March 2006 - mathfun@mailman.xmission.com

Re: [math-fun] Terminology
by dasimov＠earthlink.net 20 Mar '06

20 Mar '06

Franklin writes: << Ah, but it isn't that simple. Consider f, not as a function from X to Y, but just as a function. In set-theoretic terms, as a set of ordered pairs, such that nothing appears as the first element of any two distinct pairs. Now the domain is the set of elements appearing as the first element of any pair, and this is exactly the domain in any other sense. But the only reasonable definition for the range is the set of all elements appearing in the second element of any ordered pair; and this is the image in the other sense. Another way to look at it: if Z is the image of f: X -> Y, then f can be regarded as a function X -> Z (or X -> W for any W containing Y as a subset), and this is really the same function. So the image (often called the range) is a property of the function, while the "codomain" is a property of the function as viewed in a particular context, namely as a function X -> Y. >> I don't know about this. (For me, "just a function" always includes a domain & range, in the traditional terminology -- but I know what you're saying.) If every function had its codomain by definition shrunken down to its image, then every function would be onto, and questions about whether certain functions are, or could be, onto would become meaningless. But above all: In order to define a specific function, one needs to begin with a set to which the second element of its ordered pairs belong; it is not until after the function is defined that one can even ask what its image is. At that point, you have the option of defining a new function whose ordered pairs have their second elements in the image of the first function. But you also have the option of retaining the original such set (codomain) if there is some reason to do so. E.g., this makes it much easier to ask questions like this: Given a polynomial function of degree d, in n variables, with integer coefficients P: Z^n -> Z, characterize the set I(n,d) of all possible images. (I don't know whether this problem has been solved or not. Anyone know if it has?) --Dan

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Re: [math-fun] Terminology
by dasimov＠earthlink.net 20 Mar '06

20 Mar '06

Franklin writes: << Regarding defining a specific function, I am thinking in terms of ZF set theory. There, a function is just a set of ordered pairs (of sets) with the requisite uniqueness property. Such a function does not bring with it any concept of its codomain, just the image - which is normally called the range. The only general range would be the set of all sets - which does not exist. >> I would be interested to learn on what basis you state that ZF has a definition of function that agrees with what you write above. To the best of my knowledge the only kind of object in ZF is a set; the word "function" does not play a role. I am also not clear about what question your last sentence about "general range" is attempting to answer. --Dan

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Re: [math-fun] Home on the Range
by Henry Baker 19 Mar '06

19 Mar '06

Dan, I too noticed this back in the 1960's, when I compared my calculus (engineering-oriented) book with my algebra book. Unfortunately, the calculus textbook people haven't been particularly careful about this particular issue -- it's only a matter of time before some major standardized test gets this particular thing wrong & creates more embarrassment for the mathematics community. Before condemning the calculus people too severely, it would be good to check historical usage -- I'm not sure that the algebra people would win the historical usage argument (even among mathematicians). After all, "range" for a gun or an airplane is the actual achievable distance, not the entire shooting range, or the list of all airports in the world. At 08:17 PM 3/18/2006, dasimov(a)earthlink.net wrote: >When I first learned the rigorous definition of a function f: X -> Y, >the terminology was unequivocal: X is the domain of f, and Y is the >range of f. The latter is unconditionally the case, irrespective of whether >f is onto. > >Nowadays, so many high school course and ill-considered calculus texts have >redefined "range" to mean f(X) -- what I was taught is called the image of f -- >that in teaching I've leaned away from using the word range to mean either thing. > >Now I call the classical range by the term "codomain" (and I still call image "image"). > >It's a shame when the cognoscenti feel obliged to follow the ignorami. > >--Dan

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[math-fun] Terminology
by dasimov＠earthlink.net 19 Mar '06

19 Mar '06

When I first learned the rigorous definition of a function f: X -> Y, the terminology was unequivocal: X is the domain of f, and Y is the range of f. The latter is unconditionally the case, irrespective of whether f is onto. Nowadays, so many high school course and ill-considered calculus texts have redefined "range" to mean f(X) -- what I was taught is called the image of f -- that in teaching I've leaned away from using the word range to mean either thing. Now I call the classical range by the term "codomain" (and I still call image "image"). It's a shame when the cognoscenti feel obliged to follow the ignorami. --Dan

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Re: [math-fun] etymology
by Phil Carmody 19 Mar '06

