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(* /Archive: someone's homework problem)
Oleg Alexandrov 01:34, 16 Jun 2005 (UTC)
Any better way to write the operator than the current f o g? ?
There is an error in the definition of negative functional powers. -- Wasseralm 19:51, 17 August 2005 (UTC)
OK, you are right. But in this case I would claim we should not try to be too general. The condition f:X->X is in my opinion a very reasonable one to talk about negative functional powers. If the negative functional powers were a really important concept, then maybe it would be worth be general. But since it is just a curiosity, I would think it is not worth the trouble putting the most general condition.
But it is up to you. If you feel like going back to If f:X->Y with Y a subset X, be my guest. :) Oleg Alexandrov 17:34, 20 August 2005 (UTC)
We say:
Does anyone have a reference for this? I'm sure I recall the left-to-right notation being used in my undergrad maths degree, which was only three years ago. — Matt Crypto 11:55, 19 August 2005 (UTC)
Sometmes I wish god had only given us only one hand. I suffer from terrible left/right confusion. A kind of dyslexia I suppose, for example I'm always confusing east and west. Which reminds me of the old conundrum: "why does a mirror reverse left and right, but not top and bottom?" Paul August ☎ 17:30, 7 October 2005 (UTC)
Having functions act on the right of their arguments is still quite common among algebraists. See, for instance, Smith's Postmodern Algebra for a recent book that does this. Since there are obviously some folks who have put a lot of time into this, I will not edit the paragraph myself, but I strongly recommend that it be rewritten to suggest that the other convention is still in common use. Mkinyon 21:22, 21 February 2006 (UTC)
JA: As for typography, these tricks work in some settings:
JA: Jon Awbrey 03:20, 27 January 2006 (UTC)
Would a derivative of a composite function be a "composite derivative"? ~Kaimbridge~ 20:23, 30 January 2006 (UTC)
Consider the graph of the following function:
As the exponent converges to infinity, the function assumes a quizzical shape. A cyclical zig-zag made up of boxes set side by side. Wach one peaks at about 1.5614, and has a width of about 1.52. The latter seems to is about pi/2, which makes sense, since that makes pi its period. As for the former, I have been trying to find a connection between it and other known constants, and have yet to find anything. Any suggestions? He Who Is 21:54, 4 June 2006 (UTC)
Woops... Wasn't thinking when I wrote that. Tangent and cosine. Also, I looked at it more closely and realized that 1 is the maximum of cosx, and the peak of tancostancos...x is tan1. But I still think it is a rather interesting operation, since for everything between pi/4 + npi/2, for all integer n, it converges to tan1. Also, if one looks closely, you can see it has no zeroes, nor does it converge to zero. It actually grows to a value of about .002, shich I assume equals:
. He Who Is 22:00, 4 June 2006 (UTC)
I'm a bit confused by the line "For example, only when ; for all negative , the first expression is undefined." For x < 0 don't we have ? TooMuchMath 05:06, 20 September 2006 (UTC)
I'm wondering if there is something more general than function composition.
Example, I have a function f that maps elements of X onto real numbers. I use this function to define another function g that maps subsets of X onto real numbers--perhaps g gives an average, median, total, minimum, maximum etc.
How can I describe the relationship between f and g? Clearly I cannot say g is composed of f. I want to say something like g is 'based on' f. Anyone know of anything in the literature? If so, I guess there should be a link to funciton composition... —The preceding unsigned comment was added by 220.253.86.44 ( talk) 00:03, 24 April 2007 (UTC).
Thanks for the suggestion and links. So, in my example, g would be an operator and f its operand. In this article composition is an operator, as is the function g o f. My main problem with this is that "operators" appear to be poorly defined and have multiple--conflicting--meanings. 220.253.85.77 03:35, 26 April 2007 (UTC)
...the functions f: X → Y and g: Y → Z can be composed by computing the output of f when it has an argument of g(x) instead of x.
Should this not be computing the output of g when it has an argument of f(x) ? Tobz1000 ( talk) 17:01, 19 May 2009 (UTC)
For instance, the functions f : X → Y and g : Y → Z can be composed by computing the output of g when it has an argument of f(x) instead of x.
