

Question for Functional spaces

Let be I a set in functional space of all the functions f(x) defined on the interval $[0,1]$ my question is if we can get some $I_1 , I_2,I_3,......$ subsets of I, or if the case is more complicate dealing with numbers than with functions.--85.85.100.144 09:11, 19 February 2007 (UTC)

Types

In the section "Higher mathematics," intervals that are closed at infinity should be mentioned as well. (e.g. $\left[ \infty , - \infty \right] = \mathbb{R}^+$ the extended reals.)

Interval Arithmetic in Fortran and C++

Should it be noted that the Sun Studio compilers implement Interval Arithmetic? --rchrd 03:13, 11 July 2006 (UTC)

Done.--Patrick 09:39, 22 May 2007 (UTC)

Use of German Wikipedia Entry as a "See Also"

It should be considered bad form to ask the user to reference another language edition of Wikipedia for more information, can anyone correct this? Anpheus (talk) 19:32, 20 February 2008 (UTC)

Perhaps link to a google or babelfish translation.Tailsfan2 (talk) 16:19, 12 May 2008 (UTC)

Rewrite required

I see several serious issues with the article, and I think it requires a comprehensive revision. Before going ahead and implementing this, I thought that I'd start a discussion. Following are problems with this article:

• Lede: The definition given excludes unbounded intervals such as $[0,\infty)$.
• Also: interval is not a concept from Algebra. It has more to do with analysis, geometry and topology than with algebra.
• The section labeled informal definition basically talks about notation, not definition.
• The section labeled Formal definition speaks about intervals in ordered sets. When mathematicians speak of intervals, they usually mean intervals of real numbers. The mathematical definition of an interval is just a convex subset of the real numbers.

Often, maybe. Usually, can't say. Always - absolutely NOT. SteveWoolf (talk) 05:38, 14 November 2008 (UTC)

• There is another section on intervals in partially ordered set, which should be merged with the Formal definition section.
• It is unclear to me what the operation $x \cdot y$ means in the section on relational operations. Therefore, the whole section is unclear.

Oded (talk) 23:24, 15 April 2008 (UTC)

Seems like a good idea to me. If I have time, I'll post with some more details. But in the mean time, my suggestion is to be BOLD! Cheers, silly rabbit (talk) 23:29, 15 April 2008 (UTC)

Rewrite accomplished. Oded (talk) 01:45, 21 April 2008 (UTC)

Why not define for ANY poset?

The concept of interval is both meaningful and useful for partially ordered sets in general. This should top the article. Then give Real numbers as a special case. As for the case of non-comparable elements a and b, the interval [a,b] will be ø, the empty set. SLWoolf (talk) 16:25, 30 October 2008 (UTC)

I agree to the extent that the general definition should be mentioned in the lede, not exclusively in a tiny section towards the end of the article. On the other hand, intervals of real numbers is what the vast majority of our users will expect here. And everything interesting that can be said about intervals in order theory really belongs into other articles anyway. Therefore in my opinion a single sentence at the end of the lede, pointing away to the appropriate article (partially ordered set) is enough. Not sure if I have the time right now; if I do, I will try to fix the lede and the little paragraph further down. --Hans Adler (talk) 19:09, 30 October 2008 (UTC)

OK, it's slightly more work than I thought because we need to change the definition (case by case for open, closed and half-open intervals). So I can't do it now. --Hans Adler (talk) 19:24, 30 October 2008 (UTC)

You don't have to change any definitions - but they could be much clearer. See the definition of Interval in Partially ordered set. In any case, your article is primarily about REAL intervals and should be renamed that. Not to do so is both incorrect and problematic for those of us who work with partial orders - we need to link to a general Interval definition. S0 - as you are busy, I have tidied things up for now - but technically this article should be renamed. An article specifically about gorillas, for example, should not be named "Great Apes". Hope you agree! Regards SteveWoolf (talk) 05:29, 14 November 2008 (UTC)

The title of the article doesn't have to be perfect, as long as it's easy for the reader to find the article they actually want. In this case, the note at the beginning that points to the article on posets should be enough to clarify the subject of this article. — Carl (CBM · talk) 14:54, 14 November 2008 (UTC)

As I said once, an article specifically about gorillas, for example, should not be named "Great Apes". It isn't correct to call an article Interval and then only talk about a subset of the subject. Of course, I realise renaming it is alot of work, since it has so many links to it. SteveWoolf (talk) 16:30, 14 November 2008 (UTC)

On the other hand, as a mathematician, almost every time I say "interval" I mean "real interval"; this is probably true for everybody except order theorists. And readers who come to the article are most likely to be looking for the definition from their algebra or calculus class. So it's not quite the same situation here as with great apes, and I think that "isn't correct" is too strong a phrase for what's going on. As a somewhat parallel example, the article on concrete is about concrete bound with cement, even though asphalt concrete is also a type of concrete. — Carl (CBM · talk) 16:49, 14 November 2008 (UTC)

Okay, Maybe you're right. Be interesting if we actually knew what % of hits on the article were Real Intervals; but as you say, probably most. I'm happy with the article as it now is, so lets concrete it in and keep the apes out. (Just kidding) SteveWoolf (talk) 07:24, 17 November 2008 (UTC)

Recent Vandalism

To whoever anonymously wrote how he hates Calculus: If you vandalise again, you will be reported. Why not try to make a postive contribution to WP on a topic that interests you? Even sex, drugs and rock'n'roll are all valid topics. I also don't care much for calculus, but I expect you like using things whose production requires advanced mathematics, such as cars and planes and DVD's. SteveWoolf (talk) 12:08, 22 November 2008 (UTC)