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Topic: Simple analytical properties of n/d
Replies: 20   Last Post: Mar 11, 2013 11:01 PM

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 ross.finlayson@gmail.com Posts: 2,700 Registered: 2/15/09
Re: Simple analytical properties of n/d
Posted: Mar 8, 2013 11:29 AM

On Mar 7, 12:38 am, William Elliot <ma...@panix.com> wrote:
> On Wed, 6 Mar 2013, Ross A. Finlayson wrote:
> > > Exercise.  Graph f:(R^2 - Rx{0}) -> R, (x,y) -> x/y.
> > Mapping R^2 \ (0,0) the pointed disc to x/y, it is a many-to-one
> > function where for each x =/= 0, for each r =/= 0 there is y s.t. for
> > each x/y = r.  So its image is the four quadrants minus the axes.

>
> What's a pointed disk?  R^2\(0,0) is neither a disk nor pointed;  it's
> a punctured plain.
> No. The image of f is R.
> R^2 - both axis is not the image nor the graph.
> It's not even the projection of the graph onto the xy-plain.
> . . which is R^2 - the y-axis.
> The graph is a surface within xyz 3-space.
>

> > What then of each r/y or x/r, as r ranges?
>
> g(x) = x/a is linear;  h(x) = b/x is hyperbolic.

Putting aside these remarkable features of n/d, as evaluated in the
limit, twice, that the resulting triangle with two-sided points has
exactly unit area, there is plenty to consider with regards to ran(f)
for f: N -> R_[0,1].

Obviously and directly as N is countable, were ZF consistent and R a
set in ZF, vis-a-vis its having an arbitrarily large cardinal, ran(f)
is countable, in as to whether ran(f) = R_[0,1]. Here, to say that
ran(f) = R_[0,1], has that it would be sufficient to represent each
real in R_[0,1], in particular systems of analytical interest,
compared to: that it IS R_[0,1], with a concomitant following
construction of R for its use as the complete ordered field (unique,
up to isomorphism, in infinite ordered fields).

Then, this is about detailing the analytical properties of n/d, with
the natural integers defined, (or as above as a natural continuum of
natural integers): before the properties of rings, fields, and the
ordered fields and the complete ordered field are defined. Monoids in
algebra are formal structures, but they are built upon a more
primitive and here the primeval structure of the continuum, first as
distinct integers then to the divisions and partitions of one integer,
the unit, then via extensions among each, all in the positive
(Bourbaki), and only then to the ring of integers, fields of fractions
and rationals and reals, and at once: ring of reals.

Then for ran(f) being f: N -> ran(f), and that having structural
qualities for use as a building block of then higher level algebraic
structures, a variety of results unavailable to the standard, are
formalizable.

Is ran(f) = [0,1]? Is ran(g) for g = x for x from 0 to 1, [0,1]? Are
all the ranges of functions with image [0,1] equal to [0,1]? Then,
[0,1] isn't just the points in a set, it's all of those. In as to
types, then, there's a notion that: ran(f) = [0,1].

Regards,

Ross Finlayson