Shouldn't that last bit be instead of y, now that the example has been changed? I don't feel qualified to make an edit, but I conferred with a friend and we agreed that it seemed like the anecdote was g(f(x)), where f(x) replaces y in g(y). GeoffHadlington ( talk) 03:27, 29 August 2013 (UTC)
A further variation encountered in computer science is the Z notation: is used to denote the traditional (right) composition, but ⨾ (a fat semicolon with Unicode code point U+2A3E[2]) denotes left composition.
The unicode notation is left untranslated on my computer, even though I have Unicode Arial which works well most of the time. I suggest that someone with knowledge of this add an explanation about where to get the font that would render this symbol. Better yet, why not just refer to it as ";"? The details of Z code is a very special subject that may not belong in this article. SixWingedSeraph ( talk) 14:58, 31 August 2009 (UTC)
The statement that function composition is always associative is obviously false. The referenced page on associativity gives serveral examples of non-associative functions, including substraction over the integers and cross product of vectors. —Preceding unsigned comment added by 132.198.98.23 ( talk) 22:54, 23 March 2010 (UTC)
Sorry, I was momentarily confused with terminology. Of course there is a difference between the order in which one collaspes a chain of maps (associativity of function composition) and the order in which one creates pairs in a sequence of operands for the application of a function of the form A X A -> A (associativity of a binary operator). The later is often but not alway associative. —Preceding unsigned comment added by 132.198.98.23 ( talk) 23:55, 23 March 2010 (UTC)
A function may be defined by multiplication: f(x) = 2x for example. I prefer to apply products to the right for agreement with reading order. Now let’s take ((ab)c)d as a sequence of functions on hyperbolic quaternions using a = 1, b = i, c = j, d = j. The first function is "multiply on the right by i", and the second and third functions are "multiply on the right by j". Now if the second and third functions are composed first, then the final result is i, not the –i of the other composition order.
Comprehending such a difference requires entry into a non-associative structure. The lack of symmetry makes such structures unappealing except for mathematicians like those studying octonions or Lie algebras or some other structure with enough order to accommodate non-associativity. But these studies do not arise in secondary school, so hand waving grants "composition of functions is always associative". The Encyclopedia should not perpetuate a false promise. Rgdboer ( talk) 00:42, 30 April 2017 (UTC)
Thank you for spelling out that fz∘fw = fwz presumption and its invalidity. — Rgdboer ( talk) 02:40, 1 May 2017 (UTC)
As noted above in #typography about six years ago, the appropriate symbol appears to be Unicode U+2218: ∘. I find it displays correctly on IE9 and Mozilla Firefox 8, and is used in List of mathematical symbols. The large circle symbol used in this article is a disconcertingly large workaround. Is there any reason (now that browsers may reasonably be expected to support the more common Unicode symbols) not to update this accordingly in the article? — Quondum ☏ 18:42, 11 May 2012 (UTC)
<math>...</math>
markup is used. --
Beland (
talk)
18:40, 24 January 2021 (UTC)Regarding function composition for which g ∘ f = f ∘ g holds: Do composed functions which have this property have a dedicated name? Can they be considered symmetric functions? -- Abdull ( talk) 11:59, 18 July 2013 (UTC)
The term "Composite function" would often be interpreted in mathematics as a "Piecewise" function. The closest wikipedia page to the desired result that turns up in a search of the term "Composite function" is "Function composition". Should someone add something like "Not to be confused with piecewise functions (piecewise)"?
Micsthepick ( talk) 01:12, 5 June 2016 (UTC)
Please read 'New expression of multivariate function composition', Is it easy to be understand? Can you accept it?
For multivariate function composition
We will give it three expressions like (f.g) for unary function composition. In the expression of (f.g), '.' can be considered as a binary operation taking f and g as its operands or a binary function taking f and g as its variables.
For multivariate function, the first expression is like an operation:
The second one is like a function:
The third one is like a fraction:
Why do we use these forms? We can describe any expression in a fire-new way. For example,,first we denote it as , in which and . In addition, we denote subtraction as , multiplication as , division as , root as and logarithm as respectively. We want give an expression like in which the left part is called bare function containing only symbolics of function and the right part contains only variables.
is an expression of a function of three variables. We consider and as especial functions of three variables too and introduce unary operator to express these especial functions of three variables.
Here or is transitional variable and ..
By these examples we know the meaning of superscript and subscript of and we call it function promotion.
It is clear that we obtain by substituting and in by and respectively. So can be written in:
or
or
We never mind how complex they are. We consider them as
multivariate functions being composition results of two other
multivariate functions being composition results and or promotion results. These new expressions are different from . Actually we had departed bare
function from
variables in these new expressions and there is only one "x" in them. This is what we want to do when we solve
transcendental equations like .
