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Replies: 115   Last Post: Feb 15, 2010 5:36 AM

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 T.H. Ray Posts: 1,107 Registered: 12/13/04
Posted: Feb 5, 2010 7:33 AM

Ostap Bender wrote

> On Feb 3, 3:48 am, "T.H. Ray" <thray...@aol.com>
> wrote:

> > Ostap Bender wrote
> >
> >

> > > > > I don't know about that, but to me, a
> function is
> > > an
> > > > > assignment of
> > > > > properties to the members of your set, like

> the
> > > > > weather report:
> >
> > > > > New York -> 30 degrees, snow
> > > > > San Francisco -> 55 degrees, rain
> > > > > Los Angeles -> 69 degrees, sunny

> >
> > > > > etc.
> >
> > > > Ah, now I see the difficulty.  No--the set of
> > > cities
> > > > with corresponding temperature and weather
> > > conditions
> > > > are not defined by a function until you define
> a
> > > > relation among them.  A simple relation to
> define
> > > the
> > > > function is the season of the year; If these
> > > initial
> > > > conditions exist in winter, then in summer this
> set
> > > > of numbers will change at some particular time
> to
> > > > something like
> >
> > > I was talking about the instantaneous weather
> report
> > > on the Weather
> > > Channel or CNN: At this moment, the temperatures

> are:
> >
> > > New York -> 30 degrees, snow
> > > San Francisco -> 55 degrees, rain
> > > Los Angeles -> 69 degrees, sunny

> >
> > > > New York, 78 degrees, sunny
> > > > San Francisco, 65 degrees, rain
> > > > Los Angeles, 85 degrees, partly cloudy

> >
> > > > A function necessarily transforms...
> >
> > > What do you mean by "transforms"? Does the
> weather
> > > report transform
> > > the set of cities into a set of temperatures? I

> would
> > > use the term
> > > "assigns" instead of "transforms".

> >
> > No.  The weather report is the result of the

> function
> > that transforms one set of data into another.  It
> is
> > certainly a transformation, not an assignment of
> values,
> > because it is not arbitrary.
> >
> >
> >  There do exist arbitrary
> > assigments of values that are functions; e.g.,

> ordering
> > some group of people by height.  Even in such
> cases,
> > however, transformation applies--the order function
> > transforms a random distribution of values into an
> > ordered sequence.
> >

> > > > ... one set of data to
> > > > another.  The above is an example of a

> continuous
> > > > function;
> >
> > > That would depend on your domain. If your domain
> is
> > > the Earth' surface
> > > - then the current temperature is continuous. if

> your
> > > domain is the
> > > set of 100 big cities - there can be no notion of
> > > continuity.

> >
> > The domain for climate is, of course, the earth's
> > surface and atmospheres--what else would it be?
> >

>
> Many things. For example, the set of weather stations
> that measure the
> outside temperature. Any set can serve as the domain
> for a function.
>

Sure. However, the function does not exist without
defining a relation between sets. Simply assigning
properties to a set does not confer properties of a
function. A function is always characterized by a
transformation; i.e., a map, a relation.

> >
> >  The set
> > of cities is most certainly embedded in that

> surface
> > and their climate (by the fixed point theorem)is a
> > function of changes in that surface and atmosphere.
> >

>
> Isn't a restricted function still a function in its
> own right?
>
> http://en.wikipedia.org/wiki/Function_(mathematics)
>
> More precisely, if ? is a function from a X to Y, and
> S is any subset
> of X, the restriction of ? to S is the function ?|S
> from S to Y such
> that ?|S(s) = ?(s) for all s in S
>

> >
I don't know what you're getting at. "Restricted" in
this context does not obviate transformation (map,
relation).

> > Brouwer, who gave us the fixed point theorem, also
> gave
> > us the deeper result: all real functions are
> continuous.
> >
>
> Excuse me? What do you mean by "real functions"?
>

Functions in the range of the real numbers.

Tom

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