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• CommentAuthorAnixx
• CommentTimeJan 12th 2011 edited

I was surprised that many people in http://mathoverflow.net/questions/51829/are-there-linear-operators-which-do-not-belong-to-the-following-classes said that the notation was bad, ands some even did not understand it. Especially surprising was one comment that said that f(z) is not a function, but a number (dispite z was defined as a free variable). Even more surprising that this comment received much votes.

It that case people how would you write the following expression?

$$frac{x^3+2x^2+\sin x}{x^5-x^3+cos(x)+5}\circ \frac{x^3-5x + \log x}{x^7+3x^2+10}$$
1.
(Not sure why the LaTeX is not rendering here...)

I don't think anyone would use the composition symbol $\circ$ with expressions like that. You want to do it like this:

\begin{equation}
f: \mathbb R \to \mathbb R
\end{equation}
\begin{equation}
f: x \mapsto f(x) = \frac{x^3+2x^2+\sin x}{x^5-x^3+cos(x)+5}
\end{equation}

\begin{equation}
g: \mathbb R \to \mathbb R
\end{equation}
\begin{equation}
g: x \mapsto g(x) = \frac{x^3-5x + \log x}{x^7+3x^2+10}
\end{equation}

Then you can say $(f \circ g) : \mathbb R \to \mathbb R$ is the function defined by $(f \circ g) : x \mapsto (f \circ g)(x) = f(g(x))$.

The point is that $f$ and $g$ and $f\circ g$ are FUNCTIONS, and $f(x)$, $g(x)$, and $(f\circ g)(x)$ are the VALUES of these functions at the point $x$, which happen to be real numbers in the case of your example.

Of course in freshman calculus, we alway abuse notation and say things like "the function $f(x)$" but strictly speaking, that is sloppy and potentially confusing notation. In higher mathematics we always try to be much more careful.
2.

I would write,

Let $f,g:\mathbb{C}\rightarrow\mathbb{C}$ be defined by
$$f(x)=\frac{x^3+2x^2+\sin(x)}{x^5-x^3+\cos(x)+5}$$
and
$$g(x)=\frac{x^3-5x+\log(x)}{x^7+3x^2+10}.$$
Consider $f\circ g$.

Giving complicated things names (and using English/your native language to explain them) is a good thing.

• CommentAuthorAnixx
• CommentTimeJan 12th 2011

I thought that x in the context of functions of one variable represents identity function. If one wants a particular value he uses x_0 or something. Otherwise such expressions as (\sin x/x)' would have no meaning.
• CommentAuthorAnixx
• CommentTimeJan 12th 2011

And yes, I always use x in that meaning, sometimes making it bold.
3.

Oddly enough, there was some discussion about this sort of notation earlier: here on meta and here on MO. I tend to favor the more verbose choices of notation given above, because I think they are less likely to cause confusion for the reader.

4.
No, $x$ is a real number, whether you consider it as a fixed constant, or as a variable real number. The identity function $i : \mathbb R \to \mathbb R$ is defined by $i(x) = x$ for all $x \in \mathbb R$. All the $x$'s in this post are real numbers, not functions.
• CommentAuthorAnixx
• CommentTimeJan 12th 2011

So far I see that the poll shows that the majority of people here support such notation. That makes me curious why there was such massive outcry when I used such netation.
5.
I agree that (\sin x/x)' has no meaning if you are being careful about notation. We always make that abuse of notation in freshman calculus, where there is no danger of confusion. (In fact, when you try to explain the distinction between the function f and its values f(x) to such students, that usually *does* confuse them, so it's best to avoid that.)

Note that with your convention of x as being the identity function, what does $\frac{\sin x}{x}$ even mean? You are implicitly mixing up both your notation and the one described above by myself and Zev. To DIVIDE sin(x) by x, you have to think of these two things as NUMBERS.
6.
What poll? Everyone other than you who has posted on this discussion does not support your ambiguous notation. We agree that this is done sometimes in freshman level courses, but it should always be avoided by professional mathematicians in any other context.
• CommentAuthorAnixx
• CommentTimeJan 12th 2011

• CommentAuthorAnixx
• CommentTimeJan 12th 2011

> To DIVIDE sin(x) by x, you have to think of these two things as NUMBERS.