For an unary function promotion, . In special,, in which 'e' is the identity function.
In if and
Note,there is no in the expression.
is called oblique projection of f. Actually it is a function of n-1 variables and is dependent on only f and i,j so we denote it as . For example, — Preceding unsigned comment added by Woodschain175 ( talk • contribs) 22:34, 25 June 2017 (UTC)
You're right, my edit was not needed. Background:
[4] lists all irregular parameters. Especially "="-sign and "|"-sign may cause trouble in regular parameter input. e.g. when entering {{math||x| = 12}}
has both errors.
Now that list has listed these two instances unnecessarily (because, a pipe in a wikilink works fine). When cleaning up that list, I assumed this was a problem. And since the list is recreated every month, I did so twice ;-). THe only advantage of using {{!}} would be, that it does not show up on the list again, in January. - DePiep ( talk) 10:24, 4 December 2017 (UTC)
This article defines something called "generalized composition" which ... strikes me as bizarre. I'm widely read in math, but have never encountered this definition before -- normally, one does not (cannot) assume that the arity of the composed multivariate functions are all identical, like that. Normally, one has just and I'm wondering where the definition here is used, and what it's used for. It feels very linear-algebra-ish, without the linear. The reference on it says "universal algebra" .. I've gone through Paul Cohen's book "Universal Algebra", and I can't even begin to imagine how such a definition of "generalized composition" would appear in there. (I looked: the index points at page 113 which states that the universal functor on the category of sets is the free composition of canonical morphisms. That's not only a mouthful, but is also like a totally different universe than the one here...) Am I being stupid? What is this thing used for? It looks pretty... 67.198.37.16 ( talk) 03:21, 21 December 2017 (UTC)
I agree to Alexey Muranov that "pointwise" doesn't make sense in the first lead sentence. Although the article pointwise is not very clear, I take from it that an operation o: Y × Y → Y on a set Y can be lifted pointwise to an operation O: (X→Y) × (X→Y) → (X→Y) on the set X→Y of all functions from X to Y, by defining (O(f,g))(x) = o(f(x),g(x)) for all x∈X. Commonly, o and O are denoted by the same symbol. The article gives addition and multiplication (apparently of real numbers) as examples. This agrees with what I'd learned as a student. In function composition, we needn't have two functions with same domain and range set, so the notion of "pointwise" isn't applicable. - Jochen Burghardt ( talk) 08:31, 17 January 2019 (UTC)
The context for this discussion: currently, the first sentence reads "[...] function composition is the pointwise application of one function to the result of another [...]". This is not exact and makes no sense because the results of the second function are not necessary functions themselves, but can be numbers, and "pointwise application" of something to a number makes no sense.
Here is an example of the actual pointwise application: let be the function on real numbers that for each yields the operation of multiplication by , and let be the identity function on the reals. Then the pointwise application of to yields the function .
The operation of pointwise application is basically the combinator S of lambda-calculus and combinatory logic: .
Also, one should not say "poinwise sum of the result of and the result of " when what one means is the poinwise sum of and .-- Alexey Muranov ( talk) 09:19, 17 January 2019 (UTC)
Yes, the use of 'pointwise' in that sentence is a clear conceptual error. Was first introduced here [5] as the sentence
In mathematics, function composition is the pointwise application of one function to another to produce a third function.
which is still wrong, but at least one can reinterpret the meaning of 'pointwise' to be exactly what is done during composition and the sentence would be correct, although circular. It was made worse in [6] as
In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.
for which there is no salvaging. It doesn't look like there is any conflict here. Even the editor who introduced it agrees. Everyone can see that composition is not a result of any pointwise application of two functions, or worse of a function and an element, even if the latter is seen as the inclusion map. Cactus0192837465 ( talk) 13:03, 17 January 2019 (UTC)
POV rambling resulting in nothing because Wikipedia is based on properly sourced knowledge and not in opinions from the missinformed
|
---|
|
Ooops! When I started this talk section, I thought you'd all agree to my understanding of "pointwise" - but it seems there are many different understandings around. I think it would be helpful if some people could give a formal definition of their understanding of "pointwise opertion", "pointwise definition" etc. I think the page Talk:pointwise would be most appropriate for this, since the article " pointwise" could well be improved by adding such definitions.
