
Re: Brainstorming about STEM (was About Functions)
Posted:
Dec 13, 2011 1:48 AM


> On Mon, Dec 12, 2011 at 11:21 AM, Dan Christensen > <dc@dcproof.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. >
California? > >> 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 multidimensional 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 builtin 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.
Dan www.dcproof.com