No.
7.
Okay, I see what poll you refer to- I hadn't seen Scott's post. It's an interesting poll. I'd be curious to know the background of the voters. You'd probably get different results if all the voters were professional pure mathematicians, with no students or applied mathematicians. (I'm not insinuating that there's anything wrong with students or applied mathematicians, merely that I believe that an audience of exclusively pure mathematicians would vote differently. That's my opinion, and you are free to have other opinions....)
• CommentAuthorAnixx
• CommentTimeJan 12th 2011

By the way, how would you define an operator U that transforms f(x) into sin x / f(x) for any x and f? Would you write sine without argument, just like

U=\frac{sin}{f} ?

weird...
8.
Anixx: If $f: A \to B$ and $g A \to B$ are two functions from a domain $A$ to a codomain $B$, and $a$ is an element of the set $A$, then we can only make sense of the expression $f(a) / g(a)$ if the set $B$ has a division operation. For example, if $B$ is the real or complex numbers. So in general you cannot divide one function by another, you can only divide two elements of a field or of a division algebra in general.
9.
You cannot unambiguously define a function that takes the point f(x) to the point sin x / f(x) unless f was invertible. In that case I would write the function that sends f(x) to sin x / f(x) as

y \mapsto \sin (f ^{-1} (y)) / y

No confusion.
10.
I'm actually tired of this. Sorry, I have other things to do now.
• CommentAuthorAnixx
• CommentTimeJan 12th 2011

Division of two functions is just superposition of division function and both f and g. One can argue that the domain of definition of the resulting function differs from the domain of definition of f and g, but nothing more.
11.

The person who just voted and left a comment on the poll seems to have missed the part that said, "This poll has now run its course".

• CommentAuthorAnixx
• CommentTimeJan 12th 2011 edited

> You cannot unambiguously define a function that takes the point f(x) to the point sin x / f(x) unless f was invertible.

I cannot get what are you writing about. This is division, not applying an inverse function. Or do you use slash for denoting inverse function? :-/ How then you denote division?
12.
Anixx: you missed Spiro's point. He is used to (as am I) working in sets where division is not defined.

Generally in pure mathematics, a function has a domain which is fixed a priori. Changing the domain of definition is usually cheating and not allowed, and, in any case, should not be taken lightly.

Sprio: Yes the distinction between a function and its values is confusing to first year students. However, learning this IS a fundamental part of learning calculus, and I think failing to explain this to them is a dereliction of duty, as is giving an A to anyone who does not understand this.
• CommentAuthorAnixx
• CommentTimeJan 12th 2011 edited

> Generally in pure mathematics, a function has a domain which is fixed a priori.

Well I do not know which branch of mathematics do you mean, but in calculus, analysis, linear algebra constructing new functions by dividing one by another is a common practice.

> Changing the domain of definition is usually cheating and not allowed, and, in any case, should not be taken lightly.

It is always treated by analyzing the denominators of the fraction so it not to be zero. This is practiced always and the most widely used approach. One can also define an extended real line. So what's the problem with deviding functions?

It seems that you are working in a domain where you very rarely use actual expressions for functions, just use names of functions and their domains of definition at best. But when you often manipulate with expressions for functions, it is very difficult to exclude the free variable from such expressions.

If you are dealing with functions of multiple variables, series, integrals of them, it becomes virtually impossible.
• CommentAuthorEmerton
• CommentTimeJan 12th 2011

I don't understand the fuss being made by people in this discussion. I have no objection to writing $f(x)$ as a function. I frequently do this in the context of $f$ being a polynomial. In other contexts I don't. Just as I sometimes write $\int f(x) d\mu(x)$ when integrating a function $f$ with respect to a measure $\mu$, and other times I just write $\int f d\mu.$ It depends on the context, and what I plan to do next with the formula.

I am also one of the people who voted in Andrew Stacy's poll, who has no objection to writing $f(x)$ to denote a function.

In the same vein, writing $f(x)/x$ (say) to denote the quotient of one function by another seems perfectly fine to me, in the appropriate context. I can see that there are times when it may be a source of confusion (because of issues of the precise domain), and then one would avoid it, but there are lots of other contexts in which it would be perfectly okay (the most common, for me, being if $f(x)$ was a rational function, or more generally an analytic function, of $x$).