As for the " function composition" article, I take from your above discussion that "pointwise" should be avoided in the lead. While it might be possible to define composition "in a pointwise way" (this vague notion reflecting my lack of understanding) by employing lambda calculus and higher-order functions, this would be far too complicated for the beginners' audience the article is intended for. - Jochen Burghardt ( talk) 19:26, 19 January 2019 (UTC)
Here is a relevant opinion on math.SE. -- Alexey Muranov ( talk) 21:38, 19 January 2019 (UTC)
@ Cactus0192837465: While I appreciate your mathematical knowledge, in my opinion you could try harder to adhere to WP:POLITE. As for sources, I do consider it admissable on a talk page to contribute some statements without having a source at hand. Glancing through the current section, I don't find any source that you gave here; however you were far from scamming. Moreover, please note that I didn't just say "I don't know"! The only thing that comes close to a formal definition of "pointwise" here is mine. Your (unsourced) description "an operation defined to commute with all point evaluations" can probably turned into one, but you didn't do that. - Anyway, I think the issue of "pointwise" is settled as far as it is relevant here, but further clarification on that is useful for the " pointwise" article. - Jochen Burghardt ( talk) 10:08, 20 January 2019 (UTC)
I'm a big fan of the concept of providing real life examples to illustrate a point, so:
If an airplane's elevation at time t is given by the function h(t), and the oxygen concentration at elevation x is given by the function c(x), then (c ° h)(t) describes the oxygen concentration around the plane at time t.
Seems initially promising. However, my understanding of oxygen concentration is that it is roughly constant by elevation. Perhaps it is not but I suggest that the average layperson doesn't have a good sense of the meaning of this function, and therefore gets distracted by issues that have nothing to do with the point being illustrated. (I can go into more details if desired). the good news is there is a very simple solution. I think the average person knows that temperature tends to vary by elevation, so the concept of a function describing temperature as a function of elevation is reasonably well understood. There is no problem with the concept of them airplanes elevation expressed as a function of time t, so this example would work better if we talked about:
If an airplane's elevation at time t is given by the function h(t), and the temperature at elevation x is given by the function c(x), then (c ° h)(t) describes the temperature around the plane at time t.
Any disagreement?
The composition example:
Composition of functions on a finite set: If f = {(1, 3), (2, 1), (3, 4), (4, 6)}, and g = {(1, 5), (2, 3), (3, 4), (4, 1), (5, 3), (6, 2)}, then g ° f = {(1, 4), (2, 5), (3, 1), (4, 2)}}
isn't wrong but it's hardly intuitive. I suggested people new to the concept of composition functions won't find this example insightful.
In contrast, is a wonderful example on the right side of the page, expressed as a graphic with the caption "concrete example for the composition of two functions". I suggest moving notthat graphic up so it appears to the right of the composition example, then change the values in the composition example to match that graphic. Then readers who are not perfectly clear on what's going on in the text and formula portrayal can look at the graphic and gain insight.
This also a graphic with the caption
g ∘ f , the composition of f and g. For example, (g ∘ f )(c) = #.
I think someone was trying to be clever using symbols in Z, but in an introductory exposition this just adds a level of confusion that isn't matched by useful insight. while slightly more general than the concrete example, I don't see that it adds anything useful and propose simply removing it.
I don't believe this graphic is referred to in the text, so unless it had something that I'm missing, I don't think it deserves to remain.-- S Philbrick (Talk) 20:24, 20 July 2020 (UTC)
S Philbrick (Talk) 22:53, 21 July 2020 (UTC)If an airplane's elevation at time t is given by the function h(t), and the pressure at elevation x is given by the function p(x), then (p ° h)(t) describes the pressure around the plane at time t.
If an airplane's elevation (or height above the ground) at time t is given by the function h(t), and the pressure at elevation x is given by the function p(x), then (p ° h)(t) describes the pressure around the plane at time t.
If an airplane's elevation at time t is given by the function e(t), and the pressure at elevation x is given by the function p(x), then (p ° e)(t) describes the pressure around the plane at time t.
I made the changes. I'll copy the graphic removed here, in case anyone is interested.
-- S Philbrick (Talk) 14:15, 22 July 2020 (UTC)
I propose to insert at the very beginning the present SVG image and its caption, with two meanings of “composition”: the binary operation, also a result of the operation.