19 Mar '06

From: "Fred lunnon" <fred.lunnon(a)gmail.com> > Subject: Re: [math-fun] etymology > > One way and another, this terminology is rather overworked. > I'm currently trying to recast elementary geometry in terms of > Clifford algebra, so frequently trying to avoid referring simultaneously > to normal subgroups, vectors normal to a surface, and (the only > remotely justifiable use of the word) normalised coordinates. What issue do you have with vectors being normal to a surface? That term is not just entirely justifiable, but I'd contend that it's the single best justified use of the term out of the three you give. The OED dates that usage back to 1650. No other use of the word as an adjective has a longer history. Phil () ASCII ribbon campaign () Hopeless ribbon campaign /\ against HTML mail /\ against gratuitous bloodshed [stolen with permission from Daniel B. Cristofani] __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com

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Re: [math-fun] Re: proofs
by Schroeppel, Richard 18 Mar '06

18 Mar '06

Dylan Thurston <dthurston(a)barnard.edu> sent this. -- Rich ---------------- On Sun, Mar 05, 2006 at 02:35:19PM -0500, dasimov(a)earthlink.net wrote: > In Fall, 2002, the Russian mathematician Grigori Perelman posted to > the math ArXiv website some papers claiming to prove W. Thurston's > geometrization conjecture (TGC) > <http://en.wikipedia.org/wiki/Geometrization_conjecture>. > ... > (I believe there is at least one piece of the TGC proof that Perelman has > promised to post on the ArXiv, but has not yet done so.) That's right; furthermore, it seems unlikely he will post it at this point. > ... That October, John Morgan of Columbia published an article > written in August, 2004 that discussed Perelman's work, but he dd > not affirm that the proof was correct. John Morgan is now more optimistic, and is currently co-writing an article that will have the proof; this article will be peer reviewed. > Assuming the absence of reliable automated theorem-checkers for this > kind of proof, how do we decide when Perelman's work is correct? As someone else said: it's a sociological thing. > (Incidentally, he has apparently never submitted it to a > peer-reviewed journal, ... He now apparently says that he's not interested in getting it published. > but some sources say it's already gotten much more careful scrutiny > than most such submissions would.) Yes, but it's also harder than many other papers. Peace, Dylan Thurston

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Re: [math-fun] Cardinality puzzle - answer
by dasimov＠earthlink.net 18 Mar '06

18 Mar '06

Frankliin wrote: << Here's a related problem. Can you describe a *maximal* uncountable antichain? More difficult: describe an (uncountable) maximal antichain where every set is infinite. (I don't know the answer to this second question. It might be: no, no such description is possible.) >> This lasts kind of antichain can be shown to exist using Zorn's lemma, but here's an construction of an uncountable maximal antichain using only infinite subsets of Z: It consists of a) all subsets intersecting every set {2n, 2n+1} in exactly one element and b) all subsets of the form S_n = Z - {2n, 2n+1}. This is clearly an antichain. If any other subset X could be added, X would contain at least one element from each {2n, 2n+1} to avoid being a subset of one of the S_n's. But this means X has a subset of type a), contradiction. --Dan P.S. Here's an old cardinality puzzle in the same vein: What's the largest size of a collection of subsets of Z such that any two of them intersect in a finite set?

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[math-fun] G ö del meets Schoenberg meets Dylan
by Paul Reiners 18 Mar '06

18 Mar '06

Gödel Leaves Vienna: http://www.leipzig48.com/compositions/kurt/ A dodecaphonic, musical impression I wrote of Kurt Gödel leaving Austria for the U.S.

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Re: [math-fun] Stirling Numbers?
by dasimov＠earthlink.net 15 Mar '06

15 Mar '06

Hah! I hadn't even noticed my post hadn't gone through. After posting I found a recursion formula & calculated the terms up through 8 by hand, which were enough numbers to find the sequence in OEIS. (So I wrote a follow-up post saying Never mind, I learned that they're Bell numbers. Apparently that post is stuck, too.) And Jim, thanks for recommending the Rota book, but if you happen to know an asymptotic formula I'd love if you'd post it. --Dan I wrote about 10 days ago: << Suppose you want to count all possible partitions of the set X_n = {1,2,...,n}. I.e., you want to know how many [collections of mutually disjoint nonempty subsets of X_n] have union = X_n. . . . >> Michael Kleber wrote: << These are the Bell numbers, B(n). A000110 in the EIS, where you can find formulas and references to your heart's content. >>

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[math-fun] etymology
by Cordwell, William R 14 Mar '06

14 Mar '06

Does anyone know why "normal subgroups" are "normal"? Is it because they are the kernel of a homomorphism, and that's the way it's supposed to be? Also, does the use of "normal" for a field come from the relationship to a normal subgroup, in Galois Theory? Thanks, Bill Cordwell

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