I agree that I wouldn't normally use the composition symbol $\circ$ in the way that Anixx did in his post at the top of this thread, just because I think it looks a little strange, but it's meaning is certainly clear, and I don't see what is so objectionable about it.

Incidentally, I am a pure mathematician.

13.
Anixx: I am an algebraist. I do work with actual polynomials, but only within the ring of polynomials of several variables, where there is no division. In any case, I rarely need to think about the polynomials as functions - only as formal elements in a ring. Functions for me are rarely functions of numbers - usually they are maps taking points in one (usually abstractly defined) geometric object to another, or elements in one ring to another, or one combinatorial object to another.

I have a feeling this may be a conflict between those who view mathematics as the study of structure and those who view mathematics as the study of number. To those who see mathematics as the study of structure, functions must be reified (i.e. treated as objects in their own right) and distinguished from their values; to those who see mathematics as the study of number, functions are second class objects having no existence apart from the numbers they are applied to.

I am biased towards carefully distinguishing functions from their values in part because I see the reification of the function as one of the most important intellectual advances of the past half-millenium, with important implications not only in mathematics but also in computer science, philosophy, and to a lesser extent the humanities in general.

ps - Look up 'reify' in the dictionary. It's an important word.
• CommentAuthorAnixx
• CommentTimeJan 12th 2011 edited

Nobody argues against it. But excluding free variable from the expression of function is similar to exclusion of zero from expression of numbers. Some people use "id" symbol, but it is not always optimal. I sometimes regret that there is no commonly accepted symbol for identity function that is suitable for expressions. How would you like sin(id) as a reference to sine function?
14.
I would hate sin(id) as a reference to the sine function, because () are applied to numbers, and composition of functions should be done with $\circ$.

I am happy with $\sin$ being used to referred to the sine function. Just like $f$ is a function, and $f(x)$ is the number you get by applying $f$ to $x$, $\sin$ is a function, and $\sin(x)$ is the number you get by applying $\sin$ to $x$.

My personal preference would be for a notation like $f = [ x \mapsto \frac{\sin(x)}{x} ]$, read as "f equals the function mapping x to sin(x)/x" (even for teaching calculus), but I realize that is not standard.
• CommentAuthorAnixx
• CommentTimeJan 12th 2011

> and composition of functions should be done with $\circ$.

How would you make composition of functions with circ if the left function is function of two variables?
• CommentAuthorAnixx
• CommentTimeJan 12th 2011 edited

> I am happy with $\sin$ being used to referred to the sine function.

What about $f(x)=B_x(a)$ ? Would you just use it B in expressions, without argument?
15.

Anixx, a function of two real variables is a function of one variable from R^2. For example, if I had f:R->R^2 and g:R^2->R, there's nothing wrong with writing g \circ f.

16.

You are too focused on writing the functions with their arguments "on display" - I think you would do well to think of functions more categorically, with the domain and codomain built into the definition of the function, and without there necessarily being "arguments" to work with because the objects you are working with are not necessarily sets. In this understanding, the function f:R->R defined by f(x)=0 and the function g:R->C defined by g(x)=0 are different functions, for example. Furthermore, it is part of the definitions of a category that one can only compose functions f and g when we have f\in Hom(A,B) and g\in Hom(B,C), where A, B, and C are objects of your category.

I very much agree with Alexander Woo, in that I think of a function as an object in and of itself, and not as a mere relation between independent and dependent variables.

As to your second comment, it just depends on what I feel like emphasizing. It's obvious that a subscript is just another place to put an argument, i.e. defining the function $f:X->Z$ by $f(x)=B_x(a)$, where {B_x:Y->Z}_{x\in X} is a collection of functions indexed by X and $a\in Y$, is equivalent to defining $f:x->Z$ by $f(x)=B(x,a)$, where $B:X x Y ->Z$ and $a\in Y$, and $B(x,y)=B_x(y)$ where $B_x$ are the functions from before. If for some reason I felt like keeping the setup of an indexed family of functions, it would be perfectly fine to write B_x for a specific x\in X, as that is the name of a specific function. If I converted the indexed family of functions into a big function B (again, they are equivalent), then I would be fine with just talking about B, without requiring myself to write in an x in the first argument of B. However, I expect that people would for the most part understand what I meant if I never explicitly mentioned the change but still wrote B without a subscript.