Arthur Baelde (
talk)
13:55, 1 March 2023 (UTC)
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(* /Archive: someone's homework problem)
Oleg Alexandrov 01:34, 16 Jun 2005 (UTC)
Any better way to write the operator than the current f o g? ?
There is an error in the definition of negative functional powers. -- Wasseralm 19:51, 17 August 2005 (UTC)
OK, you are right. But in this case I would claim we should not try to be too general. The condition f:X->X is in my opinion a very reasonable one to talk about negative functional powers. If the negative functional powers were a really important concept, then maybe it would be worth be general. But since it is just a curiosity, I would think it is not worth the trouble putting the most general condition.
But it is up to you. If you feel like going back to If f:X->Y with Y a subset X, be my guest. :) Oleg Alexandrov 17:34, 20 August 2005 (UTC)
We say:
Does anyone have a reference for this? I'm sure I recall the left-to-right notation being used in my undergrad maths degree, which was only three years ago. — Matt Crypto 11:55, 19 August 2005 (UTC)
Sometmes I wish god had only given us only one hand. I suffer from terrible left/right confusion. A kind of dyslexia I suppose, for example I'm always confusing east and west. Which reminds me of the old conundrum: "why does a mirror reverse left and right, but not top and bottom?" Paul August ☎ 17:30, 7 October 2005 (UTC)
Having functions act on the right of their arguments is still quite common among algebraists. See, for instance, Smith's Postmodern Algebra for a recent book that does this. Since there are obviously some folks who have put a lot of time into this, I will not edit the paragraph myself, but I strongly recommend that it be rewritten to suggest that the other convention is still in common use. Mkinyon 21:22, 21 February 2006 (UTC)
JA: As for typography, these tricks work in some settings:
JA: Jon Awbrey 03:20, 27 January 2006 (UTC)
Would a derivative of a composite function be a "composite derivative"? ~Kaimbridge~ 20:23, 30 January 2006 (UTC)
Consider the graph of the following function:
As the exponent converges to infinity, the function assumes a quizzical shape. A cyclical zig-zag made up of boxes set side by side. Wach one peaks at about 1.5614, and has a width of about 1.52. The latter seems to is about pi/2, which makes sense, since that makes pi its period. As for the former, I have been trying to find a connection between it and other known constants, and have yet to find anything. Any suggestions? He Who Is 21:54, 4 June 2006 (UTC)
Woops... Wasn't thinking when I wrote that. Tangent and cosine. Also, I looked at it more closely and realized that 1 is the maximum of cosx, and the peak of tancostancos...x is tan1. But I still think it is a rather interesting operation, since for everything between pi/4 + npi/2, for all integer n, it converges to tan1. Also, if one looks closely, you can see it has no zeroes, nor does it converge to zero. It actually grows to a value of about .002, shich I assume equals:
. He Who Is 22:00, 4 June 2006 (UTC)
I'm a bit confused by the line "For example, only when ; for all negative , the first expression is undefined." For x < 0 don't we have ? TooMuchMath 05:06, 20 September 2006 (UTC)
I'm wondering if there is something more general than function composition.
Example, I have a function f that maps elements of X onto real numbers. I use this function to define another function g that maps subsets of X onto real numbers--perhaps g gives an average, median, total, minimum, maximum etc.
How can I describe the relationship between f and g? Clearly I cannot say g is composed of f. I want to say something like g is 'based on' f. Anyone know of anything in the literature? If so, I guess there should be a link to funciton composition... —The preceding unsigned comment was added by 220.253.86.44 ( talk) 00:03, 24 April 2007 (UTC).
Thanks for the suggestion and links. So, in my example, g would be an operator and f its operand. In this article composition is an operator, as is the function g o f. My main problem with this is that "operators" appear to be poorly defined and have multiple--conflicting--meanings. 220.253.85.77 03:35, 26 April 2007 (UTC)
...the functions f: X → Y and g: Y → Z can be composed by computing the output of f when it has an argument of g(x) instead of x.
Should this not be computing the output of g when it has an argument of f(x) ? Tobz1000 ( talk) 17:01, 19 May 2009 (UTC)
For instance, the functions f : X → Y and g : Y → Z can be composed by computing the output of g when it has an argument of f(x) instead of x.
