| T.R | Title | User | Personal
Name | Date | Lines |
|---|
| 1871.1 | Right, a circle's not a function | MOVIES::HANCOCK | | Sun May 22 1994 01:18 | 26 |
|
Definition of a Function:
A real-valued function f defined on a set D of real numbers is a rule
that assigns to each number x in D exactly one real number f(x).
I have a problem with this because I see in my mind's eye a circle, for
example, which assigns two values of the function to a single value of
x!
A circle is not a function - for the reason you give. It's a relation
between x's and y's, which for a given x is sometimes satisfied by
no y's, sometimes by just one, and sometimes by two.
I don't think the definition of `function' is that great. The notion
of a rule assigning something to a real number is murky.
It's more common to define a function to be a particular kind of relation,
and a relation to be a set of (x,y) ordered pairs. A relation is a function
if for each x there is at most one related y.
Actually, whether functions are best thought of as rules, or as sets of
ordered pairs is quite a deep question, I think.
Hank
|
| 1871.2 | | RUSURE::EDP | Always mount a scratch monkey. | Mon May 23 1994 12:18 | 16 |
| Re .0:
All sorts of math discussion is welcome here. Notes conferences are
quite useful for having different discussions going at once, so there's
no reason mathematics cannot be discussed in the conference on a
variety of levels.
-- edp
Eric Postpischil
current Math conference moderator
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To get PGP, FTP /pub/unix/security/crypt/pgp23A.zip from ftp.funet.fi.
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|
| 1871.3 | | KERNEL::JACKSON | Peter Jackson - UK CSC TP/IM | Mon May 23 1994 13:07 | 6 |
| Re .0
I did a degree with the OU concentrating on maths courses. I enjoyed
it, and recommend it.
Peter
|
| 1871.4 | | HANNAH::OSMAN | see HANNAH::IGLOO$:[OSMAN]ERIC.VT240 | Tue May 24 1994 15:58 | 5 |
|
Well, if we describe a circle as x=cos(t) and y=sin(t) then it is a function.
/Eric
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| 1871.5 | | STAR::ABBASI | chess is cool ! | Tue May 24 1994 19:18 | 21 |
|
i think of a function as box, you stick something inside it from
one side and pull the output from the other side.
go pick something and stick it in the box, the box inside does
something to the input and pushes the result from the other side, pull
the result out, do this process again , sticking the SAME thing inside and
pulling out the result again, if the result is now different from before
even though we did stick the same thing inside, then this box is
not a function.
because this is confusing, you can stick the same thing inside and
get different things as outputs, this box is then ambiguous and
iam not sure how any one can describe how it behaves.
this is also why a line parallel to the y-axis is not a function, assuming
y=f(x).
\bye
\nasser
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| 1871.6 | more on the box concept of a function | STAR::ABBASI | chess is cool ! | Tue May 24 1994 19:34 | 11 |
|
i just to add that with this box, if you stick in different things
from once side and get the same thing out every time, then this is
ok, this is a function box, this is like a box that say paint every
thing you give blue, so you stick in red fork, it paint it blue and
put it out, you give it an orange fork, it paint it blue too, and so
forth, so one can describe it. so it is a function.
\bye
\nasser
|
| 1871.7 | | AUSSIE::GARSON | achtentachtig kacheltjes | Thu May 26 1994 08:14 | 6 |
| re .4
Beg to differ although the language is a little loose. Even using a
parametric description y is not a function of x. On the other hand x
*is* a function of t and so is y. Geometrically though (x,y) as a
function of t is a helix and the single-valued property holds.
|
| 1871.8 | coordinate systems also | UTROP1::BEL_M | Michel Bel@UTO - Telecommie | Thu May 26 1994 13:20 | 4 |
| And of course it depends on the coordinate system as well. Using polar
coordinates (on 0-pi*2 ) the function r = 1 describes a circle.
|
| 1871.9 | | MOVIES::HANCOCK | | Tue May 31 1994 07:31 | 10 |
| A circle is a "locus" (of points equidistant from the centre).
As far as I know, "locus" just means "set".
The set is the range of many functions, and in some coordinate
systems, a set of ordered pairs which is functional.
It now seems equally silly to say a circle is a function and to
say that it isn't.
|
| 1871.10 | | AUSSIE::GARSON | achtentachtig kacheltjes | Wed Jun 01 1994 02:52 | 4 |
| re .8
Even in polar coordinates r is not a function of theta if the origin
lies outside the circle.
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| 1871.11 | choices | UTROP1::BEL_M | Michel Bel@UTO - Telecommie | Thu Jun 02 1994 08:45 | 12 |
| What one can really say is that a specific set of values may be /described/
/ real \
by a choice of- complex - function depending on the coordinates, the
\ whatever/
x set and the y set ( ring, field, algebra ).
Long time since I mastered in math - and then to think my last exam
before graduation was functional analysis...
( not my main topic though, that was intuitionistic logic)
|