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From: Rob Johnson on 19 Jun 2010 14:46 In article <l2csk4j5631.fsf(a)shaggy.csail.mit.edu>, Bill Dubuque <wgd(a)nestle.csail.mit.edu> wrote: >rob(a)trash.whim.org (Rob Johnson) wrote: >> Bill Dubuque <wgd(a)nestle.csail.mit.edu> wrote: >>>rob(a)trash.whim.org (Rob Johnson) wrote: >>>>Bill Dubuque <wgd(a)nestle.csail.mit.edu> wrote: >>>>>rob(a)trash.whim.org (Rob Johnson) wrote: >>>>>>David C. Ullrich <ullrich(a)math.okstate.edu> wrote: >>>>>>> >>>>>>>Say P is a structure with an addition and >>>>>>>multiplication satisfying >>>>>>> >>>>>>> (i) x+y = y+x >>>>>>> (ii) x+(y+z) = (x+y)+z >>>>>>> (iii) If x+z = y+z then x = y. >>>>>>> (iv) (P, multiplication) is an abelian group >>>>>>> (v) x(y+z) = xy+xz. >>>>>>> >>>>>>>I gather that although the definitions are not quite standard, >>>>>>>this might be called a semi-ring plus cancellation. >>>>>>> >>>>>>>Now suppose that w + x' = w' + x and y + z' = y' + z. >>>>>>>Does it follow that >>>>>>> >>>>>>>(*) wz + xy + x'y' + w'z' = w'z' + x'y' + xy + wz ? >>>>>>> >>>>>>>If we had subtraction then this would be clear, >>>>>>>since (*) just says that (w-x)(y-z) = (w'-x')(y'-z') >>>>>>>and we're given that w-x = w'-x' and y-z = y'-z' >>>>>>>(btw if (*) is _not_ the same as (w-x)(y-z) = (w'-x')(y'-z') >>>>>>>then it's a typo in (*)). Seems like the cancellation >>>>>>>property (iii) should be a substitute for subtraction, >>>>>>>but I've gone around in circles with no luck. >>>>>>> >>>>>>>(If anyone's curious, this has to do with showing >>>>>>>that the obvious definition of multiplication is >>>>>>>well-defined if we apply the Grothendiek construction >>>>>>>to try to get a ring from the semi-ring. The question >>>>>>>doesn't really matter, since in the context I have >>>>>>>in mind there are more axioms that help). >>>>>> >>>>>> Now that I have seen the reply from Bastian Erdnuess, I understand >>>>>> that the question is really to show that >>>>>> >>>>>> wy + xz + w'z' + x'y' = w'y' + x'z' + wz + xy (*) >>>>>> >>>>>> After having worked this through, I realize that it is the samw as >>>>>> Bill Dubuque's answer, but I will post it anyway since a second way >>>>>> of looking at something is sometimes helpful. >>>>>> >>>>>> (wy + xz + w'z' + x'y') + (x'y + w'z + w'y + x'z) >>>>>> >>>>>> = (wy + x'y) + (xz + w'z) + (w'z' + w'y) + (x'y' + x'z) >>>>>> >>>>>> = (w'y + xy) + (x'z + wz) + (w'z + w'y') + (x'y + x'z') >>>>>> >>>>>> = (xy + wz + w'y' + x'z') + (w'y + x'z + w'z + x'y) >>>>>> >>>>>> Now we just cancel the right summand from the far-left and far-right >>>>>> sides to get (*): >>>>>> >>>>>> wy + xz + w'z' + x'y' = xy + wz + w'y' + x'z' >>>>> >>>>>By removing the motivational remarks from my proof, and by inlining >>>>>the lemmas, you've completely obfuscated the essence of the matter. >>>>>In case it was not clear, my proof is just a special case of the >>>>>well-known proof that congruences can be multiplied, e.g. >>>>> >>>>>LEMMA a = A, b = B (mod n) => ab = AB (mod n) >>>>> >>>>>PROOF n|a-A,b-B => n | (a-A)b + A(b-B) = ab - AB >>>> >>>> I removed nothing because I wrote my article based on the typo noted >>>> by Bastian Erdnuess [...] in fact, I had not even read your post >>>> until just before I posted >>> >>>Ok, then I'm quite interested in the motivation behind your proof. >>>Why did you choose those particular expressions? Why add them, etc? >>>As you said, a second way of looking at it may be helpful, so I'm >>>interested in understanding the key idea(s) of your "second way". >> >> As I mentioned in a follow-up message, I had viewed this problem as >> a catalytic process; that is, add something that effects a change but >> itself remains unchanged. Then we can remove the catalyst using the >> cancellation property (iii). In the equations above, I tried to >> exhibit this process using parentheses. The four partial processes >> (annotated in the follow-up) are given by >> >> (wy + x'y) = (w + x')y = (w' + x)y = (w'y + xy) [1] >> >> Equation [1] converts wy to xy while rotating catalyst x'y to w'y >> >> (xz + w'z) = (x + w')z = (x' + w)z = (x'z + wz) [2] >> >> Equation [2] converts xz to wz while rotating catalyst w'z to x'z >> >> (w'z' + w'y) = w'(z' + y) = w'(z + y') = (w'z + w'y') [3] >> >> Equation [3] converts w'z' to w'y' while rotating catalyst w'y to w'z >> >> (x'y' + x'z) = x'(y' + z) = x'(y + z') = (x'y + x'z') [4] >> >> Equation [4] converts x'y' to x'z' while rotating catalyst x'z to x'y >> [...] > >But of course there are obvious symmetries that can be exploited >to streamline the proof. But that does not answer my questions. >Why did you choose those particular expression to start with? >Why did you add them? As-is your proof is pulled out of a hat >with no motivation at all for its genesis. That is why I had >assumed that you had started with one of the earlier posted >proofs when you mentioned that you saw them before posting. I didn't realize that David Ullrich needed a motivational proof; I had the impression that he was simply