Shouldn't that last bit be instead of y, now that the example has been changed? I don't feel qualified to make an edit, but I conferred with a friend and we agreed that it seemed like the anecdote was g(f(x)), where f(x) replaces y in g(y). GeoffHadlington ( talk) 03:27, 29 August 2013 (UTC)
A further variation encountered in computer science is the Z notation: is used to denote the traditional (right) composition, but ⨾ (a fat semicolon with Unicode code point U+2A3E[2]) denotes left composition.
The unicode notation is left untranslated on my computer, even though I have Unicode Arial which works well most of the time. I suggest that someone with knowledge of this add an explanation about where to get the font that would render this symbol. Better yet, why not just refer to it as ";"? The details of Z code is a very special subject that may not belong in this article. SixWingedSeraph ( talk) 14:58, 31 August 2009 (UTC)
The statement that function composition is always associative is obviously false. The referenced page on associativity gives serveral examples of non-associative functions, including substraction over the integers and cross product of vectors. —Preceding unsigned comment added by 132.198.98.23 ( talk) 22:54, 23 March 2010 (UTC)
Sorry, I was momentarily confused with terminology. Of course there is a difference between the order in which one collaspes a chain of maps (associativity of function composition) and the order in which one creates pairs in a sequence of operands for the application of a function of the form A X A -> A (associativity of a binary operator). The later is often but not alway associative. —Preceding unsigned comment added by 132.198.98.23 ( talk) 23:55, 23 March 2010 (UTC)
A function may be defined by multiplication: f(x) = 2x for example. I prefer to apply products to the right for agreement with reading order. Now let’s take ((ab)c)d as a sequence of functions on hyperbolic quaternions using a = 1, b = i, c = j, d = j. The first function is "multiply on the right by i", and the second and third functions are "multiply on the right by j". Now if the second and third functions are composed first, then the final result is i, not the –i of the other composition order.
Comprehending such a difference requires entry into a non-associative structure. The lack of symmetry makes such structures unappealing except for mathematicians like those studying octonions or Lie algebras or some other structure with enough order to accommodate non-associativity. But these studies do not arise in secondary school, so hand waving grants "composition of functions is always associative". The Encyclopedia should not perpetuate a false promise. Rgdboer ( talk) 00:42, 30 April 2017 (UTC)
Thank you for spelling out that fz∘fw = fwz presumption and its invalidity. — Rgdboer ( talk) 02:40, 1 May 2017 (UTC)
As noted above in #typography about six years ago, the appropriate symbol appears to be Unicode U+2218: ∘. I find it displays correctly on IE9 and Mozilla Firefox 8, and is used in List of mathematical symbols. The large circle symbol used in this article is a disconcertingly large workaround. Is there any reason (now that browsers may reasonably be expected to support the more common Unicode symbols) not to update this accordingly in the article? — Quondum ☏ 18:42, 11 May 2012 (UTC)
<math>...</math>
markup is used. --
Beland (
talk)
18:40, 24 January 2021 (UTC)Regarding function composition for which g ∘ f = f ∘ g holds: Do composed functions which have this property have a dedicated name? Can they be considered symmetric functions? -- Abdull ( talk) 11:59, 18 July 2013 (UTC)
The term "Composite function" would often be interpreted in mathematics as a "Piecewise" function. The closest wikipedia page to the desired result that turns up in a search of the term "Composite function" is "Function composition". Should someone add something like "Not to be confused with piecewise functions (piecewise)"?
Micsthepick ( talk) 01:12, 5 June 2016 (UTC)
Please read 'New expression of multivariate function composition', Is it easy to be understand? Can you accept it?
For multivariate function composition
We will give it three expressions like (f.g) for unary function composition. In the expression of (f.g), '.' can be considered as a binary operation taking f and g as its operands or a binary function taking f and g as its variables.
For multivariate function, the first expression is like an operation:
The second one is like a function:
The third one is like a fraction:
Why do we use these forms? We can describe any expression in a fire-new way. For example,,first we denote it as , in which and . In addition, we denote subtraction as , multiplication as , division as , root as and logarithm as respectively. We want give an expression like in which the left part is called bare function containing only symbolics of function and the right part contains only variables.
is an expression of a function of three variables. We consider and as especial functions of three variables too and introduce unary operator to express these especial functions of three variables.
Here or is transitional variable and ..
By these examples we know the meaning of superscript and subscript of and we call it function promotion.
It is clear that we obtain by substituting and in by and respectively. So can be written in:
or
or
We never mind how complex they are. We consider them as
multivariate functions being composition results of two other
multivariate functions being composition results and or promotion results. These new expressions are different from . Actually we had departed bare
function from
variables in these new expressions and there is only one "x" in them. This is what we want to do when we solve
transcendental equations like .
