Since the virtue and reliability of Wikis was brought up, I created a R Wiki page for this at http://www.sciviews.org/_rgui/wiki/doku.php?id=beginners:surprises:emptysetfuncs :-)
Anyone: please correct errors and improve it! Tony Plate Duncan Murdoch wrote: > On 1/9/2006 1:27 PM, Liaw, Andy wrote: > >>If you haven't seen this in your math courses, perhaps this would help: >> >>http://en.wikipedia.org/wiki/Empty_set >> > > > This is what is so great about Wikipedia: it gives certainty where I'd > only call it a fairly standard convention. ;-) > > Duncan Murdoch > > > >>which says, in part: >> >>Operations on the empty set >> >>Operations performed on the empty set (as a set of things to be operated >>upon) can also be confusing. (Such operations are nullary operations.) For >>example, the sum of the elements of the empty set is zero, but the product >>of the elements of the empty set is one (see empty product). This may seem >>odd, since there are no elements of the empty set, so how could it matter >>whether they are added or multiplied (since "they" do not exist)? >>Ultimately, the results of these operations say more about the operation in >>question than about the empty set. For instance, notice that zero is the >>identity element for addition, and one is the identity element for >>multiplication. >> >> >>Andy >> >> >>From: Martin Morgan >> >>>I guess I have to say yes, I'd exepct >>> >>>x <- 1:10 >>>sum(x[x>10]) ==> numeric(0) >>> >>>this would be reinforced by recongnizing that numeric(0) is not zero, >>>but nothing. I guess the summation over an empty set is an empty set, >>>rather than a set containing the number 0. Certainly these >>> >>>exp(x[x>10]) ==> numeric(0) >>>numeric(0) + 1 ==> numeric(0) >>> >>>would give me pause. >>> >>> >>>Gabor Grothendieck <[EMAIL PROTECTED]> writes: >>> >>> >>>>The way to think about it is: >>>> >>>> prod(rep(x,n)) == x^n >>>> >>>>and that works for n=0 too. >>> >>>Hmm, Not sure what to put in for x and n? do you mean x == numeric(0), >>>n == 0 (0 copies of an empty set), x == ANY n == numeric(0) (an empty >>>set of ANYthing), x == numeric(0), n == numeric(0) ? For all of these, >>>x^n evaluates to numeric(0). >>> >>>Martin (Morgan) >>> >>>Duncan Murdoch <[EMAIL PROTECTED]> writes: >>> >>> >>>>On 1/9/2006 12:40 PM, Martin Morgan wrote: >>>> >>>>>I'm a little confused. I understand that numeric(0) means an empty >>>>>numeric vector, not the number 0 expressed as numeric. As >>> >>>it is now, >>> >>>>>prod(numeric(0)) generates something -- a vector of length 1 >>>>>containing the number 1 -- from nothing. I would have expected >>>>>prod(numeric(0)) ==> numeric(0) >>>>>this is consistent with >>>>>numeric(0) ==> numeric(0) >>>>>numeric(0) * 1 ==> numeric(0) >>>>>cumprod(numeric(0)) ==> numeric(0) >>>>>and, because concatenation occus before function evaluation, >>>>>prod(c(numeric(0),1)) ==> prod( c(1) ) ==> 1 >>>>>I would expect sum() to behave the same way, e.g., sum(numeric(0)) >>>>>==> >>>>>numeric(0). From below, >>>>> >>>> >>>>I think the code below works as I'd expect. Would you >>> >>>really like the >>> >>>>last answer to be numeric(0)? >>>> >>>> > x <- 1:10 >>>> > sum(x) >>>>[1] 55 >>>> > sum(x[x>5]) >>>>[1] 40 >>>> > sum(x[x>10]) >>>>[1] 0 >>>> >>>>Duncan Murdoch >>>> >>>> >>>>>> >>>> consider exp(sum(log(numeric(0)))) ... ?) >>>>>> >> >> That's a fairly standard mathematical convention, >>>>>>which >>>>>> >> is presumably why sum and prod work that way. >>>>>> >> >> Duncan Murdoch >>>>> >>>>>I would have expected numeric(0) as the result (numeric(0) is the >>>>>result from log(numeric(0)), etc). >>>>>Martin (Morgan) >>>>>Martin Maechler <[EMAIL PROTECTED]> writes: >>>>> >>>>> >>>>>>>>>>>"Ben" == Ben Bolker <[EMAIL PROTECTED]> >>>>>>>>>>> on Sun, 08 Jan 2006 21:40:05 -0500 writes: >>>>>> >>>>>> Ben> Duncan Murdoch wrote: >>>>>> >> On 1/8/2006 9:24 PM, Ben Bolker wrote: >>>>>> >> >>> It surprised me that prod(numeric(0)) is >>> >>>1. I guess >>> >>>>>>if >>>>>> >>> you say (operation(nothing) == identity element) this >>>>>> >>> makes sense, but ?? >>>>>> >> >> >> What value were you expecting, or were you >>>>>>expecting an >>>>>> >> error? I can't think how any other value could be >>>>>> >> justified, and throwing an error would make a lot of >>>>>> >> formulas more complicated. >>>>>> >> >>> >>>>>> >> >>>> consider exp(sum(log(numeric(0)))) ... ?) >>>>>> >> >> That's a fairly standard mathematical convention, >>>>>>which >>>>>> >> is presumably why sum and prod work that way. >>>>>> >> >> Duncan Murdoch >>>>>> >>>>>> Ben> OK. I guess I was expecting NaN/NA (as opposed to >>>>>> Ben> an error), but I take the "this makes everything else >>>>>> Ben> more complicated" point. Should this be documented or >>>>>> Ben> is it just too obvious ... ? (Funny -- I'm willing to >>>>>> Ben> take gamma(1)==1 without any argument or suggestion >>>>>> Ben> that it should be documented ...) >>>>>> >>>>>>see? so it looks to me as if you have finally convinced >>>>>>yourself that '1' is the most reasonable result.. ;-) >>>>>> >>>>>>Anyway, I've added a sentence to help(prod) {which matches >>>>>>the sentence in help(sum), BTW}. >>>>>> >>>>>>Martin >>>>>> >>>>>>______________________________________________ >>>>>>R-devel@r-project.org mailing list >>>>>>https://stat.ethz.ch/mailman/listinfo/r-devel >>> >>>______________________________________________ >>>R-devel@r-project.org mailing list >>>https://stat.ethz.ch/mailman/listinfo/r-devel >>> >>> >> >> >>------------------------------------------------------------------------------ >>Notice: This e-mail message, together with any attachment...{{dropped}} > > > ______________________________________________ > R-devel@r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/r-devel > ______________________________________________ R-devel@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-devel
