> On Mon, Dec 12, 2011 at 11:21 AM, Dan Christensen > <email@example.com> wrote: > > >> > >> B = [ f(x) for x in A ] > >> > > > > B is not the range of f. The range of f is as > subset of B in this case. > > > > Yes, and for my part, B would not be a set after > that > last assignment, but a list, another object type. > [snip]
How do you change a set into something that is not a set? You have to be consistent.
> > > > There is a Set predicate in my system. Only if you > declare an object to be a set can you apply the > various axioms of set theory on it, e.g. selecting an > arbitrary subset of it. This setup avoids a number of > theoretical difficlties that makes the system easier > to use and more "mathematical." > > > > We can use sets to purge dupes out of lists.
You can look at lists as functions mapping N (the set of natural numbers) to some set, possibly N itself. I suppose, you could look at the range (or codomain) of these functions as a "list" purged of duplicates.
> That's > one of the exercises I score. There's a distance > education > component. This stuff about the print( ) function is > going > out there, but not with as much math as it could. >
You are using the word "function" in at least two different ways. It might confuse students. There are biological or mechanical functions and computer software functions -- all involving actions of some kind, be they actions of electrons, molecules, cells or larger parts in the physical world -- and there are mathematical functions that are abstract relationships between two or more sets of objects. (And others, I'm sure.)
> In the wings: modules to generate new kind of > science > and Game of Life style arrays, a Mandelbrot Set > generator, > lots more fun and games with permutations. If we > become > certified to offer something more like DM towards a > post- > high school degree, then we might tap more of this. > > In the mean time, I own the material under my own > name > and can dribble it out to other sources. The > material > becomes more standard that way, which advantages my > home faculty in Sebastopol (the base) if and when. >
> >> I'm from the > >> Wittgensteinian camp on > >> matters mathematical, which nets me some ridicule > >> from Hansen, but I > >> say it saves me time. > >> > >> I'm not as tempted to waste so many hours on the > >> union and > >> intersection of sets when we could also be > playing > >> with lists, tuples > >> etc. Lists of lists, or multi-dimensional lists, > are > >> worth relatively > >> more time, sets relatively less. > > > > I don't spend any time on unions and intersections > in my tutorial, but if you want to make up lessons > based on them, I do have built-in notation to handle > them. All of the rest is also possible. > > > > I've been looking over the documentation at your > site. > > I haven't done a whole lot with truth tables in my > writings here. >
I make no use of truth tables. Rather, I use my own simplified form of natural deduction.
> The geometry I develop is spatial to begin with and > only gets planar > later. Polyhedrons (wholes, shapes) before polygons > (facets, fields). >
While it certainly has its application in many areas, I don't think geometry is best way to teach proofs as encountered in advanced math courses. See "To the Educator" at my website.
> Our geometry links to geodesy (GIS / GPS). > > This is a STEM curriculum, not trying to "keep it > pure" (deliberately > bouncing around between S, T, E and M at high > frequency, a pedagogical > / andragogical strategy). > > > > Write to me with your ideas, and maybe we can come > up with something. But do have a look at the > tutorial. I think you will find that it is a nice > introduction to logic and proof. It takes you all the > way from proving if A and B is true then B and A is > also true to proof by induction and elementary number > theory in 10 lessons. > > > > I haven't found the tutorial yet. Is it a PDF?
It's part of the Help file in my program. There is a brief excerpt at my website: Click on "Features," scroll down, click on "View table of contents and excerpt."
> > I'm interesting in the segments linking boolean > expressions to logic > gates.
You can introduce propositional logic using my program, but maybe an approach based on truth tables might be a productive for circuit design.