For an unary function promotion, . In special,, in which 'e' is the identity function.
In if and
Note,there is no in the expression.
is called oblique projection of f. Actually it is a function of n-1 variables and is dependent on only f and i,j so we denote it as . For example, — Preceding unsigned comment added by Woodschain175 ( talk • contribs) 22:34, 25 June 2017 (UTC)
You're right, my edit was not needed. Background:
[4] lists all irregular parameters. Especially "="-sign and "|"-sign may cause trouble in regular parameter input. e.g. when entering {{math||x| = 12}}
has both errors.
Now that list has listed these two instances unnecessarily (because, a pipe in a wikilink works fine). When cleaning up that list, I assumed this was a problem. And since the list is recreated every month, I did so twice ;-). THe only advantage of using {{!}} would be, that it does not show up on the list again, in January. - DePiep ( talk) 10:24, 4 December 2017 (UTC)
This article defines something called "generalized composition" which ... strikes me as bizarre. I'm widely read in math, but have never encountered this definition before -- normally, one does not (cannot) assume that the arity of the composed multivariate functions are all identical, like that. Normally, one has just and I'm wondering where the definition here is used, and what it's used for. It feels very linear-algebra-ish, without the linear. The reference on it says "universal algebra" .. I've gone through Paul Cohen's book "Universal Algebra", and I can't even begin to imagine how such a definition of "generalized composition" would appear in there. (I looked: the index points at page 113 which states that the universal functor on the category of sets is the free composition of canonical morphisms. That's not only a mouthful, but is also like a totally different universe than the one here...) Am I being stupid? What is this thing used for? It looks pretty... 67.198.37.16 ( talk) 03:21, 21 December 2017 (UTC)
I agree to Alexey Muranov that "pointwise" doesn't make sense in the first lead sentence. Although the article pointwise is not very clear, I take from it that an operation o: Y × Y → Y on a set Y can be lifted pointwise to an operation O: (X→Y) × (X→Y) → (X→Y) on the set X→Y of all functions from X to Y, by defining (O(f,g))(x) = o(f(x),g(x)) for all x∈X. Commonly, o and O are denoted by the same symbol. The article gives addition and multiplication (apparently of real numbers) as examples. This agrees with what I'd learned as a student. In function composition, we needn't have two functions with same domain and range set, so the notion of "pointwise" isn't applicable. - Jochen Burghardt ( talk) 08:31, 17 January 2019 (UTC)
The context for this discussion: currently, the first sentence reads "[...] function composition is the pointwise application of one function to the result of another [...]". This is not exact and makes no sense because the results of the second function are not necessary functions themselves, but can be numbers, and "pointwise application" of something to a number makes no sense.
Here is an example of the actual pointwise application: let be the function on real numbers that for each yields the operation of multiplication by , and let be the identity function on the reals. Then the pointwise application of to yields the function .
The operation of pointwise application is basically the combinator S of lambda-calculus and combinatory logic: .
Also, one should not say "poinwise sum of the result of and the result of " when what one means is the poinwise sum of and .-- Alexey Muranov ( talk) 09:19, 17 January 2019 (UTC)
Yes, the use of 'pointwise' in that sentence is a clear conceptual error. Was first introduced here [5] as the sentence
In mathematics, function composition is the pointwise application of one function to another to produce a third function.
which is still wrong, but at least one can reinterpret the meaning of 'pointwise' to be exactly what is done during composition and the sentence would be correct, although circular. It was made worse in [6] as
In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function.
for which there is no salvaging. It doesn't look like there is any conflict here. Even the editor who introduced it agrees. Everyone can see that composition is not a result of any pointwise application of two functions, or worse of a function and an element, even if the latter is seen as the inclusion map. Cactus0192837465 ( talk) 13:03, 17 January 2019 (UTC)
POV rambling resulting in nothing because Wikipedia is based on properly sourced knowledge and not in opinions from the missinformed
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Ooops! When I started this talk section, I thought you'd all agree to my understanding of "pointwise" - but it seems there are many different understandings around. I think it would be helpful if some people could give a formal definition of their understanding of "pointwise opertion", "pointwise definition" etc. I think the page Talk:pointwise would be most appropriate for this, since the article " pointwise" could well be improved by adding such definitions.
