|
|
Re: Brainstorming about STEM (was About Functions)
Posted:
Dec 12, 2011 2:21 PM
|
|
> On Sun, Dec 11, 2011 at 10:10 PM, Dan Christensen
> <dc@dcproof.com> wrote:
> > Using another kind of computer-like language, DC
> Proof, we can represent a function f mapping set A to
> set B as follows:
> >
> > ALL(x):[x in A => f(x) in B] (replace "in" by
> epsilon)
> >
>
> 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.
> would work for me but I'd still to flesh out f and
> set A for this to
> work. set type supports intersection, union and so
> on.
>
Intersections and unions are supported in my system as well.
> One question is how much to stress the set concept,
> big in the 1960s
> thanks to New Math and still a feature in Saxon,
> Singapore etc.
> (inheritors of much Dolciani -type thinking,
> subclasses we might say).
>
> Why not just treat sets as one more type and stop
> trying to consider
> which type is "most primitive"?
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."
> 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.
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.
Dan
>
> Along the DM track, we're free to make these
> base-level pioneering
> adjustments because we're not hampered by the AM
> (analog math)
> detritus, the cultural baggage, the flotsam and
> jetsam. A new kind
> of freedom, helps build ubuntu (community spirit).
>
> DC Proof looks interesting. I used to play WFF 'n
> Proof a lot, used
> that RPN style if that's what it was, not unlike LISP
> which they'll
> hasten to tell you is different etc. Fun world.
>
> Kirby
>
>
> > In words: For all x, if x is an element of A, then
> f(x) is an element B.
> >
> > This simple notation completely characterizes all
> functions of 1 variable.
> >
> > In the tutorial included with my program is a
> worked example (#6) illustrating the composition of
> functions, e.g. f(g(x)), along with suitable
> exercises with hints and full solutions.
> >
> > Dan
> > Download my DC Proof 2.0 freeware at
> http://www.dcproof.com
|
|