looking for a justification using the hypotheses he supplied. However, I did not pull my proof from a hat. I started with the equations that convert terms from wy + xz + w'z' + x'y' to terms in xy + wz + w'y' + x'z'. Thus, I looked at the following equations (derived as in [1] - [4] above): (wy + x'y) = (w'y + xy) [converts wy to xy] (xz + w'z) = (x'z + wz) [converts xz to wz] (w'z' + w'y) = (w'z + w'y') [converts w'z' to w'y'] (x'y' + x'z) = (x'y + x'z') [converts x'y' to x'z'] These were the motivation for the following equations in my article > = (wy + x'y) + (xz + w'z) + (w'z' + w'y) + (x'y' + x'z) > > = (w'y + xy) + (x'z + wz) + (w'z + w'y') + (x'y + x'z') I figured that if the equation to be proved was true, the catalytic terms would cancel. >When you said you had a "second way" I thought you meant >that you had a second conceptual way of viewing the proof, >not merely a syntactic rearrangement gotten by stripping >the motivational remarks and inlining the lemmas. Why >such obfuscations appeal to you continues to puzzle me, >just as it has in our many earlier exchanges on such. If you won't take my word that I didn't just rearrange your proof and remove your "motivational remarks", I don't know what I can say that would change your mind. When I first saw your hint, it was not immediately evident to me how the two hypotheses (w + x' = w' + x and y + z' = y' + z) were used. That is why I decided to post what I had done even though you had already posted your hint. Of course, after a little thought, I saw that what you had done was work from the motivational comment that David made regarding subtraction (w-x = W-X and y-z = Y-Z). However, it still takes a bit of work to get > wy+xz+xY+wZ = wY+xZ+xy+wz ie. (w-x)(y-z) = (w-x)(Y-Z) and > wY+xZ+XY+WZ = WY+XZ+xY+wZ ie. (w-x)(Y-Z) = (W-X)(Y-Z) (after changing primes to capitals) from the two hypotheses w + X = W + x and y + Z = Y + z I based my article (in particular, identities [1]-[4] above) on those two hypotheses, not on your hint. >> take out the trash before replying > >I've tried everything, even teaching you "green" proof methods, >with energy-saving reusable lemmas, ROHS-compliant construction >materials, and insightful abstractions that promote optimal reuse. >However, you continue to generate more trash than I can take out. >Where did you learn those transfinite trash generation techniques? If undergraduate education at UCLA and graduate education at Princeton have not forced me to prove things the way you like, then don't feel bad that your educational efforts on usenet have failed. I guess I am just a lost cause. Rob Johnson <rob(a)trash.whim.org> take out the trash before replying to view any ASCII art, display article in a monospaced font
From: master1729 on 21 Jun 2010 10:53
> On Jun 16, 5:32 pm, Gerry Myerson > <ge...(a)maths.mq.edi.ai.i2u4email> > wrote: > > In article > <g3nh16hp0mr5g3f0apdqnfq38b83750...(a)4ax.com>, > > David C. Ullrich <ullr...(a)math.okstate.edu> wrote: > > > (Why? Because constructing the _positive_ > > > reals using Dedekind cuts (sets of positive > > > rationals) works out much more elegantly > > > than constructing the reals using Dedekind > > > cuts; negative numbers introduce unfortunate > > > complications in multiplication. No fair working > > > this out and publishing it, btw... you heard it > > > here first.) > > No, I read it (or something very much like it) > first on pages 25-26 of > > Conway's On Numbers And Games. He says the best > path from the positive > > integers to the reals is via the positive rationals > and then the > > positive reals, and the reason he gives is > precisely that if you go > > from the rationals to the reals by Dedekind cuts > you have to make lots > > of special cases depending on signs when you get to > the multiplication. > > Believe it or not, even _I_ have heard of > constructing > the reals via D-cuts of _positive_ rationals rather > than all the rationals. > > And where have I heard it from? Metamath (a computer > theorem prover created by Norm Megill). Megill > writes: > > "To construct the complex numbers, we start with the > finite ordinals (natural numbers) of set theory and > successively build temporary positive integers, > temporary positive rationals, temporary positive > reals > (based on Dedekind cuts), temporary signed reals, and > finally the actual complex numbers." > > Not only is multiplication easier to define on the > positive D-cuts rather than signed D-cuts, but it's > easier to define positive _rationals_ as equivalence > classes of ordered pairs of positive _naturals_, then > have to include special cases to ensure that the > second > elements (denominator) of the rationals aren't zero. > So > Megill avoids the problems with negatives and the > problem with zero until after the positive D-cuts. > > (Of course, if we were to use Cauchy sequences > instead > of D-cuts, we can at least avoid the problem of > defining multiplication for negative reals.) timothy golden and myself have proposed that too. |