Re: Cantor Confusion

WM schrieb:
On 10 Mai, 04:20, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <1178723452.432944.32...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> WM <mueck...@xxxxxxxxxxxxxxxxx> writes:
> On 9 Mai, 15:50, "*** T. Winter" <***.Win...@xxxxxx> wrote:
...
> > But that is not the definition, their formal definition is:
> > "A binary relation F is called a function (or mapping, correspondence), if
> > aFb_1 and aFb_2 imply b_1 = b_2 for any a, b_1, and b_2."
>
> And how do you think the a's and b's are selected? And from what sets
> do you think are they selected?

The first thing you should do is find how they define "binary relation".
But if you skip introductory material you can be lead to errors.

Be sure, I studied every page very carefully, so carefully that I
typed every letter (some time ago).

That does not seem to have lead to much understanding.

Some results are available here.

2. RELATIONS

Mathematicians often study relations between mathematical objects.
Relations between objects of two sorts occur most frequently; we call
them binary relations.
...
A binary relation is, therefore, determined by giving all ordered
pairs of objects in that relation; it dos not matter by what property
the set of these ordered pairs is described.

!!!!!!!!!!!! But it obviously does matter *that* it is
described. !!!!!!!!!!!!!!!

No, it does not.

...
2.3 Definition Let R be a binary relation.

Is there not a definition of a binary relation, or have you omitted it?
It seems to me that it would have been significant for a discussion on
what a relation is. Another sign of dishonesty or just carelessness?

(a) The set of all x which are in relation R with some y is called the
domain of R and denoted by dom R = {x | there exists y such that xRy}.
dom R is the set of all first coordinates of ordered pairs in R.
(b) The set of all y such that, for some x, x is in relation R with y
is called the range of R, denoted by ran R. So ran R = {y | there
exists x such that xRy}
[...]
Functions in set theory are *not* the same as functions in analysis.

I talked about mathematics, not about set theory. Nevertheless
functions *in set theory* consist of a prescription (formula, rule,
whatever), domain, and range.

Exercises

2.1 Let R be a binary relation; show that dom R c U(UR), ran R c
U(UR). Conclude from this that dom R and ran R exist.

The last exercise is a good homework for such "mathematicians" who
studied in Oxford or Harvard or at home and, therefore, do not know
that dom R and ran R do exist for any binary relation, or in other
words: without domain and range R would not be a binary relation.

Playing semantic games again?

Every natural number x has a square, x*x. (At least in mathematics.)
There is no natural number which does not have a square. Would you
therefore say that a natural number consists of a number and its square?
Do I have to give the square of a number when I define a number?

But you are good at it. Since you never state /exactly/ what you mean,
you can always claim to have meant something else.

--
Carsten Schultz (2:38, 33:47)
http://carsten.codimi.de/
PGP/GPG key on the pgp.net key servers,
fingerprint on my home page.
.

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