• CommentAuthorEmerton
• CommentTimeJan 13th 2011

Dear Zev,

I think that one should be careful making categorical statements of the form "You are too focused on writing the functions with their argments "on display"". Anixx uses a certain system of notation which is completely reasonable, possibly slightly idiosyncratic when considered in one or two of its details, but which has much in common with systems of notation used by many other mathematicians and scientists who use mathematics as a tool. Furthermore, my impression based on Anixx's posting history is that her/his topics of interest are all closely related to classical analysis. I don't see that there is any particular need "to think of functions more categorically" in that context.

Incidentally, how do you denote a sequence? A sequence is a function from the natural numbers to the real numbers (say), and almost everyone I know denotes such a thing $a_n$, not just $a$, i.e. they put the variable $n$ on display in their notation. This is convenient for many purposes, even if it is occasionally bothersome (in that certain expressions and constructions become more convoluted when forced into this notational convention). Furthermore, most people write $a_{n_i}$ for a subsequence, rather than something more functorial such as $a \circ j$ where $j$ is a monotone injection from the natural numbers to itself. The notation $f(x)$ is simiarly traditional in many circles (even if it is not as universal as the sequence notation $a_n$). Like sequence notation, it serves some purposes, and can be bothersome in others.

Finally, I do often denote the identity function from an algebraically closed field (say) to itself by $x$. This is very standard in algebraic geometry, where $x$ stands for both a polynomial variable and the (identity) function on the field of coefficients that it induces. I don't think this reflects a lack of functorial thinking; it is just a particular notation tradition.

Regards,

Matthew

17.

Emerton, my impression so far was that Anixx has not agreed that the categorical way of thinking of functions has any value - I was attempting to identify the point on which he is being stubborn, or at least the point on which our disagreement lies, so that we could effectively discuss it. The "you" was referring only to Anixx. I didn't mean to come across as claiming that writing f(x) (or the expression for f(x), if there is one) for the function f was universally bad - I don't think that at all - I just mean that Anixx is for whatever reason very intent on writing it that way, that this stubbornness is causing some confusion for him, and that considering an alternative point of view would be good for him, personally. For example, I expect that Anixx would not have had concerns about how to write the composition of functions with more than 1 variable if he had considered composition from a domain/codomain/categorical standpoint. I do agree that Anixx's questions are on a subject where the categorical approach is not required, but again, I feel that it would be helpful for Anixx to at least appreciate what Alexander Woo eloquently describes as "the reification of the function".

I stand by my answer in the case of the particular setup that Anixx proposed, namely a family of functions indexed by an arbitrary set. I completely agree that it is customary, and quite often useful, to write a sequence as a_n, not a:N->R. However, I would point out that it's common, at least in my experience, to use "a" as a name for the sequence (at least when it's not being used for the limit of the sequence), i.e. $a={a_n}_{n\in N}$, which I consider an acceptable middle ground between only ever writing a_n's (and ignoring the sequence as an object itself), and defining "a" to be a function a:N->R.

I will attempt to think of a better way of phrasing my previous post, but I've got to go to sleep for now.

18.

@Emerton- I have to disagree with you on reasonableness. The issue isn't that Anixx uses the notation f(z), but that s/he uses it in a way which doesn't make sense. Writing f(z) \circ g(z) for f(g(z)) is not a reasonable thing to do, and it was notational choices like that which spawned this thread.

• CommentAuthorAnixx
• CommentTimeJan 13th 2011

> Anixx, a function of two real variables is a function of one variable from R^2. For example, if I had f:R->R^2 and g:R^2->R, there's nothing wrong with writing g \circ f.

Of course, if you use both arguments as one vector. In reality different arguments of a function have little in common (for example, Bernoulli polynomial, what's the reason to represent its argument and order as something united?), in fact, in most cases in analysis one argument is taken constant while the other is variable.
• CommentAuthorEmerton
• CommentTimeJan 13th 2011

Dear Ben,

I agree that $f(z) \circ g(z)$ is not good notation, and so "completely reasonable" was too strong a claim. The point remains, though, that there is nothing wrong with writing $f(x)$ to denote a function, or $\sin(x)/x$ for that matter.

Best wishes,

Matt