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Re: Brainstorming about STEM (was About Functions)
Posted:
Dec 13, 2011 1:48 AM
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> 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 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.
Dan
www.dcproof.com
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