As for the " function composition" article, I take from your above discussion that "pointwise" should be avoided in the lead. While it might be possible to define composition "in a pointwise way" (this vague notion reflecting my lack of understanding) by employing lambda calculus and higher-order functions, this would be far too complicated for the beginners' audience the article is intended for. - Jochen Burghardt ( talk) 19:26, 19 January 2019 (UTC)
Here is a relevant opinion on math.SE. -- Alexey Muranov ( talk) 21:38, 19 January 2019 (UTC)
@ Cactus0192837465: While I appreciate your mathematical knowledge, in my opinion you could try harder to adhere to WP:POLITE. As for sources, I do consider it admissable on a talk page to contribute some statements without having a source at hand. Glancing through the current section, I don't find any source that you gave here; however you were far from scamming. Moreover, please note that I didn't just say "I don't know"! The only thing that comes close to a formal definition of "pointwise" here is mine. Your (unsourced) description "an operation defined to commute with all point evaluations" can probably turned into one, but you didn't do that. - Anyway, I think the issue of "pointwise" is settled as far as it is relevant here, but further clarification on that is useful for the " pointwise" article. - Jochen Burghardt ( talk) 10:08, 20 January 2019 (UTC)
I'm a big fan of the concept of providing real life examples to illustrate a point, so:
If an airplane's elevation at time t is given by the function h(t), and the oxygen concentration at elevation x is given by the function c(x), then (c ° h)(t) describes the oxygen concentration around the plane at time t.
Seems initially promising. However, my understanding of oxygen concentration is that it is roughly constant by elevation. Perhaps it is not but I suggest that the average layperson doesn't have a good sense of the meaning of this function, and therefore gets distracted by issues that have nothing to do with the point being illustrated. (I can go into more details if desired). the good news is there is a very simple solution. I think the average person knows that temperature tends to vary by elevation, so the concept of a function describing temperature as a function of elevation is reasonably well understood. There is no problem with the concept of them airplanes elevation expressed as a function of time t, so this example would work better if we talked about:
If an airplane's elevation at time t is given by the function h(t), and the temperature at elevation x is given by the function c(x), then (c ° h)(t) describes the temperature around the plane at time t.
Any disagreement?
The composition example:
Composition of functions on a finite set: If f = {(1, 3), (2, 1), (3, 4), (4, 6)}, and g = {(1, 5), (2, 3), (3, 4), (4, 1), (5, 3), (6, 2)}, then g ° f = {(1, 4), (2, 5), (3, 1), (4, 2)}}
isn't wrong but it's hardly intuitive. I suggested people new to the concept of composition functions won't find this example insightful.
In contrast, is a wonderful example on the right side of the page, expressed as a graphic with the caption "concrete example for the composition of two functions". I suggest moving notthat graphic up so it appears to the right of the composition example, then change the values in the composition example to match that graphic. Then readers who are not perfectly clear on what's going on in the text and formula portrayal can look at the graphic and gain insight.
This also a graphic with the caption
g ∘ f , the composition of f and g. For example, (g ∘ f )(c) = #.
I think someone was trying to be clever using symbols in Z, but in an introductory exposition this just adds a level of confusion that isn't matched by useful insight. while slightly more general than the concrete example, I don't see that it adds anything useful and propose simply removing it.
I don't believe this graphic is referred to in the text, so unless it had something that I'm missing, I don't think it deserves to remain.-- S Philbrick (Talk) 20:24, 20 July 2020 (UTC)
S Philbrick (Talk) 22:53, 21 July 2020 (UTC)If an airplane's elevation at time t is given by the function h(t), and the pressure at elevation x is given by the function p(x), then (p ° h)(t) describes the pressure around the plane at time t.
If an airplane's elevation (or height above the ground) at time t is given by the function h(t), and the pressure at elevation x is given by the function p(x), then (p ° h)(t) describes the pressure around the plane at time t.
If an airplane's elevation at time t is given by the function e(t), and the pressure at elevation x is given by the function p(x), then (p ° e)(t) describes the pressure around the plane at time t.
I made the changes. I'll copy the graphic removed here, in case anyone is interested.
-- S Philbrick (Talk) 14:15, 22 July 2020 (UTC)
I propose to insert at the very beginning the present SVG image and its caption, with two meanings of “composition”: the binary operation, also a result of the operation.
Arthur Baelde (
talk)
13:55, 1 March 2023 (UTC)